Theorem addsrmo 7663

Description: There is at most one result from adding signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)

Assertion

Ref	Expression
addsrmo	⊢ ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ))

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

Proof of Theorem addsrmo

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

Step	Hyp	Ref	Expression
1		enrer 7655	. . . . . . . . . . . . . . . 16 ⊢ ~R Er (P × P)
2	1	a1i 9	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → ~R Er (P × P))
3		prsrlem1 7662	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → ((((𝑤 ∈ P ∧ 𝑣 ∈ P) ∧ (𝑠 ∈ P ∧ 𝑓 ∈ P)) ∧ ((𝑢 ∈ P ∧ 𝑡 ∈ P) ∧ (𝑔 ∈ P ∧ ℎ ∈ P))) ∧ ((𝑤 +P 𝑓) = (𝑣 +P 𝑠) ∧ (𝑢 +P ℎ) = (𝑡 +P 𝑔))))
4		addcmpblnr 7659	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑤 ∈ P ∧ 𝑣 ∈ P) ∧ (𝑠 ∈ P ∧ 𝑓 ∈ P)) ∧ ((𝑢 ∈ P ∧ 𝑡 ∈ P) ∧ (𝑔 ∈ P ∧ ℎ ∈ P))) → (((𝑤 +P 𝑓) = (𝑣 +P 𝑠) ∧ (𝑢 +P ℎ) = (𝑡 +P 𝑔)) → ⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩ ~R ⟨(𝑠 +P 𝑔), (𝑓 +P ℎ)⟩))
5	4	imp 123	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑤 ∈ P ∧ 𝑣 ∈ P) ∧ (𝑠 ∈ P ∧ 𝑓 ∈ P)) ∧ ((𝑢 ∈ P ∧ 𝑡 ∈ P) ∧ (𝑔 ∈ P ∧ ℎ ∈ P))) ∧ ((𝑤 +P 𝑓) = (𝑣 +P 𝑠) ∧ (𝑢 +P ℎ) = (𝑡 +P 𝑔))) → ⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩ ~R ⟨(𝑠 +P 𝑔), (𝑓 +P ℎ)⟩)
6	3, 5	syl 14	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → ⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩ ~R ⟨(𝑠 +P 𝑔), (𝑓 +P ℎ)⟩)
7	2, 6	erthi 6526	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R = [⟨(𝑠 +P 𝑔), (𝑓 +P ℎ)⟩] ~R )
8	7	adantrlr 477	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R = [⟨(𝑠 +P 𝑔), (𝑓 +P ℎ)⟩] ~R )
9	8	adantrrr 479	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨(𝑠 +P 𝑔), (𝑓 +P ℎ)⟩] ~R ))) → [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R = [⟨(𝑠 +P 𝑔), (𝑓 +P ℎ)⟩] ~R )
10		simprlr 528	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨(𝑠 +P 𝑔), (𝑓 +P ℎ)⟩] ~R ))) → 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R )
11		simprrr 530	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨(𝑠 +P 𝑔), (𝑓 +P ℎ)⟩] ~R ))) → 𝑞 = [⟨(𝑠 +P 𝑔), (𝑓 +P ℎ)⟩] ~R )
12	9, 10, 11	3eqtr4d 2200	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ (((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ) ∧ ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨(𝑠 +P 𝑔), (𝑓 +P ℎ)⟩] ~R ))) → 𝑧 = 𝑞)
13	12	expr 373	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R )) → (((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨(𝑠 +P 𝑔), (𝑓 +P ℎ)⟩] ~R ) → 𝑧 = 𝑞))
14	13	exlimdvv 1877	. . . . . . . . 9 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R )) → (∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨(𝑠 +P 𝑔), (𝑓 +P ℎ)⟩] ~R ) → 𝑧 = 𝑞))
15	14	exlimdvv 1877	. . . . . . . 8 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R )) → (∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨(𝑠 +P 𝑔), (𝑓 +P ℎ)⟩] ~R ) → 𝑧 = 𝑞))
16	15	ex 114	. . . . . . 7 ⊢ ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → (((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ) → (∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨(𝑠 +P 𝑔), (𝑓 +P ℎ)⟩] ~R ) → 𝑧 = 𝑞)))
17	16	exlimdvv 1877	. . . . . 6 ⊢ ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → (∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ) → (∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨(𝑠 +P 𝑔), (𝑓 +P ℎ)⟩] ~R ) → 𝑧 = 𝑞)))
18	17	exlimdvv 1877	. . . . 5 ⊢ ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → (∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ) → (∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨(𝑠 +P 𝑔), (𝑓 +P ℎ)⟩] ~R ) → 𝑧 = 𝑞)))
19	18	impd 252	. . . 4 ⊢ ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → ((∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ) ∧ ∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨(𝑠 +P 𝑔), (𝑓 +P ℎ)⟩] ~R )) → 𝑧 = 𝑞))
20	19	alrimivv 1855	. . 3 ⊢ ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → ∀𝑧∀𝑞((∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ) ∧ ∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨(𝑠 +P 𝑔), (𝑓 +P ℎ)⟩] ~R )) → 𝑧 = 𝑞))
21		opeq12 3743	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → ⟨𝑤, 𝑣⟩ = ⟨𝑠, 𝑓⟩)
22	21	eceq1d 6516	. . . . . . . . . 10 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → [⟨𝑤, 𝑣⟩] ~R = [⟨𝑠, 𝑓⟩] ~R )
23	22	eqeq2d 2169	. . . . . . . . 9 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → (𝐴 = [⟨𝑤, 𝑣⟩] ~R ↔ 𝐴 = [⟨𝑠, 𝑓⟩] ~R ))
24	23	anbi1d 461	. . . . . . . 8 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ↔ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R )))
25		simpl 108	. . . . . . . . . . . 12 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → 𝑤 = 𝑠)
26	25	oveq1d 5839	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → (𝑤 +P 𝑢) = (𝑠 +P 𝑢))
27		simpr 109	. . . . . . . . . . . 12 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → 𝑣 = 𝑓)
28	27	oveq1d 5839	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → (𝑣 +P 𝑡) = (𝑓 +P 𝑡))
29	26, 28	opeq12d 3749	. . . . . . . . . 10 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → ⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩ = ⟨(𝑠 +P 𝑢), (𝑓 +P 𝑡)⟩)
30	29	eceq1d 6516	. . . . . . . . 9 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R = [⟨(𝑠 +P 𝑢), (𝑓 +P 𝑡)⟩] ~R )
31	30	eqeq2d 2169	. . . . . . . 8 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → (𝑞 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ↔ 𝑞 = [⟨(𝑠 +P 𝑢), (𝑓 +P 𝑡)⟩] ~R ))
32	24, 31	anbi12d 465	. . . . . . 7 ⊢ ((𝑤 = 𝑠 ∧ 𝑣 = 𝑓) → (((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ) ↔ ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨(𝑠 +P 𝑢), (𝑓 +P 𝑡)⟩] ~R )))
33		opeq12 3743	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → ⟨𝑢, 𝑡⟩ = ⟨𝑔, ℎ⟩)
34	33	eceq1d 6516	. . . . . . . . . 10 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → [⟨𝑢, 𝑡⟩] ~R = [⟨𝑔, ℎ⟩] ~R )
35	34	eqeq2d 2169	. . . . . . . . 9 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → (𝐵 = [⟨𝑢, 𝑡⟩] ~R ↔ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))
36	35	anbi2d 460	. . . . . . . 8 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ↔ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R )))
37		simpl 108	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → 𝑢 = 𝑔)
38	37	oveq2d 5840	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → (𝑠 +P 𝑢) = (𝑠 +P 𝑔))
39		simpr 109	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → 𝑡 = ℎ)
40	39	oveq2d 5840	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → (𝑓 +P 𝑡) = (𝑓 +P ℎ))
41	38, 40	opeq12d 3749	. . . . . . . . . 10 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → ⟨(𝑠 +P 𝑢), (𝑓 +P 𝑡)⟩ = ⟨(𝑠 +P 𝑔), (𝑓 +P ℎ)⟩)
42	41	eceq1d 6516	. . . . . . . . 9 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → [⟨(𝑠 +P 𝑢), (𝑓 +P 𝑡)⟩] ~R = [⟨(𝑠 +P 𝑔), (𝑓 +P ℎ)⟩] ~R )
43	42	eqeq2d 2169	. . . . . . . 8 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → (𝑞 = [⟨(𝑠 +P 𝑢), (𝑓 +P 𝑡)⟩] ~R ↔ 𝑞 = [⟨(𝑠 +P 𝑔), (𝑓 +P ℎ)⟩] ~R ))
44	36, 43	anbi12d 465	. . . . . . 7 ⊢ ((𝑢 = 𝑔 ∧ 𝑡 = ℎ) → (((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨(𝑠 +P 𝑢), (𝑓 +P 𝑡)⟩] ~R ) ↔ ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨(𝑠 +P 𝑔), (𝑓 +P ℎ)⟩] ~R )))
45	32, 44	cbvex4v 1910	. . . . . 6 ⊢ (∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ) ↔ ∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨(𝑠 +P 𝑔), (𝑓 +P ℎ)⟩] ~R ))
46	45	anbi2i 453	. . . . 5 ⊢ ((∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R )) ↔ (∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ) ∧ ∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨(𝑠 +P 𝑔), (𝑓 +P ℎ)⟩] ~R )))
47	46	imbi1i 237	. . . 4 ⊢ (((∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R )) → 𝑧 = 𝑞) ↔ ((∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ) ∧ ∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨(𝑠 +P 𝑔), (𝑓 +P ℎ)⟩] ~R )) → 𝑧 = 𝑞))
48	47	2albii 1451	. . 3 ⊢ (∀𝑧∀𝑞((∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R )) → 𝑧 = 𝑞) ↔ ∀𝑧∀𝑞((∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ) ∧ ∃𝑠∃𝑓∃𝑔∃ℎ((𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ) ∧ 𝑞 = [⟨(𝑠 +P 𝑔), (𝑓 +P ℎ)⟩] ~R )) → 𝑧 = 𝑞))
49	20, 48	sylibr 133	. 2 ⊢ ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → ∀𝑧∀𝑞((∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R )) → 𝑧 = 𝑞))
50		eqeq1 2164	. . . . 5 ⊢ (𝑧 = 𝑞 → (𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ↔ 𝑞 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ))
51	50	anbi2d 460	. . . 4 ⊢ (𝑧 = 𝑞 → (((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ) ↔ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R )))
52	51	4exbidv 1850	. . 3 ⊢ (𝑧 = 𝑞 → (∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ) ↔ ∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R )))
53	52	mo4 2067	. 2 ⊢ (∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ) ↔ ∀𝑧∀𝑞((∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑞 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R )) → 𝑧 = 𝑞))
54	49, 53	sylibr 133	1 ⊢ ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ))

Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1333   = wceq 1335  ∃wex 1472  ∃*wmo 2007   ∈ wcel 2128  ⟨cop 3563   class class class wbr 3965   × cxp 4584  (class class class)co 5824   Er wer 6477  [cec 6478   / cqs 6479  Pcnp 7211   +_P cpp 7213   ~_R cer 7216

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-eprel 4249  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-irdg 6317  df-1o 6363  df-2o 6364  df-oadd 6367  df-omul 6368  df-er 6480  df-ec 6482  df-qs 6486  df-ni 7224  df-pli 7225  df-mi 7226  df-lti 7227  df-plpq 7264  df-mpq 7265  df-enq 7267  df-nqqs 7268  df-plqqs 7269  df-mqqs 7270  df-1nqqs 7271  df-rq 7272  df-ltnqqs 7273  df-enq0 7344  df-nq0 7345  df-0nq0 7346  df-plq0 7347  df-mq0 7348  df-inp 7386  df-iplp 7388  df-enr 7646

This theorem is referenced by:  addsrpr  7665