Theorem 2arwcatlem4 50219

Description: Lemma for 2arwcat 50221. (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 )
2arwcatlem4.0	⊢ (𝜑 → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 0 ) = 0 )
2arwcatlem4.00	⊢ (𝜑 → ( 0 (⟨𝑋, 𝑌⟩ · 𝑍) 0 ) ∈ { 0 , 1 })
2arwcatlem4.g	⊢ (𝜑 → (𝐺 = 0 ∨ 𝐺 = 1 ))

Assertion

Ref	Expression
2arwcatlem4	⊢ (𝜑 → (𝐺(⟨𝐴, 𝐵⟩ · 𝐶)𝐹) ∈ { 0 , 1 })

Proof of Theorem 2arwcatlem4

Step	Hyp	Ref	Expression
1		2arwcatlem2.a	. . . . 5 ⊢ (𝜑 → 𝐴 = 𝑋)
2		2arwcatlem2.b	. . . . 5 ⊢ (𝜑 → 𝐵 = 𝑌)
3	1, 2	opeq12d 4839	. . . 4 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ = ⟨𝑋, 𝑌⟩)
4		2arwcatlem2.c	. . . 4 ⊢ (𝜑 → 𝐶 = 𝑍)
5	3, 4	oveq12d 7414	. . 3 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ · 𝐶) = (⟨𝑋, 𝑌⟩ · 𝑍))
6	5	oveqd 7413	. 2 ⊢ (𝜑 → (𝐺(⟨𝐴, 𝐵⟩ · 𝐶)𝐹) = (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))
7		simpr 488	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 = 0 ) ∧ 𝐺 = 0 ) → 𝐺 = 0 )
8		simplr 778	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 = 0 ) ∧ 𝐺 = 0 ) → 𝐹 = 0 )
9	7, 8	oveq12d 7414	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 = 0 ) ∧ 𝐺 = 0 ) → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = ( 0 (⟨𝑋, 𝑌⟩ · 𝑍) 0 ))
10		2arwcatlem4.00	. . . . . 6 ⊢ (𝜑 → ( 0 (⟨𝑋, 𝑌⟩ · 𝑍) 0 ) ∈ { 0 , 1 })
11	10	ad2antrr 736	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 = 0 ) ∧ 𝐺 = 0 ) → ( 0 (⟨𝑋, 𝑌⟩ · 𝑍) 0 ) ∈ { 0 , 1 })
12	9, 11	eqeltrd 2862	. . . 4 ⊢ (((𝜑 ∧ 𝐹 = 0 ) ∧ 𝐺 = 0 ) → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ∈ { 0 , 1 })
13		simpr 488	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 = 0 ) ∧ 𝐺 = 1 ) → 𝐺 = 1 )
14		simplr 778	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 = 0 ) ∧ 𝐺 = 1 ) → 𝐹 = 0 )
15	13, 14	oveq12d 7414	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 = 0 ) ∧ 𝐺 = 1 ) → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 0 ))
16		2arwcatlem4.0	. . . . . . 7 ⊢ (𝜑 → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 0 ) = 0 )
17	16	ad2antrr 736	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 = 0 ) ∧ 𝐺 = 1 ) → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 0 ) = 0 )
18	15, 17	eqtrd 2797	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 = 0 ) ∧ 𝐺 = 1 ) → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = 0 )
19		ovex 7429	. . . . . . . 8 ⊢ ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 0 ) ∈ V
20	16, 19	eqeltrrdi 2871	. . . . . . 7 ⊢ (𝜑 → 0 ∈ V)
21		prid1g 4719	. . . . . . 7 ⊢ ( 0 ∈ V → 0 ∈ { 0 , 1 })
22	20, 21	syl 17	. . . . . 6 ⊢ (𝜑 → 0 ∈ { 0 , 1 })
23	22	ad2antrr 736	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 = 0 ) ∧ 𝐺 = 1 ) → 0 ∈ { 0 , 1 })
24	18, 23	eqeltrd 2862	. . . 4 ⊢ (((𝜑 ∧ 𝐹 = 0 ) ∧ 𝐺 = 1 ) → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ∈ { 0 , 1 })
25		2arwcatlem4.g	. . . . 5 ⊢ (𝜑 → (𝐺 = 0 ∨ 𝐺 = 1 ))
26	25	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝐹 = 0 ) → (𝐺 = 0 ∨ 𝐺 = 1 ))
27	12, 24, 26	mpjaodan 971	. . 3 ⊢ ((𝜑 ∧ 𝐹 = 0 ) → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ∈ { 0 , 1 })
28		simpr 488	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 = 1 ) ∧ 𝐺 = 0 ) → 𝐺 = 0 )
29		simplr 778	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 = 1 ) ∧ 𝐺 = 0 ) → 𝐹 = 1 )
30	28, 29	oveq12d 7414	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 = 1 ) ∧ 𝐺 = 0 ) → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = ( 0 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ))
31		2arwcatlem3.0	. . . . . . 7 ⊢ (𝜑 → ( 0 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 0 )
32	31	ad2antrr 736	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 = 1 ) ∧ 𝐺 = 0 ) → ( 0 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 0 )
33	30, 32	eqtrd 2797	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 = 1 ) ∧ 𝐺 = 0 ) → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = 0 )
34	22	ad2antrr 736	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 = 1 ) ∧ 𝐺 = 0 ) → 0 ∈ { 0 , 1 })
35	33, 34	eqeltrd 2862	. . . 4 ⊢ (((𝜑 ∧ 𝐹 = 1 ) ∧ 𝐺 = 0 ) → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ∈ { 0 , 1 })
36		simpr 488	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 = 1 ) ∧ 𝐺 = 1 ) → 𝐺 = 1 )
37		simplr 778	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 = 1 ) ∧ 𝐺 = 1 ) → 𝐹 = 1 )
38	36, 37	oveq12d 7414	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 = 1 ) ∧ 𝐺 = 1 ) → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ))
39		2arwcatlem2.1	. . . . . . 7 ⊢ (𝜑 → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 1 )
40	39	ad2antrr 736	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 = 1 ) ∧ 𝐺 = 1 ) → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 1 )
41	38, 40	eqtrd 2797	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 = 1 ) ∧ 𝐺 = 1 ) → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = 1 )
42		ovex 7429	. . . . . . . 8 ⊢ ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ) ∈ V
43	39, 42	eqeltrrdi 2871	. . . . . . 7 ⊢ (𝜑 → 1 ∈ V)
44		prid2g 4720	. . . . . . 7 ⊢ ( 1 ∈ V → 1 ∈ { 0 , 1 })
45	43, 44	syl 17	. . . . . 6 ⊢ (𝜑 → 1 ∈ { 0 , 1 })
46	45	ad2antrr 736	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 = 1 ) ∧ 𝐺 = 1 ) → 1 ∈ { 0 , 1 })
47	41, 46	eqeltrd 2862	. . . 4 ⊢ (((𝜑 ∧ 𝐹 = 1 ) ∧ 𝐺 = 1 ) → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ∈ { 0 , 1 })
48	25	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝐹 = 1 ) → (𝐺 = 0 ∨ 𝐺 = 1 ))
49	35, 47, 48	mpjaodan 971	. . 3 ⊢ ((𝜑 ∧ 𝐹 = 1 ) → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ∈ { 0 , 1 })
50		2arwcatlem2.f	. . 3 ⊢ (𝜑 → (𝐹 = 0 ∨ 𝐹 = 1 ))
51	27, 49, 50	mpjaodan 971	. 2 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ∈ { 0 , 1 })
52	6, 51	eqeltrd 2862	1 ⊢ (𝜑 → (𝐺(⟨𝐴, 𝐵⟩ · 𝐶)𝐹) ∈ { 0 , 1 })

Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 858   = wceq 1560   ∈ wcel 2142  Vcvv 3454  {cpr 4584  ⟨cop 4588  (class class class)co 7396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-nul 5256

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6477  df-fv 6529  df-ov 7399

This theorem is referenced by:  2arwcat  50221