Theorem addcanprlemu 7810

Description: Lemma for addcanprg 7811. (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 7670	. . . . . . 7 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
2		prnminu 7684	. . . . . . 7 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑣 ∈ (2nd ‘𝐵)) → ∃𝑟 ∈ (2nd ‘𝐵)𝑟 <Q 𝑣)
3	1, 2	sylan 283	. . . . . 6 ⊢ ((𝐵 ∈ P ∧ 𝑣 ∈ (2nd ‘𝐵)) → ∃𝑟 ∈ (2nd ‘𝐵)𝑟 <Q 𝑣)
4	3	3ad2antl2 1184	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ 𝑣 ∈ (2nd ‘𝐵)) → ∃𝑟 ∈ (2nd ‘𝐵)𝑟 <Q 𝑣)
5	4	adantlr 477	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) → ∃𝑟 ∈ (2nd ‘𝐵)𝑟 <Q 𝑣)
6		simprr 531	. . . . . 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 7604	. . . . . 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 529	. . . . . . 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 7605	. . . . . . 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 7670	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
13		prarloc2 7699	. . . . . . . . . . . . . 14 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑡 ∈ Q) → ∃𝑢 ∈ (1st ‘𝐴)(𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))
14	12, 13	sylan 283	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ P ∧ 𝑡 ∈ Q) → ∃𝑢 ∈ (1st ‘𝐴)(𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))
15	14	adantrr 479	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → ∃𝑢 ∈ (1st ‘𝐴)(𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))
16	15	3ad2antl1 1183	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → ∃𝑢 ∈ (1st ‘𝐴)(𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))
17	16	adantlr 477	. . . . . . . . . 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 477	. . . . . . . . 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 477	. . . . . . . 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 477	. . . . . . 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 533	. . . . . . . . . . . . . 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 492	. . . . . . . . . . . . 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 1033	. . . . . . . . . . . 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 1034	. . . . . . . . . . . 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 7732	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +P 𝐵) ∈ P)
26	23, 24, 25	syl2anc 411	. . . . . . . . . . 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 7670	. . . . . . . . . . 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 529	. . . . . . . . . . . . 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 7676	. . . . . . . . . . . . 13 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑢 ∈ (1st ‘𝐴)) → 𝑢 ∈ Q)
32	29, 30, 31	syl2anc 411	. . . . . . . . . . . 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 535	. . . . . . . . . . . 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 7570	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ Q ∧ 𝑡 ∈ Q) → (𝑢 +Q 𝑡) ∈ Q)
35	32, 33, 34	syl2anc 411	. . . . . . . . . . 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 529	. . . . . . . . . . . . 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 492	. . . . . . . . . . . 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 7677	. . . . . . . . . . . 12 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑟 ∈ (2nd ‘𝐵)) → 𝑟 ∈ Q)
40	36, 38, 39	syl2anc 411	. . . . . . . . . . 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 7570	. . . . . . . . . . 11 ⊢ (((𝑢 +Q 𝑡) ∈ Q ∧ 𝑟 ∈ Q) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ Q)
42	35, 40, 41	syl2anc 411	. . . . . . . . . 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 7687	. . . . . . . . . 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 411	. . . . . . . . 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 7577	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ Q ∧ 𝑡 ∈ Q ∧ 𝑟 ∈ Q) → ((𝑢 +Q 𝑡) +Q 𝑟) = (𝑢 +Q (𝑡 +Q 𝑟)))
46	32, 33, 40, 45	syl3anc 1271	. . . . . . . . . . . . . 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 7576	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 ∈ Q ∧ 𝑟 ∈ Q) → (𝑡 +Q 𝑟) = (𝑟 +Q 𝑡))
48	47	oveq2d 6023	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ Q ∧ 𝑟 ∈ Q) → (𝑢 +Q (𝑡 +Q 𝑟)) = (𝑢 +Q (𝑟 +Q 𝑡)))
49	33, 40, 48	syl2anc 411	. . . . . . . . . . . . . 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 2262	. . . . . . . . . . . . 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 276	. . . . . . . . . . . 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 535	. . . . . . . . . . . . 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 110	. . . . . . . . . . . . 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 276	. . . . . . . . . . . . . 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 1035	. . . . . . . . . . . . . . 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 276	. . . . . . . . . . . . . 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 7663	. . . . . . . . . . . . . . 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 7570	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +Q ℎ) ∈ Q)
59	57, 58	genpprecll 7709	. . . . . . . . . . . . . 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 411	. . . . . . . . . . . . 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 433	. . . . . . . . . . . 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 2306	. . . . . . . . . . 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 5629	. . . . . . . . . . . . 13 ⊢ ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → (1st ‘(𝐴 +P 𝐵)) = (1st ‘(𝐴 +P 𝐶)))
64	63	eleq2d 2299	. . . . . . . . . . . 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 501	. . . . . . . . . . 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 167	. . . . . . . . . 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 7710	. . . . . . . . . . . . . . . . . . 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 269	. . . . . . . . . . . . . . . . . 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 1041	. . . . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . 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 124	. . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . 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 400	. . . . . . . . . . . . 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 511	. . . . . . . . . . . 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 477	. . . . . . . . . . 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 276	. . . . . . . . . 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 306	. . . . . . . . 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 669	. . . . . . . 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 7670	. . . . . . . . . . 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 7602	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ Q ∧ 𝑡 ∈ Q) → 𝑡 <Q (𝑡 +Q 𝑡))
82	33, 33, 81	syl2anc 411	. . . . . . . . . . . . 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 536	. . . . . . . . . . . . 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 4109	. . . . . . . . . . . 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 7597	. . . . . . . . . . . 12 ⊢ ((𝑡 <Q 𝑤 ∧ 𝑟 ∈ Q) → (𝑟 +Q 𝑡) <Q (𝑟 +Q 𝑤))
86	84, 40, 85	syl2anc 411	. . . . . . . . . . 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 531	. . . . . . . . . . . 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 488	. . . . . . . . . . 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 4109	. . . . . . . . . 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 7686	. . . . . . . . . 10 ⊢ ((⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ∈ P ∧ (𝑟 +Q 𝑡) <Q 𝑣) → ((𝑟 +Q 𝑡) ∈ (1st ‘𝐶) ∨ 𝑣 ∈ (2nd ‘𝐶)))
91	80, 89, 90	syl2anc 411	. . . . . . . . 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 734	. . . . . . . 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 1383	. . . . . . 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 2653	. . . . . 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 2653	. . . . 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 2653	. . . 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 2653	. . 3 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) → 𝑣 ∈ (2nd ‘𝐶))
98	97	ex 115	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (𝑣 ∈ (2nd ‘𝐵) → 𝑣 ∈ (2nd ‘𝐶)))
99	98	ssrdv 3230	1 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (2nd ‘𝐵) ⊆ (2nd ‘𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 713   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  ∃wrex 2509   ⊆ wss 3197  ⟨cop 3669   class class class wbr 4083  ‘cfv 5318  (class class class)co 6007  1^st c1st 6290  2^nd c2nd 6291  Qcnq 7475   +_Q cplq 7477   <_Q cltq 7480  Pcnp 7486   +_P cpp 7488

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4380  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-1o 6568  df-2o 6569  df-oadd 6572  df-omul 6573  df-er 6688  df-ec 6690  df-qs 6694  df-ni 7499  df-pli 7500  df-mi 7501  df-lti 7502  df-plpq 7539  df-mpq 7540  df-enq 7542  df-nqqs 7543  df-plqqs 7544  df-mqqs 7545  df-1nqqs 7546  df-rq 7547  df-ltnqqs 7548  df-enq0 7619  df-nq0 7620  df-0nq0 7621  df-plq0 7622  df-mq0 7623  df-inp 7661  df-iplp 7663

This theorem is referenced by:  addcanprg  7811