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Theorem scmatrhmval 21064
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 7152 . 2 (𝑥 = 𝑋 → (𝑥 1 ) = (𝑋 1 ))
3 simpr 485 . 2 ((𝑅𝑉𝑋𝐾) → 𝑋𝐾)
4 ovexd 7180 . 2 ((𝑅𝑉𝑋𝐾) → (𝑋 1 ) ∈ V)
51, 2, 3, 4fvmptd3 6783 1 ((𝑅𝑉𝑋𝐾) → (𝐹𝑋) = (𝑋 1 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  cmpt 5137  cfv 6348  (class class class)co 7145  Basecbs 16471   ·𝑠 cvsca 16557  1rcur 19180   Mat cmat 20944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148
This theorem is referenced by:  scmatrhmcl  21065  scmatfo  21067  scmatf1  21068  scmatghm  21070  scmatmhm  21071
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