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Theorem mat1rhmval 21812
Description: The value of the ring homomorphism 𝐹. (Contributed by AV, 22-Dec-2019.)
Hypotheses
Ref Expression
mat1rhmval.k 𝐾 = (Base‘𝑅)
mat1rhmval.a 𝐴 = ({𝐸} Mat 𝑅)
mat1rhmval.b 𝐵 = (Base‘𝐴)
mat1rhmval.o 𝑂 = ⟨𝐸, 𝐸
mat1rhmval.f 𝐹 = (𝑥𝐾 ↦ {⟨𝑂, 𝑥⟩})
Assertion
Ref Expression
mat1rhmval ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → (𝐹𝑋) = {⟨𝑂, 𝑋⟩})
Distinct variable groups:   𝑥,𝐾   𝑥,𝑂   𝑥,𝐸   𝑥,𝑅   𝑥,𝑉   𝑥,𝑋
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem mat1rhmval
StepHypRef Expression
1 mat1rhmval.f . 2 𝐹 = (𝑥𝐾 ↦ {⟨𝑂, 𝑥⟩})
2 opeq2 4829 . . 3 (𝑥 = 𝑋 → ⟨𝑂, 𝑥⟩ = ⟨𝑂, 𝑋⟩)
32sneqd 4596 . 2 (𝑥 = 𝑋 → {⟨𝑂, 𝑥⟩} = {⟨𝑂, 𝑋⟩})
4 simp3 1138 . 2 ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → 𝑋𝐾)
5 snex 5386 . . 3 {⟨𝑂, 𝑋⟩} ∈ V
65a1i 11 . 2 ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → {⟨𝑂, 𝑋⟩} ∈ V)
71, 3, 4, 6fvmptd3 6968 1 ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → (𝐹𝑋) = {⟨𝑂, 𝑋⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3443  {csn 4584  cop 4590  cmpt 5186  cfv 6493  (class class class)co 7353  Basecbs 17075  Ringcrg 19950   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-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
This theorem is referenced by:  mat1rhmelval  21813  mat1rhmcl  21814  mat1mhm  21817
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