Theorem addnq0mo 7475

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 7463	. . . . . . . . . . . . . 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 7474	. . . . . . . . . . . . . 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 7471	. . . . . . . . . . . . . . 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 124	. . . . . . . . . . . . . 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 6606	. . . . . . . . . . . 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 538	. . . . . . . . . . . 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 540	. . . . . . . . . . . 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 2232	. . . . . . . . . . 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 375	. . . . . . . . . 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 1909	. . . . . . . . 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 1909	. . . . . . . 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 115	. . . . . . 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 1909	. . . . . 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 1909	. . . . 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 254	. . . 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 1886	. . 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 3795	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → ⟨𝑤, 𝑣⟩ = ⟨𝑠, 𝑓⟩)
20	19	eceq1d 6594	. . . . . . . . . 10 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → [⟨𝑤, 𝑣⟩] ~Q0 = [⟨𝑠, 𝑓⟩] ~Q0 )
21	20	eqeq2d 2201	. . . . . . . . 9 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → (𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ↔ 𝐴 = [⟨𝑠, 𝑓⟩] ~Q0 ))
22	21	anbi1d 465	. . . . . . . 8 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → ((𝐴 = [⟨𝑤, 𝑣⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ↔ (𝐴 = [⟨𝑠, 𝑓⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 )))
23		simpl 109	. . . . . . . . . . . . 13 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → 𝑤 = 𝑠)
24	23	oveq1d 5910	. . . . . . . . . . . 12 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → (𝑤 ·o 𝑡) = (𝑠 ·o 𝑡))
25		simpr 110	. . . . . . . . . . . . 13 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → 𝑣 = 𝑓)
26	25	oveq1d 5910	. . . . . . . . . . . 12 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → (𝑣 ·o 𝑢) = (𝑓 ·o 𝑢))
27	24, 26	oveq12d 5913	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → ((𝑤 ·o 𝑡) +o (𝑣 ·o 𝑢)) = ((𝑠 ·o 𝑡) +o (𝑓 ·o 𝑢)))
28	25	oveq1d 5910	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → (𝑣 ·o 𝑡) = (𝑓 ·o 𝑡))
29	27, 28	opeq12d 3801	. . . . . . . . . 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 6594	. . . . . . . . 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 2201	. . . . . . . 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 473	. . . . . . 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 3795	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → ⟨𝑢, 𝑡⟩ = ⟨𝑔, ℎ⟩)
34	33	eceq1d 6594	. . . . . . . . . 10 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → [⟨𝑢, 𝑡⟩] ~Q0 = [⟨𝑔, ℎ⟩] ~Q0 )
35	34	eqeq2d 2201	. . . . . . . . 9 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → (𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ↔ 𝐵 = [⟨𝑔, ℎ⟩] ~Q0 ))
36	35	anbi2d 464	. . . . . . . 8 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → ((𝐴 = [⟨𝑠, 𝑓⟩] ~Q0 ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~Q0 ) ↔ (𝐴 = [⟨𝑠, 𝑓⟩] ~Q0 ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~Q0 )))
37		simpr 110	. . . . . . . . . . . . 13 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → 𝑡 = ℎ)
38	37	oveq2d 5911	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → (𝑠 ·o 𝑡) = (𝑠 ·o ℎ))
39		simpl 109	. . . . . . . . . . . . 13 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → 𝑢 = 𝑔)
40	39	oveq2d 5911	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → (𝑓 ·o 𝑢) = (𝑓 ·o 𝑔))
41	38, 40	oveq12d 5913	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → ((𝑠 ·o 𝑡) +o (𝑓 ·o 𝑢)) = ((𝑠 ·o ℎ) +o (𝑓 ·o 𝑔)))
42	37	oveq2d 5911	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → (𝑓 ·o 𝑡) = (𝑓 ·o ℎ))
43	41, 42	opeq12d 3801	. . . . . . . . . 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 6594	. . . . . . . . 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 2201	. . . . . . . 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 473	. . . . . . 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 1942	. . . . . 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 457	. . . . 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 238	. . . 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 1482	. . 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 134	. 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 2196	. . . . 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 464	. . . 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 1881	. . 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 2099	. 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 134	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 104  ∀wal 1362   = wceq 1364  ∃wex 1503  ∃*wmo 2039   ∈ wcel 2160  ⟨cop 3610   class class class wbr 4018  ωcom 4607   × cxp 4642  (class class class)co 5895   +_o coa 6437   ·_o comu 6438   Er wer 6555  [cec 6556   / cqs 6557  Ncnpi 7300   ~_Q0 ceq0 7314

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-recs 6329  df-irdg 6394  df-oadd 6444  df-omul 6445  df-er 6558  df-ec 6560  df-qs 6564  df-ni 7332  df-mi 7334  df-enq0 7452

This theorem is referenced by:  addnnnq0  7477