Theorem rhmsubcALTVlem2 44383

Description: Lemma 2 for rhmsubcALTV 44386. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)

Hypotheses

Ref	Expression
rngcrescrhmALTV.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
rngcrescrhmALTV.c	⊢ 𝐶 = (RngCatALTV‘𝑈)
rngcrescrhmALTV.r	⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈))
rngcrescrhmALTV.h	⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅))

Assertion

Ref	Expression
rhmsubcALTVlem2	⊢ ((𝜑 ∧ 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌))

Proof of Theorem rhmsubcALTVlem2

Step	Hyp	Ref	Expression
1		opelxpi 5594	. . . 4 ⊢ ((𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → ⟨𝑋, 𝑌⟩ ∈ (𝑅 × 𝑅))
2	1	3adant1 1126	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → ⟨𝑋, 𝑌⟩ ∈ (𝑅 × 𝑅))
3	2	fvresd 6692	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (( RingHom ↾ (𝑅 × 𝑅))‘⟨𝑋, 𝑌⟩) = ( RingHom ‘⟨𝑋, 𝑌⟩))
4		df-ov 7161	. . 3 ⊢ (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
5		rngcrescrhmALTV.h	. . . 4 ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅))
6	5	fveq1i 6673	. . 3 ⊢ (𝐻‘⟨𝑋, 𝑌⟩) = (( RingHom ↾ (𝑅 × 𝑅))‘⟨𝑋, 𝑌⟩)
7	4, 6	eqtri 2846	. 2 ⊢ (𝑋𝐻𝑌) = (( RingHom ↾ (𝑅 × 𝑅))‘⟨𝑋, 𝑌⟩)
8		df-ov 7161	. 2 ⊢ (𝑋 RingHom 𝑌) = ( RingHom ‘⟨𝑋, 𝑌⟩)
9	3, 7, 8	3eqtr4g 2883	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌))

Syntax hints:   → wi 4   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ∩ cin 3937  ⟨cop 4575   × cxp 5555   ↾ cres 5559  ‘cfv 6357  (class class class)co 7158  Ringcrg 19299   RingHom crh 19466  RngCatALTVcrngcALTV 44236

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-xp 5563  df-res 5569  df-iota 6316  df-fv 6365  df-ov 7161

This theorem is referenced by:  rhmsubcALTVlem3  44384  rhmsubcALTVlem4  44385