Theorem addnq0mo 6985

Description: There is at most one result from adding nonnegative fractions. (Contributed by Jim Kingdon, 23-Nov-2019.)

Assertion

Ref	Expression
addnq0mo	⊢ ((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) → ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ))

Distinct variable groups:   𝑡,𝐴,𝑢,𝑣,𝑤,𝑧   𝑡,𝐵,𝑢,𝑣,𝑤,𝑧

Proof of Theorem addnq0mo

Dummy variables 𝑓 𝑔 ℎ 𝑞 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		enq0er 6973	. . . . . . . . . . . . . 14 ⊢ ~Q0 Er (ω × N)
2	1	a1i 9	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0 ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ℎ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 ℎ)⟩] ~Q0 ))) → ~Q0 Er (ω × N))
3		nnnq0lem1 6984	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0 ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ℎ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 ℎ)⟩] ~Q0 ))) → ((((𝑤 ∈ ω ∧ 𝑣 ∈ N) ∧ (𝑠 ∈ ω ∧ 𝑓 ∈ N)) ∧ ((𝑢 ∈ ω ∧ 𝑡 ∈ N) ∧ (𝑔 ∈ ω ∧ ℎ ∈ N))) ∧ ((𝑤 ·𝑜 𝑓) = (𝑣 ·𝑜 𝑠) ∧ (𝑢 ·𝑜 ℎ) = (𝑡 ·𝑜 𝑔))))
4		addcmpblnq0 6981	. . . . . . . . . . . . . . 15 ⊢ ((((𝑤 ∈ ω ∧ 𝑣 ∈ N) ∧ (𝑠 ∈ ω ∧ 𝑓 ∈ N)) ∧ ((𝑢 ∈ ω ∧ 𝑡 ∈ N) ∧ (𝑔 ∈ ω ∧ ℎ ∈ N))) → (((𝑤 ·𝑜 𝑓) = (𝑣 ·𝑜 𝑠) ∧ (𝑢 ·𝑜 ℎ) = (𝑡 ·𝑜 𝑔)) → ⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩ ~Q0 ⟨((𝑠 ·𝑜 ℎ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 ℎ)⟩))
5	4	imp 122	. . . . . . . . . . . . . 14 ⊢ (((((𝑤 ∈ ω ∧ 𝑣 ∈ N) ∧ (𝑠 ∈ ω ∧ 𝑓 ∈ N)) ∧ ((𝑢 ∈ ω ∧ 𝑡 ∈ N) ∧ (𝑔 ∈ ω ∧ ℎ ∈ N))) ∧ ((𝑤 ·𝑜 𝑓) = (𝑣 ·𝑜 𝑠) ∧ (𝑢 ·𝑜 ℎ) = (𝑡 ·𝑜 𝑔))) → ⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩ ~Q0 ⟨((𝑠 ·𝑜 ℎ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 ℎ)⟩)
6	3, 5	syl 14	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0 ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ℎ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 ℎ)⟩] ~Q0 ))) → ⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩ ~Q0 ⟨((𝑠 ·𝑜 ℎ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 ℎ)⟩)
7	2, 6	erthi 6318	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0 ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ℎ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 ℎ)⟩] ~Q0 ))) → [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 = [⟨((𝑠 ·𝑜 ℎ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 ℎ)⟩] ~Q0 )
8		simprlr 505	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0 ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ℎ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 ℎ)⟩] ~Q0 ))) → 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )
9		simprrr 507	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0 ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ℎ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 ℎ)⟩] ~Q0 ))) → 𝑞 = [⟨((𝑠 ·𝑜 ℎ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 ℎ)⟩] ~Q0 )
10	7, 8, 9	3eqtr4d 2130	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0 ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ℎ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 ℎ)⟩] ~Q0 ))) → 𝑧 = 𝑞)
11	10	expr 367	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )) → (((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0 ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ℎ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 ℎ)⟩] ~Q0 ) → 𝑧 = 𝑞))
12	11	exlimdvv 1825	. . . . . . . . 9 ⊢ (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )) → (∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0 ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ℎ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 ℎ)⟩] ~Q0 ) → 𝑧 = 𝑞))
13	12	exlimdvv 1825	. . . . . . . 8 ⊢ (((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )) → (∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0 ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ℎ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 ℎ)⟩] ~Q0 ) → 𝑧 = 𝑞))
14	13	ex 113	. . . . . . 7 ⊢ ((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) → (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) → (∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0 ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ℎ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 ℎ)⟩] ~Q0 ) → 𝑧 = 𝑞)))
15	14	exlimdvv 1825	. . . . . 6 ⊢ ((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) → (∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) → (∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0 ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ℎ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 ℎ)⟩] ~Q0 ) → 𝑧 = 𝑞)))
16	15	exlimdvv 1825	. . . . 5 ⊢ ((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) → (∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) → (∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0 ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ℎ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 ℎ)⟩] ~Q0 ) → 𝑧 = 𝑞)))
17	16	impd 251	. . . 4 ⊢ ((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) → ((∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ∧ ∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0 ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ℎ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 ℎ)⟩] ~Q0 )) → 𝑧 = 𝑞))
18	17	alrimivv 1803	. . 3 ⊢ ((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) → ∀𝑧∀𝑞((∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ∧ ∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0 ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ℎ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 ℎ)⟩] ~Q0 )) → 𝑧 = 𝑞))
19		opeq12 3619	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → ⟨𝑤, 𝑣⟩ = ⟨𝑠, 𝑓⟩)
20	19	eceq1d 6308	. . . . . . . . . 10 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → [⟨𝑤, 𝑣⟩] ~Q0 = [⟨𝑠, 𝑓⟩] ~Q0 )
21	20	eqeq2d 2099	. . . . . . . . 9 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → (𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ↔ 𝐴 = [⟨𝑠, 𝑓⟩] ~Q0 ))
22	21	anbi1d 453	. . . . . . . 8 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → ((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ↔ (𝐴 = [⟨𝑠, 𝑓⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 )))
23		simpl 107	. . . . . . . . . . . . 13 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → 𝑤 = 𝑠)
24	23	oveq1d 5649	. . . . . . . . . . . 12 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → (𝑤 ·𝑜 𝑡) = (𝑠 ·𝑜 𝑡))
25		simpr 108	. . . . . . . . . . . . 13 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → 𝑣 = 𝑓)
26	25	oveq1d 5649	. . . . . . . . . . . 12 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → (𝑣 ·𝑜 𝑢) = (𝑓 ·𝑜 𝑢))
27	24, 26	oveq12d 5652	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → ((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)) = ((𝑠 ·𝑜 𝑡) +𝑜 (𝑓 ·𝑜 𝑢)))
28	25	oveq1d 5649	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → (𝑣 ·𝑜 𝑡) = (𝑓 ·𝑜 𝑡))
29	27, 28	opeq12d 3625	. . . . . . . . . 10 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → ⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩ = ⟨((𝑠 ·𝑜 𝑡) +𝑜 (𝑓 ·𝑜 𝑢)), (𝑓 ·𝑜 𝑡)⟩)
30	29	eceq1d 6308	. . . . . . . . 9 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 = [⟨((𝑠 ·𝑜 𝑡) +𝑜 (𝑓 ·𝑜 𝑢)), (𝑓 ·𝑜 𝑡)⟩] ~Q0 )
31	30	eqeq2d 2099	. . . . . . . 8 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → (𝑞 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ↔ 𝑞 = [⟨((𝑠 ·𝑜 𝑡) +𝑜 (𝑓 ·𝑜 𝑢)), (𝑓 ·𝑜 𝑡)⟩] ~Q0 ))
32	22, 31	anbi12d 457	. . . . . . 7 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ↔ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 𝑡) +𝑜 (𝑓 ·𝑜 𝑢)), (𝑓 ·𝑜 𝑡)⟩] ~Q0 )))
33		opeq12 3619	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → ⟨𝑢, 𝑡⟩ = ⟨𝑔, ℎ⟩)
34	33	eceq1d 6308	. . . . . . . . . 10 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → [⟨𝑢, 𝑡⟩] ~Q0 = [⟨𝑔, ℎ⟩] ~Q0 )
35	34	eqeq2d 2099	. . . . . . . . 9 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → (𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ↔ 𝐵 = [⟨𝑔, ℎ⟩] ~Q0 ))
36	35	anbi2d 452	. . . . . . . 8 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ↔ (𝐴 = [⟨𝑠, 𝑓⟩] ~Q0 ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~Q0 )))
37		simpr 108	. . . . . . . . . . . . 13 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → 𝑡 = ℎ)
38	37	oveq2d 5650	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → (𝑠 ·𝑜 𝑡) = (𝑠 ·𝑜 ℎ))
39		simpl 107	. . . . . . . . . . . . 13 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → 𝑢 = 𝑔)
40	39	oveq2d 5650	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → (𝑓 ·𝑜 𝑢) = (𝑓 ·𝑜 𝑔))
41	38, 40	oveq12d 5652	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → ((𝑠 ·𝑜 𝑡) +𝑜 (𝑓 ·𝑜 𝑢)) = ((𝑠 ·𝑜 ℎ) +𝑜 (𝑓 ·𝑜 𝑔)))
42	37	oveq2d 5650	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → (𝑓 ·𝑜 𝑡) = (𝑓 ·𝑜 ℎ))
43	41, 42	opeq12d 3625	. . . . . . . . . 10 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → ⟨((𝑠 ·𝑜 𝑡) +𝑜 (𝑓 ·𝑜 𝑢)), (𝑓 ·𝑜 𝑡)⟩ = ⟨((𝑠 ·𝑜 ℎ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 ℎ)⟩)
44	43	eceq1d 6308	. . . . . . . . 9 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → [⟨((𝑠 ·𝑜 𝑡) +𝑜 (𝑓 ·𝑜 𝑢)), (𝑓 ·𝑜 𝑡)⟩] ~Q0 = [⟨((𝑠 ·𝑜 ℎ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 ℎ)⟩] ~Q0 )
45	44	eqeq2d 2099	. . . . . . . 8 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → (𝑞 = [⟨((𝑠 ·𝑜 𝑡) +𝑜 (𝑓 ·𝑜 𝑢)), (𝑓 ·𝑜 𝑡)⟩] ~Q0 ↔ 𝑞 = [⟨((𝑠 ·𝑜 ℎ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 ℎ)⟩] ~Q0 ))
46	36, 45	anbi12d 457	. . . . . . 7 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → (((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 𝑡) +𝑜 (𝑓 ·𝑜 𝑢)), (𝑓 ·𝑜 𝑡)⟩] ~Q0 ) ↔ ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0 ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ℎ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 ℎ)⟩] ~Q0 )))
47	32, 46	cbvex4v 1853	. . . . . 6 ⊢ (∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ↔ ∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0 ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ℎ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 ℎ)⟩] ~Q0 ))
48	47	anbi2i 445	. . . . 5 ⊢ ((∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )) ↔ (∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ∧ ∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0 ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ℎ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 ℎ)⟩] ~Q0 )))
49	48	imbi1i 236	. . . 4 ⊢ (((∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )) → 𝑧 = 𝑞) ↔ ((∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ∧ ∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0 ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ℎ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 ℎ)⟩] ~Q0 )) → 𝑧 = 𝑞))
50	49	2albii 1405	. . 3 ⊢ (∀𝑧∀𝑞((∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )) → 𝑧 = 𝑞) ↔ ∀𝑧∀𝑞((∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ∧ ∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0 ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑠 ·𝑜 ℎ) +𝑜 (𝑓 ·𝑜 𝑔)), (𝑓 ·𝑜 ℎ)⟩] ~Q0 )) → 𝑧 = 𝑞))
51	18, 50	sylibr 132	. 2 ⊢ ((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) → ∀𝑧∀𝑞((∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )) → 𝑧 = 𝑞))
52		eqeq1 2094	. . . . 5 ⊢ (𝑧 = 𝑞 → (𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ↔ 𝑞 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ))
53	52	anbi2d 452	. . . 4 ⊢ (𝑧 = 𝑞 → (((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ↔ ((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )))
54	53	4exbidv 1798	. . 3 ⊢ (𝑧 = 𝑞 → (∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ↔ ∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )))
55	54	mo4 2009	. 2 ⊢ (∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ↔ ∀𝑧∀𝑞((∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑞 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 )) → 𝑧 = 𝑞))
56	51, 55	sylibr 132	1 ⊢ ((𝐴 ∈ ((ω × N) / ~Q0 ) ∧ 𝐵 ∈ ((ω × N) / ~Q0 )) → ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ∧ 𝑧 = [⟨((𝑤 ·𝑜 𝑡) +𝑜 (𝑣 ·𝑜 𝑢)), (𝑣 ·𝑜 𝑡)⟩] ~Q0 ))

Syntax hints:   → wi 4   ∧ wa 102  ∀wal 1287   = wceq 1289  ∃wex 1426   ∈ wcel 1438  ∃*wmo 1949  ⟨cop 3444   class class class wbr 3837  ωcom 4395   × cxp 4426  (class class class)co 5634   +_𝑜 coa 6160   ·_𝑜 comu 6161   Er wer 6269  [cec 6270   / cqs 6271  Ncnpi 6810   ~_Q0 ceq0 6824

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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393

This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-oadd 6167  df-omul 6168  df-er 6272  df-ec 6274  df-qs 6278  df-ni 6842  df-mi 6844  df-enq0 6962

This theorem is referenced by:  addnnnq0  6987