Theorem 2arwcatlem4 49577

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 )
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 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 → (𝐺(⟨𝐴, 𝐵⟩ · 𝐶)𝐹) = (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))
7		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 = 0 ) ∧ 𝐺 = 0 ) → 𝐺 = 0 )
8		simplr 768	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 = 0 ) ∧ 𝐺 = 0 ) → 𝐹 = 0 )
9	7, 8	oveq12d 7407	. . . . 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 726	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 = 0 ) ∧ 𝐺 = 0 ) → ( 0 (⟨𝑋, 𝑌⟩ · 𝑍) 0 ) ∈ { 0 , 1 })
12	9, 11	eqeltrd 2829	. . . 4 ⊢ (((𝜑 ∧ 𝐹 = 0 ) ∧ 𝐺 = 0 ) → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ∈ { 0 , 1 })
13		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 = 0 ) ∧ 𝐺 = 1 ) → 𝐺 = 1 )
14		simplr 768	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 = 0 ) ∧ 𝐺 = 1 ) → 𝐹 = 0 )
15	13, 14	oveq12d 7407	. . . . . 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 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 = 0 ) ∧ 𝐺 = 1 ) → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 0 ) = 0 )
18	15, 17	eqtrd 2765	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 = 0 ) ∧ 𝐺 = 1 ) → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = 0 )
19		ovex 7422	. . . . . . . 8 ⊢ ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 0 ) ∈ V
20	16, 19	eqeltrrdi 2838	. . . . . . 7 ⊢ (𝜑 → 0 ∈ V)
21		prid1g 4726	. . . . . . 7 ⊢ ( 0 ∈ V → 0 ∈ { 0 , 1 })
22	20, 21	syl 17	. . . . . 6 ⊢ (𝜑 → 0 ∈ { 0 , 1 })
23	22	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 = 0 ) ∧ 𝐺 = 1 ) → 0 ∈ { 0 , 1 })
24	18, 23	eqeltrd 2829	. . . 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 480	. . . 4 ⊢ ((𝜑 ∧ 𝐹 = 0 ) → (𝐺 = 0 ∨ 𝐺 = 1 ))
27	12, 24, 26	mpjaodan 960	. . 3 ⊢ ((𝜑 ∧ 𝐹 = 0 ) → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ∈ { 0 , 1 })
28		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 = 1 ) ∧ 𝐺 = 0 ) → 𝐺 = 0 )
29		simplr 768	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 = 1 ) ∧ 𝐺 = 0 ) → 𝐹 = 1 )
30	28, 29	oveq12d 7407	. . . . . 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 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 = 1 ) ∧ 𝐺 = 0 ) → ( 0 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 0 )
33	30, 32	eqtrd 2765	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 = 1 ) ∧ 𝐺 = 0 ) → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = 0 )
34	22	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 = 1 ) ∧ 𝐺 = 0 ) → 0 ∈ { 0 , 1 })
35	33, 34	eqeltrd 2829	. . . 4 ⊢ (((𝜑 ∧ 𝐹 = 1 ) ∧ 𝐺 = 0 ) → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ∈ { 0 , 1 })
36		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 = 1 ) ∧ 𝐺 = 1 ) → 𝐺 = 1 )
37		simplr 768	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 = 1 ) ∧ 𝐺 = 1 ) → 𝐹 = 1 )
38	36, 37	oveq12d 7407	. . . . . 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 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 = 1 ) ∧ 𝐺 = 1 ) → ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ) = 1 )
41	38, 40	eqtrd 2765	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 = 1 ) ∧ 𝐺 = 1 ) → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = 1 )
42		ovex 7422	. . . . . . . 8 ⊢ ( 1 (⟨𝑋, 𝑌⟩ · 𝑍) 1 ) ∈ V
43	39, 42	eqeltrrdi 2838	. . . . . . 7 ⊢ (𝜑 → 1 ∈ V)
44		prid2g 4727	. . . . . . 7 ⊢ ( 1 ∈ V → 1 ∈ { 0 , 1 })
45	43, 44	syl 17	. . . . . 6 ⊢ (𝜑 → 1 ∈ { 0 , 1 })
46	45	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 = 1 ) ∧ 𝐺 = 1 ) → 1 ∈ { 0 , 1 })
47	41, 46	eqeltrd 2829	. . . 4 ⊢ (((𝜑 ∧ 𝐹 = 1 ) ∧ 𝐺 = 1 ) → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ∈ { 0 , 1 })
48	25	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐹 = 1 ) → (𝐺 = 0 ∨ 𝐺 = 1 ))
49	35, 47, 48	mpjaodan 960	. . 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 960	. 2 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ∈ { 0 , 1 })
52	6, 51	eqeltrd 2829	1 ⊢ (𝜑 → (𝐺(⟨𝐴, 𝐵⟩ · 𝐶)𝐹) ∈ { 0 , 1 })

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  Vcvv 3450  {cpr 4593  ⟨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  ax-nul 5263

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-ne 2927  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