Theorem mulnnnq0 7282

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 4579	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ N) → ⟨𝐴, 𝐵⟩ ∈ (ω × N))
2		enq0ex 7271	. . . . 5 ⊢ ~Q0 ∈ V
3	2	ecelqsi 6491	. . . 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 4579	. . . 4 ⊢ ((𝐶 ∈ ω ∧ 𝐷 ∈ N) → ⟨𝐶, 𝐷⟩ ∈ (ω × N))
6	2	ecelqsi 6491	. . . 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 336	. 2 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) → ([⟨𝐴, 𝐵⟩] ~Q0 ∈ ((ω × N) / ~Q0 ) ∧ [⟨𝐶, 𝐷⟩] ~Q0 ∈ ((ω × N) / ~Q0 )))
9		eqid 2140	. . . 4 ⊢ [⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0
10		eqid 2140	. . . 4 ⊢ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0
11	9, 10	pm3.2i 270	. . 3 ⊢ ([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 )
12		eqid 2140	. . 3 ⊢ [⟨(𝐴 ·o 𝐶), (𝐵 ·o 𝐷)⟩] ~Q0 = [⟨(𝐴 ·o 𝐶), (𝐵 ·o 𝐷)⟩] ~Q0
13		opeq12 3715	. . . . . 6 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩)
14		eceq1 6472	. . . . . . . . 9 ⊢ (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → [⟨𝑤, 𝑣⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0 )
15	14	eqeq2d 2152	. . . . . . . 8 ⊢ (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → ([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ↔ [⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0 ))
16	15	anbi1d 461	. . . . . . 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 2692	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
18		vex 2692	. . . . . . . . . . 11 ⊢ 𝑣 ∈ V
19	17, 18	opth 4167	. . . . . . . . . 10 ⊢ (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑤 = 𝐴 ∧ 𝑣 = 𝐵))
20		oveq1 5789	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝐴 → (𝑤 ·o 𝐶) = (𝐴 ·o 𝐶))
21	20	adantr 274	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → (𝑤 ·o 𝐶) = (𝐴 ·o 𝐶))
22		oveq1 5789	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝐵 → (𝑣 ·o 𝐷) = (𝐵 ·o 𝐷))
23	22	adantl 275	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → (𝑣 ·o 𝐷) = (𝐵 ·o 𝐷))
24	21, 23	opeq12d 3721	. . . . . . . . . 10 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ⟨(𝑤 ·o 𝐶), (𝑣 ·o 𝐷)⟩ = ⟨(𝐴 ·o 𝐶), (𝐵 ·o 𝐷)⟩)
25	19, 24	sylbi 120	. . . . . . . . 9 ⊢ (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → ⟨(𝑤 ·o 𝐶), (𝑣 ·o 𝐷)⟩ = ⟨(𝐴 ·o 𝐶), (𝐵 ·o 𝐷)⟩)
26	25	eceq1d 6473	. . . . . . . 8 ⊢ (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → [⟨(𝑤 ·o 𝐶), (𝑣 ·o 𝐷)⟩] ~Q0 = [⟨(𝐴 ·o 𝐶), (𝐵 ·o 𝐷)⟩] ~Q0 )
27	26	eqeq2d 2152	. . . . . . 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 465	. . . . . 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 2779	. . . 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 3715	. . . . . . 7 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → ⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩)
32		eceq1 6472	. . . . . . . . . 10 ⊢ (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → [⟨𝑢, 𝑡⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 )
33	32	eqeq2d 2152	. . . . . . . . 9 ⊢ (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → ([⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ↔ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ))
34	33	anbi2d 460	. . . . . . . 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 2692	. . . . . . . . . . . 12 ⊢ 𝑢 ∈ V
36		vex 2692	. . . . . . . . . . . 12 ⊢ 𝑡 ∈ V
37	35, 36	opth 4167	. . . . . . . . . . 11 ⊢ (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝑢 = 𝐶 ∧ 𝑡 = 𝐷))
38		oveq2 5790	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝐶 → (𝑤 ·o 𝑢) = (𝑤 ·o 𝐶))
39	38	adantr 274	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → (𝑤 ·o 𝑢) = (𝑤 ·o 𝐶))
40		oveq2 5790	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝐷 → (𝑣 ·o 𝑡) = (𝑣 ·o 𝐷))
41	40	adantl 275	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → (𝑣 ·o 𝑡) = (𝑣 ·o 𝐷))
42	39, 41	opeq12d 3721	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → ⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩ = ⟨(𝑤 ·o 𝐶), (𝑣 ·o 𝐷)⟩)
43	37, 42	sylbi 120	. . . . . . . . . 10 ⊢ (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → ⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩ = ⟨(𝑤 ·o 𝐶), (𝑣 ·o 𝐷)⟩)
44	43	eceq1d 6473	. . . . . . . . 9 ⊢ (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 = [⟨(𝑤 ·o 𝐶), (𝑣 ·o 𝐷)⟩] ~Q0 )
45	44	eqeq2d 2152	. . . . . . . 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 465	. . . . . . 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 2779	. . . . 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 1855	. . . 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 407	. . 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 429	. 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 6441	. . . 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 2147	. . . . . . . 8 ⊢ (𝑥 = [⟨𝐴, 𝐵⟩] ~Q0 → (𝑥 = [⟨𝑤, 𝑣⟩] ~Q0 ↔ [⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ))
55		eqeq1 2147	. . . . . . . 8 ⊢ (𝑦 = [⟨𝐶, 𝐷⟩] ~Q0 → (𝑦 = [⟨𝑢, 𝑡⟩] ~Q0 ↔ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ))
56	54, 55	bi2anan9 596	. . . . . . 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 2147	. . . . . . 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 596	. . . . . 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 1177	. . . . 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 1843	. . . 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 7280	. . . 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 7261	. . . 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 5900	. . 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 1305	. 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 103   ↔ wb 104   ∧ w3a 963   = wceq 1332  ∃wex 1469   ∈ wcel 1481  Vcvv 2689  ⟨cop 3535  ωcom 4512   × cxp 4545  (class class class)co 5782   ·_o comu 6319  [cec 6435   / cqs 6436  Ncnpi 7104   ~_Q0 ceq0 7118   ·_Q0 cmq0 7122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-oadd 6325  df-omul 6326  df-er 6437  df-ec 6439  df-qs 6443  df-ni 7136  df-mi 7138  df-enq0 7256  df-nq0 7257  df-mq0 7260

This theorem is referenced by:  mulclnq0  7284  nqnq0m  7287  nq0m0r  7288  distrnq0  7291  mulcomnq0  7292  nq02m  7297