Let $V \xrightarrow{\phi} W \xrightarrow{\psi} V$. Show that $\phi$ is injective and $\psi$ is surjective.
141166
How to show that when the linear transformation $\Phi =\psi \circ \phi : E \rightarrow G$ is injective, then $\phi$ is also injective?
46746
How to prove if $u\in W^{1,p}$, then $|u|\in W^{1,p}$?
10307
Show the inverse of a bijective function is bijective
89089
Prove that there is a unique inner product on $V$
84114
Does there exist a surjective homomorphism from $(\mathbb R,+)$ to $(\mathbb Q,+)$ ?
30327
$\ker \phi = (x_1-a_1, ..., x_n-a_n)$ for a ring homomorphism $\phi: R[x_1, ..., x_n] \to R$
118049
Show that $\mathbb{R}^{\mathbb{R}} = U_{e} \oplus U_{o}$
137982
Show that $\textbf{ZF}$ shows $\phi$ is $\Delta_1$
89514
Show that if $g \circ f$ is injective, then so is $f$.
133627
If $\phi(g)=g^3$ is a homomorphism and $3 \nmid |G|$, $G$ is abelian.
95086
Homeomorphism is bijective
70912
If $g\circ f$ is injective and $f$ is surjective then $g$ is injective
72251
$g\circ f$ is Surjective and $g$ is Injective then Prove that $f$ is surjective
38033
A surjective linear map into a finite dimensional space is open
156727
A step in the proof of Characterization of $W^{1,\infty}$
62077
Prove that if $W_1\subseteq V$ finite-dimensional, then there is $W_2\subseteq V$ such that $V=W_1\oplus W_2$
63952
Disproving: If $(\Gamma\vdash(\phi\,\lor\psi))$, either $(\Gamma\vdash\phi)$ or $(\Gamma\vdash\psi)$ is true