2arwcatlem3 - Mathbox for Zhi Wang

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Theorem 2arwcatlem3 49576

Description: Lemma for 2arwcat 49579. (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 4847	. . . 4 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ = ⟨𝑋, 𝑌⟩)
4		2arwcatlem2.c	. . . 4 ⊢ (𝜑 → 𝐶 = 𝑍)
5	3, 4	oveq12d 7407	. . 3 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ · 𝐶) = (⟨𝑋, 𝑌⟩ · 𝑍))
6	5	oveqd 7406	. 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 480	. . . 4 ⊢ ((𝜑 ∧ 𝐹 = 0 ) → ( 0 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 0 )
9		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 = 0 ) → 𝐹 = 0 )
10	9	oveq1d 7404	. . . 4 ⊢ ((𝜑 ∧ 𝐹 = 0 ) → (𝐹(⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = ( 0 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ))
11	8, 10, 9	3eqtr4d 2775	. . 3 ⊢ ((𝜑 ∧ 𝐹 = 0 ) → (𝐹(⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 𝐹)
12		2arwcatlem2.1	. . . . 5 ⊢ (𝜑 → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 1 )
13	12	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐹 = 1 ) → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 1 )
14		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 = 1 ) → 𝐹 = 1 )
15	14	oveq1d 7404	. . . 4 ⊢ ((𝜑 ∧ 𝐹 = 1 ) → (𝐹(⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ))
16	13, 15, 14	3eqtr4d 2775	. . 3 ⊢ ((𝜑 ∧ 𝐹 = 1 ) → (𝐹(⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 𝐹)
17		2arwcatlem2.f	. . 3 ⊢ (𝜑 → (𝐹 = 0 ∨ 𝐹 = 1 ))
18	11, 16, 17	mpjaodan 960	. 2 ⊢ (𝜑 → (𝐹(⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 𝐹)
19	6, 18	eqtrd 2765	1 ⊢ (𝜑 → (𝐹(⟨𝐴, 𝐵⟩ · 𝐶) 1 ) = 𝐹)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1540 ⟨cop 4597 (class class class)co 7389

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-ext 2702

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2066 df-clab 2709 df-cleq 2722 df-clel 2804 df-rab 3409 df-v 3452 df-dif 3919 df-un 3921 df-ss 3933 df-nul 4299 df-if 4491 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-br 5110 df-iota 6466 df-fv 6521 df-ov 7392

This theorem is referenced by: 2arwcat 49579

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