Theorem addnnnq0 6639

Description: Addition of non-negative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 22-Nov-2019.)

Assertion

Ref	Expression
addnnnq0	⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) → ([⟨𝐴, 𝐵⟩] ~Q0 +Q0 [⟨𝐶, 𝐷⟩] ~Q0 ) = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 )

Proof of Theorem addnnnq0

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 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0
13		opeq12 3572	. . . . . . . . 9 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ⟨𝑤, 𝑣⟩ = ⟨𝐴, 𝐵⟩)
14	13	eceq1d 6165	. . . . . . . 8 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → [⟨𝑤, 𝑣⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0 )
15	14	eqeq2d 2092	. . . . . . 7 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ↔ [⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0 ))
16	15	anbi1d 452	. . . . . 6 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 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		simpl 107	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → 𝑤 = 𝐴)
18	17	oveq1d 5547	. . . . . . . . . 10 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → (𝑤 ·𝑜 𝐷) = (𝐴 ·𝑜 𝐷))
19		simpr 108	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → 𝑣 = 𝐵)
20	19	oveq1d 5547	. . . . . . . . . 10 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → (𝑣 ·𝑜 𝐶) = (𝐵 ·𝑜 𝐶))
21	18, 20	oveq12d 5550	. . . . . . . . 9 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ((𝑤 ·𝑜 𝐷) +𝑜 (𝑣 ·𝑜 𝐶)) = ((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)))
22	19	oveq1d 5547	. . . . . . . . 9 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → (𝑣 ·𝑜 𝐷) = (𝐵 ·𝑜 𝐷))
23	21, 22	opeq12d 3578	. . . . . . . 8 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ⟨((𝑤 ·𝑜 𝐷) +𝑜 (𝑣 ·𝑜 𝐶)), (𝑣 ·𝑜 𝐷)⟩ = ⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩)
24	23	eceq1d 6165	. . . . . . 7 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → [⟨((𝑤 ·𝑜 𝐷) +𝑜 (𝑣 ·𝑜 𝐶)), (𝑣 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 )
25	24	eqeq2d 2092	. . . . . 6 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ([⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝐷) +𝑜 (𝑣 ·𝑜 𝐶)), (𝑣 ·𝑜 𝐷)⟩] ~Q0 ↔ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ))
26	16, 25	anbi12d 456	. . . . 5 ⊢ ((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) → ((([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝐷) +𝑜 (𝑣 ·𝑜 𝐶)), (𝑣 ·𝑜 𝐷)⟩] ~Q0 ) ↔ (([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 )))
27	26	spc2egv 2687	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ N) → ((([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝐴, 𝐵⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ) → ∃𝑤∃𝑣(([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝐷) +𝑜 (𝑣 ·𝑜 𝐶)), (𝑣 ·𝑜 𝐷)⟩] ~Q0 )))
28		opeq12 3572	. . . . . . . . . 10 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → ⟨𝑢, 𝑡⟩ = ⟨𝐶, 𝐷⟩)
29	28	eceq1d 6165	. . . . . . . . 9 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → [⟨𝑢, 𝑡⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 )
30	29	eqeq2d 2092	. . . . . . . 8 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → ([⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ↔ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ))
31	30	anbi2d 451	. . . . . . 7 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 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 )))
32		simpr 108	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → 𝑡 = 𝐷)
33	32	oveq2d 5548	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → (𝑤 ·𝑜 𝑡) = (𝑤 ·𝑜 𝐷))
34		simpl 107	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → 𝑢 = 𝐶)
35	34	oveq2d 5548	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → (𝑣 ·𝑜 𝑢) = (𝑣 ·𝑜 𝐶))
36	33, 35	oveq12d 5550	. . . . . . . . . 10 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → ((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)) = ((𝑤 ·𝑜 𝐷) +𝑜 (𝑣 ·𝑜 𝐶)))
37	32	oveq2d 5548	. . . . . . . . . 10 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → (𝑣 ·𝑜 𝑡) = (𝑣 ·𝑜 𝐷))
38	36, 37	opeq12d 3578	. . . . . . . . 9 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → ⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩ = ⟨((𝑤 ·𝑜 𝐷) +𝑜 (𝑣 ·𝑜 𝐶)), (𝑣 ·𝑜 𝐷)⟩)
39	38	eceq1d 6165	. . . . . . . 8 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝐷) +𝑜 (𝑣 ·𝑜 𝐶)), (𝑣 ·𝑜 𝐷)⟩] ~Q0 )
40	39	eqeq2d 2092	. . . . . . 7 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → ([⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ↔ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝐷) +𝑜 (𝑣 ·𝑜 𝐶)), (𝑣 ·𝑜 𝐷)⟩] ~Q0 ))
41	31, 40	anbi12d 456	. . . . . 6 ⊢ ((𝑢 = 𝐶 ∧ 𝑡 = 𝐷) → ((([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ↔ (([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝐷) +𝑜 (𝑣 ·𝑜 𝐶)), (𝑣 ·𝑜 𝐷)⟩] ~Q0 )))
42	41	spc2egv 2687	. . . . 5 ⊢ ((𝐶 ∈ ω ∧ 𝐷 ∈ N) → ((([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝐷) +𝑜 (𝑣 ·𝑜 𝐶)), (𝑣 ·𝑜 𝐷)⟩] ~Q0 ) → ∃𝑢∃𝑡(([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )))
43	42	2eximdv 1803	. . . 4 ⊢ ((𝐶 ∈ ω ∧ 𝐷 ∈ N) → (∃𝑤∃𝑣(([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝐶, 𝐷⟩] ~Q0 ) ∧ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝐷) +𝑜 (𝑣 ·𝑜 𝐶)), (𝑣 ·𝑜 𝐷)⟩] ~Q0 ) → ∃𝑤∃𝑣∃𝑢∃𝑡(([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )))
44	27, 43	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 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ) → ∃𝑤∃𝑣∃𝑢∃𝑡(([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )))
45	11, 12, 44	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 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ))
46		ecexg 6133	. . . 4 ⊢ ( ~Q0 ∈ V → [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ∈ V)
47	2, 46	ax-mp 7	. . 3 ⊢ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ∈ V
48		simp1 938	. . . . . . . 8 ⊢ ((𝑥 = [⟨𝐴, 𝐵⟩] ~Q0 ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ~Q0 ∧ 𝑧 = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ) → 𝑥 = [⟨𝐴, 𝐵⟩] ~Q0 )
49	48	eqeq1d 2089	. . . . . . 7 ⊢ ((𝑥 = [⟨𝐴, 𝐵⟩] ~Q0 ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ~Q0 ∧ 𝑧 = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ) → (𝑥 = [⟨𝑤, 𝑣⟩] ~Q0 ↔ [⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ))
50		simp2 939	. . . . . . . 8 ⊢ ((𝑥 = [⟨𝐴, 𝐵⟩] ~Q0 ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ~Q0 ∧ 𝑧 = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ) → 𝑦 = [⟨𝐶, 𝐷⟩] ~Q0 )
51	50	eqeq1d 2089	. . . . . . 7 ⊢ ((𝑥 = [⟨𝐴, 𝐵⟩] ~Q0 ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ~Q0 ∧ 𝑧 = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ) → (𝑦 = [⟨𝑢, 𝑡⟩] ~Q0 ↔ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ))
52	49, 51	anbi12d 456	. . . . . 6 ⊢ ((𝑥 = [⟨𝐴, 𝐵⟩] ~Q0 ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ~Q0 ∧ 𝑧 = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ) → ((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝑦 = [⟨𝑢, 𝑡⟩] ~Q0 ) ↔ ([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 )))
53		simp3 940	. . . . . . 7 ⊢ ((𝑥 = [⟨𝐴, 𝐵⟩] ~Q0 ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ~Q0 ∧ 𝑧 = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ) → 𝑧 = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 )
54	53	eqeq1d 2089	. . . . . 6 ⊢ ((𝑥 = [⟨𝐴, 𝐵⟩] ~Q0 ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ~Q0 ∧ 𝑧 = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ) → (𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ↔ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ))
55	52, 54	anbi12d 456	. . . . 5 ⊢ ((𝑥 = [⟨𝐴, 𝐵⟩] ~Q0 ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ~Q0 ∧ 𝑧 = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ) → (((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝑦 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ↔ (([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )))
56	55	4exbidv 1791	. . . 4 ⊢ ((𝑥 = [⟨𝐴, 𝐵⟩] ~Q0 ∧ 𝑦 = [⟨𝐶, 𝐷⟩] ~Q0 ∧ 𝑧 = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ) → (∃𝑤∃𝑣∃𝑢∃𝑡((𝑥 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝑦 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ↔ ∃𝑤∃𝑣∃𝑢∃𝑡(([⟨𝐴, 𝐵⟩] ~Q0 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ [⟨𝐶, 𝐷⟩] ~Q0 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )))
57		addnq0mo 6637	. . . 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 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ))
58		dfplq0qs 6620	. . . 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 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ))}
59	56, 57, 58	ovig 5642	. . 3 ⊢ (([⟨𝐴, 𝐵⟩] ~Q0 ∈ ((ω × N) / ~Q0 ) ∧ [⟨𝐶, 𝐷⟩] ~Q0 ∈ ((ω × N) / ~Q0 ) ∧ [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·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 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) → ([⟨𝐴, 𝐵⟩] ~Q0 +Q0 [⟨𝐶, 𝐷⟩] ~Q0 ) = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ))
60	47, 59	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 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) → ([⟨𝐴, 𝐵⟩] ~Q0 +Q0 [⟨𝐶, 𝐷⟩] ~Q0 ) = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 ))
61	8, 45, 60	sylc 61	1 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) → ([⟨𝐴, 𝐵⟩] ~Q0 +Q0 [⟨𝐶, 𝐷⟩] ~Q0 ) = [⟨((𝐴 ·𝑜 𝐷) +𝑜 (𝐵 ·𝑜 𝐶)), (𝐵 ·𝑜 𝐷)⟩] ~Q0 )

Syntax hints:   → wi 4   ∧ wa 102   ∧ w3a 919   = wceq 1284  ∃wex 1421   ∈ wcel 1433  Vcvv 2601  ⟨cop 3401  ωcom 4331   × cxp 4361  (class class class)co 5532   +_𝑜 coa 6021   ·_𝑜 comu 6022  [cec 6127   / cqs 6128  Ncnpi 6462   ~_Q0 ceq0 6476   +_Q0 cplq0 6479

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-plq0 6617

This theorem is referenced by:  addclnq0  6641  nqpnq0nq  6643  nqnq0a  6644  nq0a0  6647  nnanq0  6648  distrnq0  6649  addassnq0  6652