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Theorem mat1rhmval 20104
 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 . . 3 𝐹 = (𝑥𝐾 ↦ {⟨𝑂, 𝑥⟩})
21a1i 11 . 2 ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → 𝐹 = (𝑥𝐾 ↦ {⟨𝑂, 𝑥⟩}))
3 opeq2 4341 . . . 4 (𝑥 = 𝑋 → ⟨𝑂, 𝑥⟩ = ⟨𝑂, 𝑋⟩)
43sneqd 4137 . . 3 (𝑥 = 𝑋 → {⟨𝑂, 𝑥⟩} = {⟨𝑂, 𝑋⟩})
54adantl 481 . 2 (((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) ∧ 𝑥 = 𝑋) → {⟨𝑂, 𝑥⟩} = {⟨𝑂, 𝑋⟩})
6 simp3 1056 . 2 ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → 𝑋𝐾)
7 snex 4835 . . 3 {⟨𝑂, 𝑋⟩} ∈ V
87a1i 11 . 2 ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → {⟨𝑂, 𝑋⟩} ∈ V)
92, 5, 6, 8fvmptd 6197 1 ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → (𝐹𝑋) = {⟨𝑂, 𝑋⟩})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125  ⟨cop 4131   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  Ringcrg 18370   Mat cmat 20032 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812 This theorem is referenced by:  mat1rhmelval  20105  mat1rhmcl  20106  mat1mhm  20109
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