Theorem addcanprlemu 6805

Description: Lemma for addcanprg 6806. (Contributed by Jim Kingdon, 25-Dec-2019.)

Assertion

Ref	Expression
addcanprlemu	⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (2nd ‘𝐵) ⊆ (2nd ‘𝐶))

Proof of Theorem addcanprlemu

Dummy variables 𝑓 𝑔 ℎ 𝑞 𝑟 𝑠 𝑡 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prop 6665	. . . . . . 7 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
2		prnminu 6679	. . . . . . 7 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑣 ∈ (2nd ‘𝐵)) → ∃𝑟 ∈ (2nd ‘𝐵)𝑟 <Q 𝑣)
3	1, 2	sylan 277	. . . . . 6 ⊢ ((𝐵 ∈ P ∧ 𝑣 ∈ (2nd ‘𝐵)) → ∃𝑟 ∈ (2nd ‘𝐵)𝑟 <Q 𝑣)
4	3	3ad2antl2 1101	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ 𝑣 ∈ (2nd ‘𝐵)) → ∃𝑟 ∈ (2nd ‘𝐵)𝑟 <Q 𝑣)
5	4	adantlr 460	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) → ∃𝑟 ∈ (2nd ‘𝐵)𝑟 <Q 𝑣)
6		simprr 498	. . . . . 6 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) → 𝑟 <Q 𝑣)
7		ltexnqi 6599	. . . . . 6 ⊢ (𝑟 <Q 𝑣 → ∃𝑤 ∈ Q (𝑟 +Q 𝑤) = 𝑣)
8	6, 7	syl 14	. . . . 5 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) → ∃𝑤 ∈ Q (𝑟 +Q 𝑤) = 𝑣)
9		simprl 497	. . . . . . 7 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) → 𝑤 ∈ Q)
10		halfnqq 6600	. . . . . . 7 ⊢ (𝑤 ∈ Q → ∃𝑡 ∈ Q (𝑡 +Q 𝑡) = 𝑤)
11	9, 10	syl 14	. . . . . 6 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) → ∃𝑡 ∈ Q (𝑡 +Q 𝑡) = 𝑤)
12		prop 6665	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
13		prarloc2 6694	. . . . . . . . . . . . . 14 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑡 ∈ Q) → ∃𝑢 ∈ (1st ‘𝐴)(𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))
14	12, 13	sylan 277	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ P ∧ 𝑡 ∈ Q) → ∃𝑢 ∈ (1st ‘𝐴)(𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))
15	14	adantrr 462	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → ∃𝑢 ∈ (1st ‘𝐴)(𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))
16	15	3ad2antl1 1100	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → ∃𝑢 ∈ (1st ‘𝐴)(𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))
17	16	adantlr 460	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → ∃𝑢 ∈ (1st ‘𝐴)(𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))
18	17	adantlr 460	. . . . . . . . 9 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → ∃𝑢 ∈ (1st ‘𝐴)(𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))
19	18	adantlr 460	. . . . . . . 8 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → ∃𝑢 ∈ (1st ‘𝐴)(𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))
20	19	adantlr 460	. . . . . . 7 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → ∃𝑢 ∈ (1st ‘𝐴)(𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))
21		simplll 499	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) → (𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P))
22	21	ad3antrrr 475	. . . . . . . . . . . . 13 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → (𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P))
23	22	simp1d 950	. . . . . . . . . . . 12 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → 𝐴 ∈ P)
24	22	simp2d 951	. . . . . . . . . . . 12 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → 𝐵 ∈ P)
25		addclpr 6727	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +P 𝐵) ∈ P)
26	23, 24, 25	syl2anc 403	. . . . . . . . . . 11 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → (𝐴 +P 𝐵) ∈ P)
27		prop 6665	. . . . . . . . . . 11 ⊢ ((𝐴 +P 𝐵) ∈ P → ⟨(1st ‘(𝐴 +P 𝐵)), (2nd ‘(𝐴 +P 𝐵))⟩ ∈ P)
28	26, 27	syl 14	. . . . . . . . . 10 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → ⟨(1st ‘(𝐴 +P 𝐵)), (2nd ‘(𝐴 +P 𝐵))⟩ ∈ P)
29	23, 12	syl 14	. . . . . . . . . . . . 13 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
30		simprl 497	. . . . . . . . . . . . 13 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → 𝑢 ∈ (1st ‘𝐴))
31		elprnql 6671	. . . . . . . . . . . . 13 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑢 ∈ (1st ‘𝐴)) → 𝑢 ∈ Q)
32	29, 30, 31	syl2anc 403	. . . . . . . . . . . 12 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → 𝑢 ∈ Q)
33		simplrl 501	. . . . . . . . . . . 12 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → 𝑡 ∈ Q)
34		addclnq 6565	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ Q ∧ 𝑡 ∈ Q) → (𝑢 +Q 𝑡) ∈ Q)
35	32, 33, 34	syl2anc 403	. . . . . . . . . . 11 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → (𝑢 +Q 𝑡) ∈ Q)
36	24, 1	syl 14	. . . . . . . . . . . 12 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
37		simprl 497	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) → 𝑟 ∈ (2nd ‘𝐵))
38	37	ad3antrrr 475	. . . . . . . . . . . 12 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → 𝑟 ∈ (2nd ‘𝐵))
39		elprnqu 6672	. . . . . . . . . . . 12 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑟 ∈ (2nd ‘𝐵)) → 𝑟 ∈ Q)
40	36, 38, 39	syl2anc 403	. . . . . . . . . . 11 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → 𝑟 ∈ Q)
41		addclnq 6565	. . . . . . . . . . 11 ⊢ (((𝑢 +Q 𝑡) ∈ Q ∧ 𝑟 ∈ Q) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ Q)
42	35, 40, 41	syl2anc 403	. . . . . . . . . 10 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ Q)
43		prdisj 6682	. . . . . . . . . 10 ⊢ ((⟨(1st ‘(𝐴 +P 𝐵)), (2nd ‘(𝐴 +P 𝐵))⟩ ∈ P ∧ ((𝑢 +Q 𝑡) +Q 𝑟) ∈ Q) → ¬ (((𝑢 +Q 𝑡) +Q 𝑟) ∈ (1st ‘(𝐴 +P 𝐵)) ∧ ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (2nd ‘(𝐴 +P 𝐵))))
44	28, 42, 43	syl2anc 403	. . . . . . . . 9 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → ¬ (((𝑢 +Q 𝑡) +Q 𝑟) ∈ (1st ‘(𝐴 +P 𝐵)) ∧ ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (2nd ‘(𝐴 +P 𝐵))))
45		addassnqg 6572	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ Q ∧ 𝑡 ∈ Q ∧ 𝑟 ∈ Q) → ((𝑢 +Q 𝑡) +Q 𝑟) = (𝑢 +Q (𝑡 +Q 𝑟)))
46	32, 33, 40, 45	syl3anc 1169	. . . . . . . . . . . . . 14 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → ((𝑢 +Q 𝑡) +Q 𝑟) = (𝑢 +Q (𝑡 +Q 𝑟)))
47		addcomnqg 6571	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 ∈ Q ∧ 𝑟 ∈ Q) → (𝑡 +Q 𝑟) = (𝑟 +Q 𝑡))
48	47	oveq2d 5548	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ Q ∧ 𝑟 ∈ Q) → (𝑢 +Q (𝑡 +Q 𝑟)) = (𝑢 +Q (𝑟 +Q 𝑡)))
49	33, 40, 48	syl2anc 403	. . . . . . . . . . . . . 14 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → (𝑢 +Q (𝑡 +Q 𝑟)) = (𝑢 +Q (𝑟 +Q 𝑡)))
50	46, 49	eqtrd 2113	. . . . . . . . . . . . 13 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → ((𝑢 +Q 𝑡) +Q 𝑟) = (𝑢 +Q (𝑟 +Q 𝑡)))
51	50	adantr 270	. . . . . . . . . . . 12 ⊢ (((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑟 +Q 𝑡) ∈ (1st ‘𝐶)) → ((𝑢 +Q 𝑡) +Q 𝑟) = (𝑢 +Q (𝑟 +Q 𝑡)))
52		simplrl 501	. . . . . . . . . . . . 13 ⊢ (((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑟 +Q 𝑡) ∈ (1st ‘𝐶)) → 𝑢 ∈ (1st ‘𝐴))
53		simpr 108	. . . . . . . . . . . . 13 ⊢ (((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑟 +Q 𝑡) ∈ (1st ‘𝐶)) → (𝑟 +Q 𝑡) ∈ (1st ‘𝐶))
54	23	adantr 270	. . . . . . . . . . . . . 14 ⊢ (((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑟 +Q 𝑡) ∈ (1st ‘𝐶)) → 𝐴 ∈ P)
55	22	simp3d 952	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → 𝐶 ∈ P)
56	55	adantr 270	. . . . . . . . . . . . . 14 ⊢ (((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑟 +Q 𝑡) ∈ (1st ‘𝐶)) → 𝐶 ∈ P)
57		df-iplp 6658	. . . . . . . . . . . . . . 15 ⊢ +P = (𝑞 ∈ P, 𝑠 ∈ P ↦ ⟨{𝑓 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (1st ‘𝑞) ∧ ℎ ∈ (1st ‘𝑠) ∧ 𝑓 = (𝑔 +Q ℎ))}, {𝑓 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (2nd ‘𝑞) ∧ ℎ ∈ (2nd ‘𝑠) ∧ 𝑓 = (𝑔 +Q ℎ))}⟩)
58		addclnq 6565	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +Q ℎ) ∈ Q)
59	57, 58	genpprecll 6704	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → ((𝑢 ∈ (1st ‘𝐴) ∧ (𝑟 +Q 𝑡) ∈ (1st ‘𝐶)) → (𝑢 +Q (𝑟 +Q 𝑡)) ∈ (1st ‘(𝐴 +P 𝐶))))
60	54, 56, 59	syl2anc 403	. . . . . . . . . . . . 13 ⊢ (((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑟 +Q 𝑡) ∈ (1st ‘𝐶)) → ((𝑢 ∈ (1st ‘𝐴) ∧ (𝑟 +Q 𝑡) ∈ (1st ‘𝐶)) → (𝑢 +Q (𝑟 +Q 𝑡)) ∈ (1st ‘(𝐴 +P 𝐶))))
61	52, 53, 60	mp2and 423	. . . . . . . . . . . 12 ⊢ (((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑟 +Q 𝑡) ∈ (1st ‘𝐶)) → (𝑢 +Q (𝑟 +Q 𝑡)) ∈ (1st ‘(𝐴 +P 𝐶)))
62	51, 61	eqeltrd 2155	. . . . . . . . . . 11 ⊢ (((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑟 +Q 𝑡) ∈ (1st ‘𝐶)) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (1st ‘(𝐴 +P 𝐶)))
63		fveq2 5198	. . . . . . . . . . . . 13 ⊢ ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → (1st ‘(𝐴 +P 𝐵)) = (1st ‘(𝐴 +P 𝐶)))
64	63	eleq2d 2148	. . . . . . . . . . . 12 ⊢ ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → (((𝑢 +Q 𝑡) +Q 𝑟) ∈ (1st ‘(𝐴 +P 𝐵)) ↔ ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (1st ‘(𝐴 +P 𝐶))))
65	64	ad7antlr 484	. . . . . . . . . . 11 ⊢ (((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑟 +Q 𝑡) ∈ (1st ‘𝐶)) → (((𝑢 +Q 𝑡) +Q 𝑟) ∈ (1st ‘(𝐴 +P 𝐵)) ↔ ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (1st ‘(𝐴 +P 𝐶))))
66	62, 65	mpbird 165	. . . . . . . . . 10 ⊢ (((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑟 +Q 𝑡) ∈ (1st ‘𝐶)) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (1st ‘(𝐴 +P 𝐵)))
67	57, 58	genppreclu 6705	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (((𝑢 +Q 𝑡) ∈ (2nd ‘𝐴) ∧ 𝑟 ∈ (2nd ‘𝐵)) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (2nd ‘(𝐴 +P 𝐵))))
68	67	ancomsd 265	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑟 ∈ (2nd ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴)) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (2nd ‘(𝐴 +P 𝐵))))
69	68	3adant3 958	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝑟 ∈ (2nd ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴)) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (2nd ‘(𝐴 +P 𝐵))))
70	69	ad2antrr 471	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) → ((𝑟 ∈ (2nd ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴)) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (2nd ‘(𝐴 +P 𝐵))))
71	70	imp 122	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (2nd ‘(𝐴 +P 𝐵)))
72	71	adantrlr 468	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ ((𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (2nd ‘(𝐴 +P 𝐵)))
73	72	anassrs 392	. . . . . . . . . . . . 13 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴)) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (2nd ‘(𝐴 +P 𝐵)))
74	73	ad2ant2rl 494	. . . . . . . . . . . 12 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (2nd ‘(𝐴 +P 𝐵)))
75	74	adantlr 460	. . . . . . . . . . 11 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (2nd ‘(𝐴 +P 𝐵)))
76	75	adantr 270	. . . . . . . . . 10 ⊢ (((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑟 +Q 𝑡) ∈ (1st ‘𝐶)) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (2nd ‘(𝐴 +P 𝐵)))
77	66, 76	jca 300	. . . . . . . . 9 ⊢ (((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑟 +Q 𝑡) ∈ (1st ‘𝐶)) → (((𝑢 +Q 𝑡) +Q 𝑟) ∈ (1st ‘(𝐴 +P 𝐵)) ∧ ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (2nd ‘(𝐴 +P 𝐵))))
78	44, 77	mtand 623	. . . . . . . 8 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → ¬ (𝑟 +Q 𝑡) ∈ (1st ‘𝐶))
79		prop 6665	. . . . . . . . . . 11 ⊢ (𝐶 ∈ P → ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ∈ P)
80	55, 79	syl 14	. . . . . . . . . 10 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ∈ P)
81		ltaddnq 6597	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ Q ∧ 𝑡 ∈ Q) → 𝑡 <Q (𝑡 +Q 𝑡))
82	33, 33, 81	syl2anc 403	. . . . . . . . . . . . 13 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → 𝑡 <Q (𝑡 +Q 𝑡))
83		simplrr 502	. . . . . . . . . . . . 13 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → (𝑡 +Q 𝑡) = 𝑤)
84	82, 83	breqtrd 3809	. . . . . . . . . . . 12 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → 𝑡 <Q 𝑤)
85		ltanqi 6592	. . . . . . . . . . . 12 ⊢ ((𝑡 <Q 𝑤 ∧ 𝑟 ∈ Q) → (𝑟 +Q 𝑡) <Q (𝑟 +Q 𝑤))
86	84, 40, 85	syl2anc 403	. . . . . . . . . . 11 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → (𝑟 +Q 𝑡) <Q (𝑟 +Q 𝑤))
87		simprr 498	. . . . . . . . . . . 12 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) → (𝑟 +Q 𝑤) = 𝑣)
88	87	ad2antrr 471	. . . . . . . . . . 11 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → (𝑟 +Q 𝑤) = 𝑣)
89	86, 88	breqtrd 3809	. . . . . . . . . 10 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → (𝑟 +Q 𝑡) <Q 𝑣)
90		prloc 6681	. . . . . . . . . 10 ⊢ ((⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ∈ P ∧ (𝑟 +Q 𝑡) <Q 𝑣) → ((𝑟 +Q 𝑡) ∈ (1st ‘𝐶) ∨ 𝑣 ∈ (2nd ‘𝐶)))
91	80, 89, 90	syl2anc 403	. . . . . . . . 9 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → ((𝑟 +Q 𝑡) ∈ (1st ‘𝐶) ∨ 𝑣 ∈ (2nd ‘𝐶)))
92	91	orcomd 680	. . . . . . . 8 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → (𝑣 ∈ (2nd ‘𝐶) ∨ (𝑟 +Q 𝑡) ∈ (1st ‘𝐶)))
93	78, 92	ecased 1280	. . . . . . 7 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → 𝑣 ∈ (2nd ‘𝐶))
94	20, 93	rexlimddv 2481	. . . . . 6 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → 𝑣 ∈ (2nd ‘𝐶))
95	11, 94	rexlimddv 2481	. . . . 5 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤 ∈ Q ∧ (𝑟 +Q 𝑤) = 𝑣)) → 𝑣 ∈ (2nd ‘𝐶))
96	8, 95	rexlimddv 2481	. . . 4 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) ∧ (𝑟 ∈ (2nd ‘𝐵) ∧ 𝑟 <Q 𝑣)) → 𝑣 ∈ (2nd ‘𝐶))
97	5, 96	rexlimddv 2481	. . 3 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) → 𝑣 ∈ (2nd ‘𝐶))
98	97	ex 113	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (𝑣 ∈ (2nd ‘𝐵) → 𝑣 ∈ (2nd ‘𝐶)))
99	98	ssrdv 3005	1 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (2nd ‘𝐵) ⊆ (2nd ‘𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103   ∨ wo 661   ∧ w3a 919   = wceq 1284   ∈ wcel 1433  ∃wrex 2349   ⊆ wss 2973  ⟨cop 3401   class class class wbr 3785  ‘cfv 4922  (class class class)co 5532  1^st c1st 5785  2^nd c2nd 5786  Qcnq 6470   +_Q cplq 6472   <_Q cltq 6475  Pcnp 6481   +_P cpp 6483

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-eprel 4044  df-id 4048  df-po 4051  df-iso 4052  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-1o 6024  df-2o 6025  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-pli 6495  df-mi 6496  df-lti 6497  df-plpq 6534  df-mpq 6535  df-enq 6537  df-nqqs 6538  df-plqqs 6539  df-mqqs 6540  df-1nqqs 6541  df-rq 6542  df-ltnqqs 6543  df-enq0 6614  df-nq0 6615  df-0nq0 6616  df-plq0 6617  df-mq0 6618  df-inp 6656  df-iplp 6658

This theorem is referenced by:  addcanprg  6806