Theorem scmatrhmval 22492

Description: The value of the ring homomorphism 𝐹. (Contributed by AV, 22-Dec-2019.)

Hypotheses

Ref	Expression
scmatrhmval.k	⊢ 𝐾 = (Base‘𝑅)
scmatrhmval.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
scmatrhmval.o	⊢ 1 = (1r‘𝐴)
scmatrhmval.t	⊢ ∗ = ( ·𝑠 ‘𝐴)
scmatrhmval.f	⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ (𝑥 ∗ 1 ))

Assertion

Ref	Expression
scmatrhmval	⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝐹‘𝑋) = (𝑋 ∗ 1 ))

Distinct variable groups:   𝑥,𝐾   𝑥,𝑅   𝑥,𝑉   𝑥,𝑋   𝑥, 1   𝑥, ∗

Allowed substitution hints:   𝐴(𝑥)   𝐹(𝑥)   𝑁(𝑥)

Proof of Theorem scmatrhmval

Step	Hyp	Ref	Expression
1		scmatrhmval.f	. 2 ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ (𝑥 ∗ 1 ))
2		oveq1 7374	. 2 ⊢ (𝑥 = 𝑋 → (𝑥 ∗ 1 ) = (𝑋 ∗ 1 ))
3		simpr 484	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → 𝑋 ∈ 𝐾)
4		ovexd 7402	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝑋 ∗ 1 ) ∈ V)
5	1, 2, 3, 4	fvmptd3 6971	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝐹‘𝑋) = (𝑋 ∗ 1 ))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3429   ↦ cmpt 5166  ‘cfv 6498  (class class class)co 7367  Basecbs 17179   ·_𝑠 cvsca 17224  1_rcur 20162   Mat cmat 22372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6454  df-fun 6500  df-fv 6506  df-ov 7370

This theorem is referenced by:  scmatrhmcl  22493  scmatfo  22495  scmatf1  22496  scmatghm  22498  scmatmhm  22499