Theorem scmatrhmval 22665

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 7419	. 2 ⊢ (𝑥 = 𝑋 → (𝑥 ∗ 1 ) = (𝑋 ∗ 1 ))
3		simpr 489	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → 𝑋 ∈ 𝐾)
4		ovexd 7447	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝑋 ∗ 1 ) ∈ V)
5	1, 2, 3, 4	fvmptd3 7015	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝐹‘𝑋) = (𝑋 ∗ 1 ))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ↦ cmpt 5193  ‘cfv 6538  (class class class)co 7412  Basecbs 17270   ·_𝑠 cvsca 17315  1_rcur 20264   Mat cmat 22545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7415

This theorem is referenced by:  scmatrhmcl  22666  scmatfo  22668  scmatf1  22669  scmatghm  22671  scmatmhm  22672