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Theorem rhmsubclem2 20582
Description: Lemma 2 for rhmsubc 20585. (Contributed by AV, 2-Mar-2020.)
Hypotheses
Ref Expression
rngcrescrhm.u (πœ‘ β†’ π‘ˆ ∈ 𝑉)
rngcrescrhm.c 𝐢 = (RngCatβ€˜π‘ˆ)
rngcrescrhm.r (πœ‘ β†’ 𝑅 = (Ring ∩ π‘ˆ))
rngcrescrhm.h 𝐻 = ( RingHom β†Ύ (𝑅 Γ— 𝑅))
Assertion
Ref Expression
rhmsubclem2 ((πœ‘ ∧ 𝑋 ∈ 𝑅 ∧ π‘Œ ∈ 𝑅) β†’ (π‘‹π»π‘Œ) = (𝑋 RingHom π‘Œ))

Proof of Theorem rhmsubclem2
StepHypRef Expression
1 opelxpi 5706 . . . 4 ((𝑋 ∈ 𝑅 ∧ π‘Œ ∈ 𝑅) β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ (𝑅 Γ— 𝑅))
213adant1 1127 . . 3 ((πœ‘ ∧ 𝑋 ∈ 𝑅 ∧ π‘Œ ∈ 𝑅) β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ (𝑅 Γ— 𝑅))
32fvresd 6905 . 2 ((πœ‘ ∧ 𝑋 ∈ 𝑅 ∧ π‘Œ ∈ 𝑅) β†’ (( RingHom β†Ύ (𝑅 Γ— 𝑅))β€˜βŸ¨π‘‹, π‘ŒβŸ©) = ( RingHom β€˜βŸ¨π‘‹, π‘ŒβŸ©))
4 df-ov 7408 . . 3 (π‘‹π»π‘Œ) = (π»β€˜βŸ¨π‘‹, π‘ŒβŸ©)
5 rngcrescrhm.h . . . 4 𝐻 = ( RingHom β†Ύ (𝑅 Γ— 𝑅))
65fveq1i 6886 . . 3 (π»β€˜βŸ¨π‘‹, π‘ŒβŸ©) = (( RingHom β†Ύ (𝑅 Γ— 𝑅))β€˜βŸ¨π‘‹, π‘ŒβŸ©)
74, 6eqtri 2754 . 2 (π‘‹π»π‘Œ) = (( RingHom β†Ύ (𝑅 Γ— 𝑅))β€˜βŸ¨π‘‹, π‘ŒβŸ©)
8 df-ov 7408 . 2 (𝑋 RingHom π‘Œ) = ( RingHom β€˜βŸ¨π‘‹, π‘ŒβŸ©)
93, 7, 83eqtr4g 2791 1 ((πœ‘ ∧ 𝑋 ∈ 𝑅 ∧ π‘Œ ∈ 𝑅) β†’ (π‘‹π»π‘Œ) = (𝑋 RingHom π‘Œ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ∩ cin 3942  βŸ¨cop 4629   Γ— cxp 5667   β†Ύ cres 5671  β€˜cfv 6537  (class class class)co 7405  Ringcrg 20138   RingHom crh 20371  RngCatcrngc 20512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-res 5681  df-iota 6489  df-fv 6545  df-ov 7408
This theorem is referenced by:  rhmsubclem3  20583  rhmsubclem4  20584
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