Theorem rhmsubclem2 44378

Description: Lemma 2 for rhmsubc 44381. (Contributed by AV, 2-Mar-2020.)

Hypotheses

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

Assertion

Ref	Expression
rhmsubclem2	⊢ ((𝜑 ∧ 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌))

Proof of Theorem rhmsubclem2

Step	Hyp	Ref	Expression
1		opelxpi 5592	. . . 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 6690	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (( RingHom ↾ (𝑅 × 𝑅))‘⟨𝑋, 𝑌⟩) = ( RingHom ‘⟨𝑋, 𝑌⟩))
4		df-ov 7159	. . 3 ⊢ (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
5		rngcrescrhm.h	. . . 4 ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅))
6	5	fveq1i 6671	. . 3 ⊢ (𝐻‘⟨𝑋, 𝑌⟩) = (( RingHom ↾ (𝑅 × 𝑅))‘⟨𝑋, 𝑌⟩)
7	4, 6	eqtri 2844	. 2 ⊢ (𝑋𝐻𝑌) = (( RingHom ↾ (𝑅 × 𝑅))‘⟨𝑋, 𝑌⟩)
8		df-ov 7159	. 2 ⊢ (𝑋 RingHom 𝑌) = ( RingHom ‘⟨𝑋, 𝑌⟩)
9	3, 7, 8	3eqtr4g 2881	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌))

Syntax hints:   → wi 4   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ∩ cin 3935  ⟨cop 4573   × cxp 5553   ↾ cres 5557  ‘cfv 6355  (class class class)co 7156  Ringcrg 19297   RingHom crh 19464  RngCatcrngc 44248

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 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

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 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-xp 5561  df-res 5567  df-iota 6314  df-fv 6363  df-ov 7159

This theorem is referenced by:  rhmsubclem3  44379  rhmsubclem4  44380