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Theorem rhmsubcALTVlem2 46493
Description: Lemma 2 for rhmsubcALTV 46496. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
Hypotheses
Ref Expression
rngcrescrhmALTV.u (πœ‘ β†’ π‘ˆ ∈ 𝑉)
rngcrescrhmALTV.c 𝐢 = (RngCatALTVβ€˜π‘ˆ)
rngcrescrhmALTV.r (πœ‘ β†’ 𝑅 = (Ring ∩ π‘ˆ))
rngcrescrhmALTV.h 𝐻 = ( RingHom β†Ύ (𝑅 Γ— 𝑅))
Assertion
Ref Expression
rhmsubcALTVlem2 ((πœ‘ ∧ 𝑋 ∈ 𝑅 ∧ π‘Œ ∈ 𝑅) β†’ (π‘‹π»π‘Œ) = (𝑋 RingHom π‘Œ))

Proof of Theorem rhmsubcALTVlem2
StepHypRef Expression
1 opelxpi 5674 . . . 4 ((𝑋 ∈ 𝑅 ∧ π‘Œ ∈ 𝑅) β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ (𝑅 Γ— 𝑅))
213adant1 1131 . . 3 ((πœ‘ ∧ 𝑋 ∈ 𝑅 ∧ π‘Œ ∈ 𝑅) β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ (𝑅 Γ— 𝑅))
32fvresd 6866 . 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 7364 . . 3 (π‘‹π»π‘Œ) = (π»β€˜βŸ¨π‘‹, π‘ŒβŸ©)
5 rngcrescrhmALTV.h . . . 4 𝐻 = ( RingHom β†Ύ (𝑅 Γ— 𝑅))
65fveq1i 6847 . . 3 (π»β€˜βŸ¨π‘‹, π‘ŒβŸ©) = (( RingHom β†Ύ (𝑅 Γ— 𝑅))β€˜βŸ¨π‘‹, π‘ŒβŸ©)
74, 6eqtri 2761 . 2 (π‘‹π»π‘Œ) = (( RingHom β†Ύ (𝑅 Γ— 𝑅))β€˜βŸ¨π‘‹, π‘ŒβŸ©)
8 df-ov 7364 . 2 (𝑋 RingHom π‘Œ) = ( RingHom β€˜βŸ¨π‘‹, π‘ŒβŸ©)
93, 7, 83eqtr4g 2798 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 1088   = wceq 1542   ∈ wcel 2107   ∩ cin 3913  βŸ¨cop 4596   Γ— cxp 5635   β†Ύ cres 5639  β€˜cfv 6500  (class class class)co 7361  Ringcrg 19972   RingHom crh 20153  RngCatALTVcrngcALTV 46346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-xp 5643  df-res 5649  df-iota 6452  df-fv 6508  df-ov 7364
This theorem is referenced by:  rhmsubcALTVlem3  46494  rhmsubcALTVlem4  46495
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