Theorem 2arwcatlem2 49334

Description: Lemma for 2arwcat 49338. (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 )
2arwcatlem2.0	⊢ (𝜑 → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 0 ) = 0 )

Assertion

Ref	Expression
2arwcatlem2	⊢ (𝜑 → ( 1 (⟨𝐴, 𝐵⟩ · 𝐶)𝐹) = 𝐹)

Proof of Theorem 2arwcatlem2

Step	Hyp	Ref	Expression
1		2arwcatlem2.a	. . . . 5 ⊢ (𝜑 → 𝐴 = 𝑋)
2		2arwcatlem2.b	. . . . 5 ⊢ (𝜑 → 𝐵 = 𝑌)
3	1, 2	opeq12d 4855	. . . 4 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ = ⟨𝑋, 𝑌⟩)
4		2arwcatlem2.c	. . . 4 ⊢ (𝜑 → 𝐶 = 𝑍)
5	3, 4	oveq12d 7418	. . 3 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ · 𝐶) = (⟨𝑋, 𝑌⟩ · 𝑍))
6	5	oveqd 7417	. 2 ⊢ (𝜑 → ( 1 (⟨𝐴, 𝐵⟩ · 𝐶)𝐹) = ( 1 (⟨𝑋, 𝑌⟩ · 𝑍)𝐹))
7		2arwcatlem2.0	. . . . 5 ⊢ (𝜑 → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 0 ) = 0 )
8	7	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐹 = 0 ) → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 0 ) = 0 )
9		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 = 0 ) → 𝐹 = 0 )
10	9	oveq2d 7416	. . . 4 ⊢ ((𝜑 ∧ 𝐹 = 0 ) → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 0 ))
11	8, 10, 9	3eqtr4d 2779	. . 3 ⊢ ((𝜑 ∧ 𝐹 = 0 ) → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = 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	oveq2d 7416	. . . 4 ⊢ ((𝜑 ∧ 𝐹 = 1 ) → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ))
16	13, 15, 14	3eqtr4d 2779	. . 3 ⊢ ((𝜑 ∧ 𝐹 = 1 ) → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = 𝐹)
17		2arwcatlem2.f	. . 3 ⊢ (𝜑 → (𝐹 = 0 ∨ 𝐹 = 1 ))
18	11, 16, 17	mpjaodan 960	. 2 ⊢ (𝜑 → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = 𝐹)
19	6, 18	eqtrd 2769	1 ⊢ (𝜑 → ( 1 (⟨𝐴, 𝐵⟩ · 𝐶)𝐹) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1539  ⟨cop 4605  (class class class)co 7400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-rab 3414  df-v 3459  df-dif 3927  df-un 3929  df-ss 3941  df-nul 4307  df-if 4499  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4882  df-br 5118  df-iota 6481  df-fv 6536  df-ov 7403

This theorem is referenced by:  2arwcat  49338