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Theorem scmatrhmval 21860
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 = (1r𝐴)
scmatrhmval.t = ( ·𝑠𝐴)
scmatrhmval.f 𝐹 = (𝑥𝐾 ↦ (𝑥 1 ))
Assertion
Ref Expression
scmatrhmval ((𝑅𝑉𝑋𝐾) → (𝐹𝑋) = (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
StepHypRef Expression
1 scmatrhmval.f . 2 𝐹 = (𝑥𝐾 ↦ (𝑥 1 ))
2 oveq1 7360 . 2 (𝑥 = 𝑋 → (𝑥 1 ) = (𝑋 1 ))
3 simpr 485 . 2 ((𝑅𝑉𝑋𝐾) → 𝑋𝐾)
4 ovexd 7388 . 2 ((𝑅𝑉𝑋𝐾) → (𝑋 1 ) ∈ V)
51, 2, 3, 4fvmptd3 6968 1 ((𝑅𝑉𝑋𝐾) → (𝐹𝑋) = (𝑋 1 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3443  cmpt 5186  cfv 6493  (class class class)co 7353  Basecbs 17075   ·𝑠 cvsca 17129  1rcur 19904   Mat cmat 21738
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6445  df-fun 6495  df-fv 6501  df-ov 7356
This theorem is referenced by:  scmatrhmcl  21861  scmatfo  21863  scmatf1  21864  scmatghm  21866  scmatmhm  21867
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