Theorem mulnnnq0 7764

Description: Multiplication of nonnegative 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 4780	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ N) → ⟨𝐴, 𝐵⟩ ∈ (ω × N))
2		enq0ex 7753	. . . . 5 ⊢ ~Q0 ∈ V
3	2	ecelqsi 6822	. . . 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 4780	. . . 4 ⊢ ((𝐶 ∈ ω ∧ 𝐷 ∈ N) → ⟨𝐶, 𝐷⟩ ∈ (ω × N))
6	2	ecelqsi 6822	. . . 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 338	. 2 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) → ([⟨𝐴, 𝐵⟩] ~Q0 ∈ ((ω × N) / ~Q0 ) ∧ [⟨𝐶, 𝐷⟩] ~Q0 ∈ ((ω × N) / ~Q0 )))
9		eqid 2232	. . . 4 ⊢ [⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0
10		eqid 2232	. . . 4 ⊢ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0
11	9, 10	pm3.2i 272	. . 3 ⊢ ([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 )
12		eqid 2232	. . 3 ⊢ [⟨(𝐴 ·o 𝐶), (𝐵 ·o 𝐷)⟩] ~Q0 = [⟨(𝐴 ·o 𝐶), (𝐵 ·o 𝐷)⟩] ~Q0
13		opeq12 3884	. . . . . 6 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩)
14		eceq1 6801	. . . . . . . . 9 ⊢ (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → [⟨𝑤, 𝑣⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0 )
15	14	eqeq2d 2244	. . . . . . . 8 ⊢ (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → ([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ↔ [⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0 ))
16	15	anbi1d 465	. . . . . . 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 2815	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
18		vex 2815	. . . . . . . . . . 11 ⊢ 𝑣 ∈ V
19	17, 18	opth 4352	. . . . . . . . . 10 ⊢ (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑤 = 𝐴 ∧ 𝑣 = 𝐵))
20		oveq1 6056	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝐴 → (𝑤 ·o 𝐶) = (𝐴 ·o 𝐶))
21	20	adantr 276	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → (𝑤 ·o 𝐶) = (𝐴 ·o 𝐶))
22		oveq1 6056	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝐵 → (𝑣 ·o 𝐷) = (𝐵 ·o 𝐷))
23	22	adantl 277	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → (𝑣 ·o 𝐷) = (𝐵 ·o 𝐷))
24	21, 23	opeq12d 3890	. . . . . . . . . 10 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ⟨(𝑤 ·o 𝐶), (𝑣 ·o 𝐷)⟩ = ⟨(𝐴 ·o 𝐶), (𝐵 ·o 𝐷)⟩)
25	19, 24	sylbi 121	. . . . . . . . 9 ⊢ (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → ⟨(𝑤 ·o 𝐶), (𝑣 ·o 𝐷)⟩ = ⟨(𝐴 ·o 𝐶), (𝐵 ·o 𝐷)⟩)
26	25	eceq1d 6802	. . . . . . . 8 ⊢ (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → [⟨(𝑤 ·o 𝐶), (𝑣 ·o 𝐷)⟩] ~Q0 = [⟨(𝐴 ·o 𝐶), (𝐵 ·o 𝐷)⟩] ~Q0 )
27	26	eqeq2d 2244	. . . . . . 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 473	. . . . . 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 2906	. . . 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 3884	. . . . . . 7 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → ⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩)
32		eceq1 6801	. . . . . . . . . 10 ⊢ (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → [⟨𝑢, 𝑡⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 )
33	32	eqeq2d 2244	. . . . . . . . 9 ⊢ (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → ([⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ↔ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ))
34	33	anbi2d 464	. . . . . . . 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 2815	. . . . . . . . . . . 12 ⊢ 𝑢 ∈ V
36		vex 2815	. . . . . . . . . . . 12 ⊢ 𝑡 ∈ V
37	35, 36	opth 4352	. . . . . . . . . . 11 ⊢ (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝑢 = 𝐶 ∧ 𝑡 = 𝐷))
38		oveq2 6057	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝐶 → (𝑤 ·o 𝑢) = (𝑤 ·o 𝐶))
39	38	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → (𝑤 ·o 𝑢) = (𝑤 ·o 𝐶))
40		oveq2 6057	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝐷 → (𝑣 ·o 𝑡) = (𝑣 ·o 𝐷))
41	40	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → (𝑣 ·o 𝑡) = (𝑣 ·o 𝐷))
42	39, 41	opeq12d 3890	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → ⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩ = ⟨(𝑤 ·o 𝐶), (𝑣 ·o 𝐷)⟩)
43	37, 42	sylbi 121	. . . . . . . . . 10 ⊢ (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → ⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩ = ⟨(𝑤 ·o 𝐶), (𝑣 ·o 𝐷)⟩)
44	43	eceq1d 6802	. . . . . . . . 9 ⊢ (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 = [⟨(𝑤 ·o 𝐶), (𝑣 ·o 𝐷)⟩] ~Q0 )
45	44	eqeq2d 2244	. . . . . . . 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 473	. . . . . . 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 2906	. . . . 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 1931	. . . 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 409	. . 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 432	. 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 6770	. . . 4 ⊢ ( ~Q0 ∈ V → [⟨(𝐴 ·o 𝐶), (𝐵 ·o 𝐷)⟩] ~Q0 ∈ V)
53	2, 52	ax-mp 5	. . 3 ⊢ [⟨(𝐴 ·o 𝐶), (𝐵 ·o 𝐷)⟩] ~Q0 ∈ V
54		eqeq1 2239	. . . . . . . 8 ⊢ (𝑥 = [⟨𝐴, 𝐵⟩] ~Q0 → (𝑥 = [⟨𝑤, 𝑣⟩] ~Q0 ↔ [⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ))
55		eqeq1 2239	. . . . . . . 8 ⊢ (𝑦 = [⟨𝐶, 𝐷⟩] ~Q0 → (𝑦 = [⟨𝑢, 𝑡⟩] ~Q0 ↔ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ))
56	54, 55	bi2anan9 610	. . . . . . 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 2239	. . . . . . 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 610	. . . . . 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 1221	. . . . 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 1919	. . . 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 7762	. . . 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 7743	. . . 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 6174	. . 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 1363	. 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 62	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 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398  ∃wex 1541   ∈ wcel 2203  Vcvv 2812  ⟨cop 3691  ωcom 4711   × cxp 4746  (class class class)co 6049   ·_o comu 6644  [cec 6764   / cqs 6765  Ncnpi 7586   ~_Q0 ceq0 7600   ·_Q0 cmq0 7604

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-oadd 6650  df-omul 6651  df-er 6766  df-ec 6768  df-qs 6772  df-ni 7618  df-mi 7620  df-enq0 7738  df-nq0 7739  df-mq0 7742

This theorem is referenced by:  mulclnq0  7766  nqnq0m  7769  nq0m0r  7770  distrnq0  7773  mulcomnq0  7774  nq02m  7779