Theorem scmatrhmval 22483

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 7375	. 2 ⊢ (𝑥 = 𝑋 → (𝑥 ∗ 1 ) = (𝑋 ∗ 1 ))
3		simpr 484	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → 𝑋 ∈ 𝐾)
4		ovexd 7403	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝑋 ∗ 1 ) ∈ V)
5	1, 2, 3, 4	fvmptd3 6973	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝐹‘𝑋) = (𝑋 ∗ 1 ))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3442   ↦ cmpt 5181  ‘cfv 6500  (class class class)co 7368  Basecbs 17148   ·_𝑠 cvsca 17193  1_rcur 20128   Mat cmat 22363

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 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371

This theorem is referenced by:  scmatrhmcl  22484  scmatfo  22486  scmatf1  22487  scmatghm  22489  scmatmhm  22490