Theorem mulnnnq0 7258

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 4571	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ N) → ⟨𝐴, 𝐵⟩ ∈ (ω × N))
2		enq0ex 7247	. . . . 5 ⊢ ~Q0 ∈ V
3	2	ecelqsi 6483	. . . 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 4571	. . . 4 ⊢ ((𝐶 ∈ ω ∧ 𝐷 ∈ N) → ⟨𝐶, 𝐷⟩ ∈ (ω × N))
6	2	ecelqsi 6483	. . . 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 2139	. . . 4 ⊢ [⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0
10		eqid 2139	. . . 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 2139	. . 3 ⊢ [⟨(𝐴 ·o 𝐶), (𝐵 ·o 𝐷)⟩] ~Q0 = [⟨(𝐴 ·o 𝐶), (𝐵 ·o 𝐷)⟩] ~Q0
13		opeq12 3707	. . . . . 6 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩)
14		eceq1 6464	. . . . . . . . 9 ⊢ (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → [⟨𝑤, 𝑣⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0 )
15	14	eqeq2d 2151	. . . . . . . 8 ⊢ (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → ([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ↔ [⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0 ))
16	15	anbi1d 460	. . . . . . 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 2689	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
18		vex 2689	. . . . . . . . . . 11 ⊢ 𝑣 ∈ V
19	17, 18	opth 4159	. . . . . . . . . 10 ⊢ (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑤 = 𝐴 ∧ 𝑣 = 𝐵))
20		oveq1 5781	. . . . . . . . . . . 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 5781	. . . . . . . . . . . 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 3713	. . . . . . . . . 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 6465	. . . . . . . 8 ⊢ (⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩ → [⟨(𝑤 ·o 𝐶), (𝑣 ·o 𝐷)⟩] ~Q0 = [⟨(𝐴 ·o 𝐶), (𝐵 ·o 𝐷)⟩] ~Q0 )
27	26	eqeq2d 2151	. . . . . . 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 464	. . . . . 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 2775	. . . 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 3707	. . . . . . 7 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → ⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩)
32		eceq1 6464	. . . . . . . . . 10 ⊢ (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → [⟨𝑢, 𝑡⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 )
33	32	eqeq2d 2151	. . . . . . . . 9 ⊢ (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → ([⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ↔ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ))
34	33	anbi2d 459	. . . . . . . 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 2689	. . . . . . . . . . . 12 ⊢ 𝑢 ∈ V
36		vex 2689	. . . . . . . . . . . 12 ⊢ 𝑡 ∈ V
37	35, 36	opth 4159	. . . . . . . . . . 11 ⊢ (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝑢 = 𝐶 ∧ 𝑡 = 𝐷))
38		oveq2 5782	. . . . . . . . . . . . 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 5782	. . . . . . . . . . . . 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 3713	. . . . . . . . . . 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 6465	. . . . . . . . 9 ⊢ (⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩ → [⟨(𝑤 ·o 𝑢), (𝑣 ·o 𝑡)⟩] ~Q0 = [⟨(𝑤 ·o 𝐶), (𝑣 ·o 𝐷)⟩] ~Q0 )
45	44	eqeq2d 2151	. . . . . . . 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 464	. . . . . . 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 2775	. . . . 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 1854	. . . 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 406	. . 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 428	. 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 6433	. . . 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 2146	. . . . . . . 8 ⊢ (𝑥 = [⟨𝐴, 𝐵⟩] ~Q0 → (𝑥 = [⟨𝑤, 𝑣⟩] ~Q0 ↔ [⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ))
55		eqeq1 2146	. . . . . . . 8 ⊢ (𝑦 = [⟨𝐶, 𝐷⟩] ~Q0 → (𝑦 = [⟨𝑢, 𝑡⟩] ~Q0 ↔ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ))
56	54, 55	bi2anan9 595	. . . . . . 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 2146	. . . . . . 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 595	. . . . . 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 1176	. . . . 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 1842	. . . 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 7256	. . . 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 7237	. . . 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 5892	. . 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 1304	. 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 962   = wceq 1331  ∃wex 1468   ∈ wcel 1480  Vcvv 2686  ⟨cop 3530  ωcom 4504   × cxp 4537  (class class class)co 5774   ·_o comu 6311  [cec 6427   / cqs 6428  Ncnpi 7080   ~_Q0 ceq0 7094   ·_Q0 cmq0 7098

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-mi 7114  df-enq0 7232  df-nq0 7233  df-mq0 7236

This theorem is referenced by:  mulclnq0  7260  nqnq0m  7263  nq0m0r  7264  distrnq0  7267  mulcomnq0  7268  nq02m  7273