Theorem 2arwcatlem3 50434

Description: Lemma for 2arwcat 50437. (Contributed by Zhi Wang, 5-Nov-2025.)

Hypotheses

Ref	Expression
2arwcatlem2.a	⊢ (𝜑 → 𝐴 = 𝑋)
2arwcatlem2.b	⊢ (𝜑 → 𝐵 = 𝑌)
2arwcatlem2.c	⊢ (𝜑 → 𝐶 = 𝑍)
2arwcatlem2.f	⊢ (𝜑 → (𝐹 = 0 ∨ 𝐹 = 1 ))
2arwcatlem2.1	⊢ (𝜑 → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 1 )
2arwcatlem3.0	⊢ (𝜑 → ( 0 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 0 )

Assertion

Ref	Expression
2arwcatlem3	⊢ (𝜑 → (𝐹(⟨𝐴, 𝐵⟩ · 𝐶) 1 ) = 𝐹)

Proof of Theorem 2arwcatlem3

Step	Hyp	Ref	Expression
1		2arwcatlem2.a	. . . . 5 ⊢ (𝜑 → 𝐴 = 𝑋)
2		2arwcatlem2.b	. . . . 5 ⊢ (𝜑 → 𝐵 = 𝑌)
3	1, 2	opeq12d 4848	. . . 4 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ = ⟨𝑋, 𝑌⟩)
4		2arwcatlem2.c	. . . 4 ⊢ (𝜑 → 𝐶 = 𝑍)
5	3, 4	oveq12d 7437	. . 3 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ · 𝐶) = (⟨𝑋, 𝑌⟩ · 𝑍))
6	5	oveqd 7436	. 2 ⊢ (𝜑 → (𝐹(⟨𝐴, 𝐵⟩ · 𝐶) 1 ) = (𝐹(⟨𝑋, 𝑌⟩ · 𝑍) 1 ))
7		2arwcatlem3.0	. . . . 5 ⊢ (𝜑 → ( 0 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 0 )
8	7	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐹 = 0 ) → ( 0 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 0 )
9		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 = 0 ) → 𝐹 = 0 )
10	9	oveq1d 7434	. . . 4 ⊢ ((𝜑 ∧ 𝐹 = 0 ) → (𝐹(⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = ( 0 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ))
11	8, 10, 9	3eqtr4d 2810	. . 3 ⊢ ((𝜑 ∧ 𝐹 = 0 ) → (𝐹(⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 𝐹)
12		2arwcatlem2.1	. . . . 5 ⊢ (𝜑 → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 1 )
13	12	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐹 = 1 ) → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 1 )
14		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 = 1 ) → 𝐹 = 1 )
15	14	oveq1d 7434	. . . 4 ⊢ ((𝜑 ∧ 𝐹 = 1 ) → (𝐹(⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ))
16	13, 15, 14	3eqtr4d 2810	. . 3 ⊢ ((𝜑 ∧ 𝐹 = 1 ) → (𝐹(⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 𝐹)
17		2arwcatlem2.f	. . 3 ⊢ (𝜑 → (𝐹 = 0 ∨ 𝐹 = 1 ))
18	11, 16, 17	mpjaodan 973	. 2 ⊢ (𝜑 → (𝐹(⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 𝐹)
19	6, 18	eqtrd 2800	1 ⊢ (𝜑 → (𝐹(⟨𝐴, 𝐵⟩ · 𝐶) 1 ) = 𝐹)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570  ⟨cop 4597  (class class class)co 7419

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6496  df-fv 6548  df-ov 7422

This theorem is used by:  2arwcat  50437