Theorem mulnnnq0 6640

Description: Multiplication of non-negative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 19-Nov-2019.)

Assertion

Ref	Expression
mulnnnq0	⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) → ([⟨𝐴, 𝐵⟩] ~Q0 ·Q0 [⟨𝐶, 𝐷⟩] ~Q0 ) = [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 )

Proof of Theorem mulnnnq0

Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opelxpi 4394	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ N) → ⟨𝐴, 𝐵⟩ ∈ (ω × N))
2		enq0ex 6629	. . . . 5 ⊢ ~Q0 ∈ V
3	2	ecelqsi 6183	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ (ω × N) → [⟨𝐴, 𝐵⟩] ~Q0 ∈ ((ω × N) / ~Q0 ))
4	1, 3	syl 14	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ N) → [⟨𝐴, 𝐵⟩] ~Q0 ∈ ((ω × N) / ~Q0 ))
5		opelxpi 4394	. . . 4 ⊢ ((𝐶 ∈ ω ∧ 𝐷 ∈ N) → ⟨𝐶, 𝐷⟩ ∈ (ω × N))
6	2	ecelqsi 6183	. . . 4 ⊢ (⟨𝐶, 𝐷⟩ ∈ (ω × N) → [⟨𝐶, 𝐷⟩] ~Q0 ∈ ((ω × N) / ~Q0 ))
7	5, 6	syl 14	. . 3 ⊢ ((𝐶 ∈ ω ∧ 𝐷 ∈ N) → [⟨𝐶, 𝐷⟩] ~Q0 ∈ ((ω × N) / ~Q0 ))
8	4, 7	anim12i 331	. 2 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) → ([⟨𝐴, 𝐵⟩] ~Q0 ∈ ((ω × N) / ~Q0 ) ∧ [⟨𝐶, 𝐷⟩] ~Q0 ∈ ((ω × N) / ~Q0 )))
9		eqid 2081	. . . 4 ⊢ [⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0
10		eqid 2081	. . . 4 ⊢ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0
11	9, 10	pm3.2i 266	. . 3 ⊢ ([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 )
12		eqid 2081	. . 3 ⊢ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0
13		opeq12 3572	. . . . . 6 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩)
14		eceq1 6164	. . . . . . . . 9 ⊢ (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → [⟨𝑤, 𝑣⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0 )
15	14	eqeq2d 2092	. . . . . . . 8 ⊢ (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → ([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ↔ [⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0 ))
16	15	anbi1d 452	. . . . . . 7 ⊢ (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → (([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ↔ ([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 )))
17		vex 2604	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
18		vex 2604	. . . . . . . . . . 11 ⊢ 𝑣 ∈ V
19	17, 18	opth 3992	. . . . . . . . . 10 ⊢ (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑤 = 𝐴 ∧ 𝑣 = 𝐵))
20		oveq1 5539	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝐴 → (𝑤 ·𝑜 𝐶) = (𝐴 ·𝑜 𝐶))
21	20	adantr 270	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → (𝑤 ·𝑜 𝐶) = (𝐴 ·𝑜 𝐶))
22		oveq1 5539	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝐵 → (𝑣 ·𝑜 𝐷) = (𝐵 ·𝑜 𝐷))
23	22	adantl 271	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → (𝑣 ·𝑜 𝐷) = (𝐵 ·𝑜 𝐷))
24	21, 23	opeq12d 3578	. . . . . . . . . 10 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ⟨(𝑤 ·𝑜 𝐶), (𝑣 ·𝑜 𝐷)⟩ = ⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩)
25	19, 24	sylbi 119	. . . . . . . . 9 ⊢ (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → ⟨(𝑤 ·𝑜 𝐶), (𝑣 ·𝑜 𝐷)⟩ = ⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩)
26	25	eceq1d 6165	. . . . . . . 8 ⊢ (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → [⟨(𝑤 ·𝑜 𝐶), (𝑣 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 )
27	26	eqeq2d 2092	. . . . . . 7 ⊢ (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → ([⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝐶), (𝑣 ·𝑜 𝐷)⟩] ~Q0 ↔ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ))
28	16, 27	anbi12d 456	. . . . . 6 ⊢ (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → ((([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝐶), (𝑣 ·𝑜 𝐷)⟩] ~Q0 ) ↔ (([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 )))
29	13, 28	syl 14	. . . . 5 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ((([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝐶), (𝑣 ·𝑜 𝐷)⟩] ~Q0 ) ↔ (([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 )))
30	29	spc2egv 2687	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ N) → ((([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ) → ∃𝑤∃𝑣(([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝐶), (𝑣 ·𝑜 𝐷)⟩] ~Q0 )))
31		opeq12 3572	. . . . . . 7 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → ⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩)
32		eceq1 6164	. . . . . . . . . 10 ⊢ (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → [⟨𝑢, 𝑡⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 )
33	32	eqeq2d 2092	. . . . . . . . 9 ⊢ (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → ([⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ↔ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ))
34	33	anbi2d 451	. . . . . . . 8 ⊢ (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → (([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ↔ ([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 )))
35		vex 2604	. . . . . . . . . . . 12 ⊢ 𝑢 ∈ V
36		vex 2604	. . . . . . . . . . . 12 ⊢ 𝑡 ∈ V
37	35, 36	opth 3992	. . . . . . . . . . 11 ⊢ (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝑢 = 𝐶 ∧ 𝑡 = 𝐷))
38		oveq2 5540	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝐶 → (𝑤 ·𝑜 𝑢) = (𝑤 ·𝑜 𝐶))
39	38	adantr 270	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → (𝑤 ·𝑜 𝑢) = (𝑤 ·𝑜 𝐶))
40		oveq2 5540	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝐷 → (𝑣 ·𝑜 𝑡) = (𝑣 ·𝑜 𝐷))
41	40	adantl 271	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → (𝑣 ·𝑜 𝑡) = (𝑣 ·𝑜 𝐷))
42	39, 41	opeq12d 3578	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → ⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩ = ⟨(𝑤 ·𝑜 𝐶), (𝑣 ·𝑜 𝐷)⟩)
43	37, 42	sylbi 119	. . . . . . . . . 10 ⊢ (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → ⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩ = ⟨(𝑤 ·𝑜 𝐶), (𝑣 ·𝑜 𝐷)⟩)
44	43	eceq1d 6165	. . . . . . . . 9 ⊢ (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝐶), (𝑣 ·𝑜 𝐷)⟩] ~Q0 )
45	44	eqeq2d 2092	. . . . . . . 8 ⊢ (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → ([⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ↔ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝐶), (𝑣 ·𝑜 𝐷)⟩] ~Q0 ))
46	34, 45	anbi12d 456	. . . . . . 7 ⊢ (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → ((([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ↔ (([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝐶), (𝑣 ·𝑜 𝐷)⟩] ~Q0 )))
47	31, 46	syl 14	. . . . . 6 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → ((([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ↔ (([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝐶), (𝑣 ·𝑜 𝐷)⟩] ~Q0 )))
48	47	spc2egv 2687	. . . . 5 ⊢ ((𝐶 ∈ ω ∧ 𝐷 ∈ N) → ((([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝐶), (𝑣 ·𝑜 𝐷)⟩] ~Q0 ) → ∃𝑢∃𝑡(([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )))
49	48	2eximdv 1803	. . . 4 ⊢ ((𝐶 ∈ ω ∧ 𝐷 ∈ N) → (∃𝑤∃𝑣(([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝐶), (𝑣 ·𝑜 𝐷)⟩] ~Q0 ) → ∃𝑤∃𝑣∃𝑢∃𝑡(([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )))
50	30, 49	sylan9 401	. . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) → ((([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ) → ∃𝑤∃𝑣∃𝑢∃𝑡(([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )))
51	11, 12, 50	mp2ani 422	. 2 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) → ∃𝑤∃𝑣∃𝑢∃𝑡(([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ))
52		ecexg 6133	. . . 4 ⊢ ( ~Q0 ∈ V → [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ∈ V)
53	2, 52	ax-mp 7	. . 3 ⊢ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ∈ V
54		eqeq1 2087	. . . . . . . 8 ⊢ (𝑥 = [⟨𝐴, 𝐵⟩] ~Q0 → (𝑥 = [⟨𝑤, 𝑣⟩] ~Q0 ↔ [⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ))
55		eqeq1 2087	. . . . . . . 8 ⊢ (𝑦 = [⟨𝐶, 𝐷⟩] ~Q0 → (𝑦 = [⟨𝑢, 𝑡⟩] ~Q0 ↔ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ))
56	54, 55	bi2anan9 570	. . . . . . 7 ⊢ ((𝑥 = [⟨𝐴, 𝐵⟩] ~Q0 ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ~Q0 ) → ((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝑦 = [⟨𝑢, 𝑡⟩] ~Q0 ) ↔ ([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 )))
57		eqeq1 2087	. . . . . . 7 ⊢ (𝑧 = [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 → (𝑧 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ↔ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ))
58	56, 57	bi2anan9 570	. . . . . 6 ⊢ (((𝑥 = [⟨𝐴, 𝐵⟩] ~Q0 ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ) → (((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝑦 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ↔ (([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )))
59	58	3impa 1133	. . . . 5 ⊢ ((𝑥 = [⟨𝐴, 𝐵⟩] ~Q0 ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ~Q0 ∧ 𝑧 = [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ) → (((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝑦 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ↔ (([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )))
60	59	4exbidv 1791	. . . 4 ⊢ ((𝑥 = [⟨𝐴, 𝐵⟩] ~Q0 ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ~Q0 ∧ 𝑧 = [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ) → (∃𝑤∃𝑣∃𝑢∃𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝑦 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ↔ ∃𝑤∃𝑣∃𝑢∃𝑡(([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )))
61		mulnq0mo 6638	. . . 4 ⊢ ((𝑥 ∈ ((ω × N) / ~Q0 ) ∧ 𝑦 ∈ ((ω × N) / ~Q0 )) → ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝑦 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ))
62		dfmq0qs 6619	. . . 4 ⊢ ·Q0 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ((ω × N) / ~Q0 ) ∧ 𝑦 ∈ ((ω × N) / ~Q0 )) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝑦 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ))}
63	60, 61, 62	ovig 5642	. . 3 ⊢ (([⟨𝐴, 𝐵⟩] ~Q0 ∈ ((ω × N) / ~Q0 ) ∧ [⟨𝐶, 𝐷⟩] ~Q0 ∈ ((ω × N) / ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ∈ V) → (∃𝑤∃𝑣∃𝑢∃𝑡(([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) → ([⟨𝐴, 𝐵⟩] ~Q0 ·Q0 [⟨𝐶, 𝐷⟩] ~Q0 ) = [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ))
64	53, 63	mp3an3 1257	. 2 ⊢ (([⟨𝐴, 𝐵⟩] ~Q0 ∈ ((ω × N) / ~Q0 ) ∧ [⟨𝐶, 𝐷⟩] ~Q0 ∈ ((ω × N) / ~Q0 )) → (∃𝑤∃𝑣∃𝑢∃𝑡(([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨(𝑤 ·𝑜 𝑢), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) → ([⟨𝐴, 𝐵⟩] ~Q0 ·Q0 [⟨𝐶, 𝐷⟩] ~Q0 ) = [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ))
65	8, 51, 64	sylc 61	1 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) → ([⟨𝐴, 𝐵⟩] ~Q0 ·Q0 [⟨𝐶, 𝐷⟩] ~Q0 ) = [⟨(𝐴 ·𝑜 𝐶), (𝐵 ·𝑜 𝐷)⟩] ~Q0 )

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   ∧ w3a 919   = wceq 1284  ∃wex 1421   ∈ wcel 1433  Vcvv 2601  ⟨cop 3401  ωcom 4331   × cxp 4361  (class class class)co 5532   ·_𝑜 comu 6022  [cec 6127   / cqs 6128  Ncnpi 6462   ~_Q0 ceq0 6476   ·_Q0 cmq0 6480

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-mi 6496  df-enq0 6614  df-nq0 6615  df-mq0 6618

This theorem is referenced by:  mulclnq0  6642  nqnq0m  6645  nq0m0r  6646  distrnq0  6649  mulcomnq0  6650  nq02m  6655