Theorem 2arwcatlem3 50260

Description: Lemma for 2arwcat 50263. (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 4850	. . . 4 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ = ⟨𝑋, 𝑌⟩)
4		2arwcatlem2.c	. . . 4 ⊢ (𝜑 → 𝐶 = 𝑍)
5	3, 4	oveq12d 7429	. . 3 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ · 𝐶) = (⟨𝑋, 𝑌⟩ · 𝑍))
6	5	oveqd 7428	. 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 485	. . . 4 ⊢ ((𝜑 ∧ 𝐹 = 0 ) → ( 0 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 0 )
9		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 = 0 ) → 𝐹 = 0 )
10	9	oveq1d 7426	. . . 4 ⊢ ((𝜑 ∧ 𝐹 = 0 ) → (𝐹(⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = ( 0 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ))
11	8, 10, 9	3eqtr4d 2814	. . 3 ⊢ ((𝜑 ∧ 𝐹 = 0 ) → (𝐹(⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 𝐹)
12		2arwcatlem2.1	. . . . 5 ⊢ (𝜑 → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 1 )
13	12	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝐹 = 1 ) → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 1 )
14		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 = 1 ) → 𝐹 = 1 )
15	14	oveq1d 7426	. . . 4 ⊢ ((𝜑 ∧ 𝐹 = 1 ) → (𝐹(⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ))
16	13, 15, 14	3eqtr4d 2814	. . 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 2804	1 ⊢ (𝜑 → (𝐹(⟨𝐴, 𝐵⟩ · 𝐶) 1 ) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1567  ⟨cop 4600  (class class class)co 7411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-iota 6493  df-fv 6545  df-ov 7414

This theorem is referenced by:  2arwcat  50263