Theorem addcanprlemu 7416

Description: Lemma for addcanprg 7417. (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 7276	. . . . . . 7 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
2		prnminu 7290	. . . . . . 7 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑣 ∈ (2nd ‘𝐵)) → ∃𝑟 ∈ (2nd ‘𝐵)𝑟 <Q 𝑣)
3	1, 2	sylan 281	. . . . . 6 ⊢ ((𝐵 ∈ P ∧ 𝑣 ∈ (2nd ‘𝐵)) → ∃𝑟 ∈ (2nd ‘𝐵)𝑟 <Q 𝑣)
4	3	3ad2antl2 1144	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ 𝑣 ∈ (2nd ‘𝐵)) → ∃𝑟 ∈ (2nd ‘𝐵)𝑟 <Q 𝑣)
5	4	adantlr 468	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) → ∃𝑟 ∈ (2nd ‘𝐵)𝑟 <Q 𝑣)
6		simprr 521	. . . . . 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 7210	. . . . . 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 520	. . . . . . 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 7211	. . . . . . 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 7276	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
13		prarloc2 7305	. . . . . . . . . . . . . 14 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑡 ∈ Q) → ∃𝑢 ∈ (1st ‘𝐴)(𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))
14	12, 13	sylan 281	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ P ∧ 𝑡 ∈ Q) → ∃𝑢 ∈ (1st ‘𝐴)(𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))
15	14	adantrr 470	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → ∃𝑢 ∈ (1st ‘𝐴)(𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))
16	15	3ad2antl1 1143	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → ∃𝑢 ∈ (1st ‘𝐴)(𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))
17	16	adantlr 468	. . . . . . . . . 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 468	. . . . . . . . 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 468	. . . . . . . 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 468	. . . . . . 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 522	. . . . . . . . . . . . . 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 483	. . . . . . . . . . . . 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 993	. . . . . . . . . . . 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 994	. . . . . . . . . . . 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 7338	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +P 𝐵) ∈ P)
26	23, 24, 25	syl2anc 408	. . . . . . . . . . 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 7276	. . . . . . . . . . 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 520	. . . . . . . . . . . . 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 7282	. . . . . . . . . . . . 13 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑢 ∈ (1st ‘𝐴)) → 𝑢 ∈ Q)
32	29, 30, 31	syl2anc 408	. . . . . . . . . . . 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 524	. . . . . . . . . . . 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 7176	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ Q ∧ 𝑡 ∈ Q) → (𝑢 +Q 𝑡) ∈ Q)
35	32, 33, 34	syl2anc 408	. . . . . . . . . . 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 520	. . . . . . . . . . . . 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 483	. . . . . . . . . . . 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 7283	. . . . . . . . . . . 12 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑟 ∈ (2nd ‘𝐵)) → 𝑟 ∈ Q)
40	36, 38, 39	syl2anc 408	. . . . . . . . . . 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 7176	. . . . . . . . . . 11 ⊢ (((𝑢 +Q 𝑡) ∈ Q ∧ 𝑟 ∈ Q) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ Q)
42	35, 40, 41	syl2anc 408	. . . . . . . . . 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 7293	. . . . . . . . . 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 408	. . . . . . . . 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 7183	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ Q ∧ 𝑡 ∈ Q ∧ 𝑟 ∈ Q) → ((𝑢 +Q 𝑡) +Q 𝑟) = (𝑢 +Q (𝑡 +Q 𝑟)))
46	32, 33, 40, 45	syl3anc 1216	. . . . . . . . . . . . . 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 7182	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 ∈ Q ∧ 𝑟 ∈ Q) → (𝑡 +Q 𝑟) = (𝑟 +Q 𝑡))
48	47	oveq2d 5783	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ Q ∧ 𝑟 ∈ Q) → (𝑢 +Q (𝑡 +Q 𝑟)) = (𝑢 +Q (𝑟 +Q 𝑡)))
49	33, 40, 48	syl2anc 408	. . . . . . . . . . . . . 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 2170	. . . . . . . . . . . . 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 274	. . . . . . . . . . . 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 524	. . . . . . . . . . . . 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 109	. . . . . . . . . . . . 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 274	. . . . . . . . . . . . . 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 995	. . . . . . . . . . . . . . 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 274	. . . . . . . . . . . . . 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 7269	. . . . . . . . . . . . . . 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 7176	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +Q ℎ) ∈ Q)
59	57, 58	genpprecll 7315	. . . . . . . . . . . . . 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 408	. . . . . . . . . . . . 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 429	. . . . . . . . . . . 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 2214	. . . . . . . . . . 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 5414	. . . . . . . . . . . . 13 ⊢ ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → (1st ‘(𝐴 +P 𝐵)) = (1st ‘(𝐴 +P 𝐶)))
64	63	eleq2d 2207	. . . . . . . . . . . 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 492	. . . . . . . . . . 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 166	. . . . . . . . . 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 7316	. . . . . . . . . . . . . . . . . . 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 267	. . . . . . . . . . . . . . . . . 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 1001	. . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . 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 123	. . . . . . . . . . . . . . 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 476	. . . . . . . . . . . . . 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 397	. . . . . . . . . . . . 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 502	. . . . . . . . . . . 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 468	. . . . . . . . . . 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 274	. . . . . . . . . 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 304	. . . . . . . . 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 654	. . . . . . . 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 7276	. . . . . . . . . . 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 7208	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ Q ∧ 𝑡 ∈ Q) → 𝑡 <Q (𝑡 +Q 𝑡))
82	33, 33, 81	syl2anc 408	. . . . . . . . . . . . 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 525	. . . . . . . . . . . . 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 3949	. . . . . . . . . . . 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 7203	. . . . . . . . . . . 12 ⊢ ((𝑡 <Q 𝑤 ∧ 𝑟 ∈ Q) → (𝑟 +Q 𝑡) <Q (𝑟 +Q 𝑤))
86	84, 40, 85	syl2anc 408	. . . . . . . . . . 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 521	. . . . . . . . . . . 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 479	. . . . . . . . . . 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 3949	. . . . . . . . . 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 7292	. . . . . . . . . 10 ⊢ ((⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ∈ P ∧ (𝑟 +Q 𝑡) <Q 𝑣) → ((𝑟 +Q 𝑡) ∈ (1st ‘𝐶) ∨ 𝑣 ∈ (2nd ‘𝐶)))
91	80, 89, 90	syl2anc 408	. . . . . . . . 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 718	. . . . . . . 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 1327	. . . . . . 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 2552	. . . . . 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 2552	. . . . 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 2552	. . . 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 2552	. . 3 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd ‘𝐵)) → 𝑣 ∈ (2nd ‘𝐶))
98	97	ex 114	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (𝑣 ∈ (2nd ‘𝐵) → 𝑣 ∈ (2nd ‘𝐶)))
99	98	ssrdv 3098	1 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (2nd ‘𝐵) ⊆ (2nd ‘𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 697   ∧ w3a 962   = wceq 1331   ∈ wcel 1480  ∃wrex 2415   ⊆ wss 3066  ⟨cop 3525   class class class wbr 3924  ‘cfv 5118  (class class class)co 5767  1^st c1st 6029  2^nd c2nd 6030  Qcnq 7081   +_Q cplq 7083   <_Q cltq 7086  Pcnp 7092   +_P cpp 7094

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-2o 6307  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-pli 7106  df-mi 7107  df-lti 7108  df-plpq 7145  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-plqqs 7150  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154  df-enq0 7225  df-nq0 7226  df-0nq0 7227  df-plq0 7228  df-mq0 7229  df-inp 7267  df-iplp 7269

This theorem is referenced by:  addcanprg  7417