Theorem scmatrhmval 21584

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 7262	. 2 ⊢ (𝑥 = 𝑋 → (𝑥 ∗ 1 ) = (𝑋 ∗ 1 ))
3		simpr 484	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → 𝑋 ∈ 𝐾)
4		ovexd 7290	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝑋 ∗ 1 ) ∈ V)
5	1, 2, 3, 4	fvmptd3 6880	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝐹‘𝑋) = (𝑋 ∗ 1 ))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ↦ cmpt 5153  ‘cfv 6418  (class class class)co 7255  Basecbs 16840   ·_𝑠 cvsca 16892  1_rcur 19652   Mat cmat 21464

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-ov 7258

This theorem is referenced by:  scmatrhmcl  21585  scmatfo  21587  scmatf1  21588  scmatghm  21590  scmatmhm  21591