Theorem addcanprleml 6804

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

Assertion

Ref	Expression
addcanprleml	⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (1st ‘𝐵) ⊆ (1st ‘𝐶))

Proof of Theorem addcanprleml

Dummy variables 𝑓 𝑔 ℎ 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		prnmaddl 6680	. . . . . . 7 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑣 ∈ (1st ‘𝐵)) → ∃𝑤 ∈ Q (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))
3	1, 2	sylan 277	. . . . . 6 ⊢ ((𝐵 ∈ P ∧ 𝑣 ∈ (1st ‘𝐵)) → ∃𝑤 ∈ Q (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))
4	3	3ad2antl2 1101	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ 𝑣 ∈ (1st ‘𝐵)) → ∃𝑤 ∈ Q (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))
5	4	adantlr 460	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) → ∃𝑤 ∈ Q (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))
6		simprl 497	. . . . . 6 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) → 𝑤 ∈ Q)
7		halfnqq 6600	. . . . . 6 ⊢ (𝑤 ∈ Q → ∃𝑡 ∈ Q (𝑡 +Q 𝑡) = 𝑤)
8	6, 7	syl 14	. . . . 5 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) → ∃𝑡 ∈ Q (𝑡 +Q 𝑡) = 𝑤)
9		simplll 499	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) → (𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P))
10	9	adantr 270	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → (𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P))
11	10	simp1d 950	. . . . . . . 8 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → 𝐴 ∈ P)
12		prop 6665	. . . . . . . 8 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
13	11, 12	syl 14	. . . . . . 7 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
14		simprl 497	. . . . . . 7 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → 𝑡 ∈ Q)
15		prarloc2 6694	. . . . . . 7 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑡 ∈ Q) → ∃𝑢 ∈ (1st ‘𝐴)(𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))
16	13, 14, 15	syl2anc 403	. . . . . 6 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → ∃𝑢 ∈ (1st ‘𝐴)(𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))
17	9	ad2antrr 471	. . . . . . . . . . . 12 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → (𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P))
18	17	simp1d 950	. . . . . . . . . . 11 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → 𝐴 ∈ P)
19	17	simp2d 951	. . . . . . . . . . 11 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → 𝐵 ∈ P)
20		addclpr 6727	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +P 𝐵) ∈ P)
21	18, 19, 20	syl2anc 403	. . . . . . . . . 10 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → (𝐴 +P 𝐵) ∈ P)
22		prop 6665	. . . . . . . . . 10 ⊢ ((𝐴 +P 𝐵) ∈ P → ⟨(1st ‘(𝐴 +P 𝐵)), (2nd ‘(𝐴 +P 𝐵))⟩ ∈ P)
23	21, 22	syl 14	. . . . . . . . 9 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → ⟨(1st ‘(𝐴 +P 𝐵)), (2nd ‘(𝐴 +P 𝐵))⟩ ∈ P)
24	18, 12	syl 14	. . . . . . . . . . 11 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
25		simprl 497	. . . . . . . . . . 11 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → 𝑢 ∈ (1st ‘𝐴))
26		elprnql 6671	. . . . . . . . . . 11 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑢 ∈ (1st ‘𝐴)) → 𝑢 ∈ Q)
27	24, 25, 26	syl2anc 403	. . . . . . . . . 10 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → 𝑢 ∈ Q)
28	19, 1	syl 14	. . . . . . . . . . . 12 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
29		simplr 496	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) → 𝑣 ∈ (1st ‘𝐵))
30	29	ad2antrr 471	. . . . . . . . . . . 12 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → 𝑣 ∈ (1st ‘𝐵))
31		elprnql 6671	. . . . . . . . . . . 12 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑣 ∈ (1st ‘𝐵)) → 𝑣 ∈ Q)
32	28, 30, 31	syl2anc 403	. . . . . . . . . . 11 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → 𝑣 ∈ Q)
33		simplrl 501	. . . . . . . . . . . 12 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → 𝑤 ∈ Q)
34	33	adantr 270	. . . . . . . . . . 11 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → 𝑤 ∈ Q)
35		addclnq 6565	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ Q ∧ 𝑤 ∈ Q) → (𝑣 +Q 𝑤) ∈ Q)
36	32, 34, 35	syl2anc 403	. . . . . . . . . 10 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → (𝑣 +Q 𝑤) ∈ Q)
37		addclnq 6565	. . . . . . . . . 10 ⊢ ((𝑢 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ Q) → (𝑢 +Q (𝑣 +Q 𝑤)) ∈ Q)
38	27, 36, 37	syl2anc 403	. . . . . . . . 9 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → (𝑢 +Q (𝑣 +Q 𝑤)) ∈ Q)
39		prdisj 6682	. . . . . . . . 9 ⊢ ((⟨(1st ‘(𝐴 +P 𝐵)), (2nd ‘(𝐴 +P 𝐵))⟩ ∈ P ∧ (𝑢 +Q (𝑣 +Q 𝑤)) ∈ Q) → ¬ ((𝑢 +Q (𝑣 +Q 𝑤)) ∈ (1st ‘(𝐴 +P 𝐵)) ∧ (𝑢 +Q (𝑣 +Q 𝑤)) ∈ (2nd ‘(𝐴 +P 𝐵))))
40	23, 38, 39	syl2anc 403	. . . . . . . 8 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → ¬ ((𝑢 +Q (𝑣 +Q 𝑤)) ∈ (1st ‘(𝐴 +P 𝐵)) ∧ (𝑢 +Q (𝑣 +Q 𝑤)) ∈ (2nd ‘(𝐴 +P 𝐵))))
41	18	adantr 270	. . . . . . . . . 10 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd ‘𝐶)) → 𝐴 ∈ P)
42	19	adantr 270	. . . . . . . . . 10 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd ‘𝐶)) → 𝐵 ∈ P)
43		simplrl 501	. . . . . . . . . 10 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd ‘𝐶)) → 𝑢 ∈ (1st ‘𝐴))
44		simplrr 502	. . . . . . . . . . 11 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))
45	44	ad2antrr 471	. . . . . . . . . 10 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd ‘𝐶)) → (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))
46		df-iplp 6658	. . . . . . . . . . . 12 ⊢ +P = (𝑟 ∈ P, 𝑠 ∈ P ↦ ⟨{𝑓 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (1st ‘𝑟) ∧ ℎ ∈ (1st ‘𝑠) ∧ 𝑓 = (𝑔 +Q ℎ))}, {𝑓 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (2nd ‘𝑟) ∧ ℎ ∈ (2nd ‘𝑠) ∧ 𝑓 = (𝑔 +Q ℎ))}⟩)
47		addclnq 6565	. . . . . . . . . . . 12 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +Q ℎ) ∈ Q)
48	46, 47	genpprecll 6704	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑢 ∈ (1st ‘𝐴) ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵)) → (𝑢 +Q (𝑣 +Q 𝑤)) ∈ (1st ‘(𝐴 +P 𝐵))))
49	48	imp 122	. . . . . . . . . 10 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) → (𝑢 +Q (𝑣 +Q 𝑤)) ∈ (1st ‘(𝐴 +P 𝐵)))
50	41, 42, 43, 45, 49	syl22anc 1170	. . . . . . . . 9 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd ‘𝐶)) → (𝑢 +Q (𝑣 +Q 𝑤)) ∈ (1st ‘(𝐴 +P 𝐵)))
51	27	adantr 270	. . . . . . . . . . . . 13 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd ‘𝐶)) → 𝑢 ∈ Q)
52	14	ad2antrr 471	. . . . . . . . . . . . 13 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd ‘𝐶)) → 𝑡 ∈ Q)
53	32	adantr 270	. . . . . . . . . . . . 13 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd ‘𝐶)) → 𝑣 ∈ Q)
54		addcomnqg 6571	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
55	54	adantl 271	. . . . . . . . . . . . 13 ⊢ (((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd ‘𝐶)) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
56		addassnqg 6572	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑓 +Q 𝑔) +Q ℎ) = (𝑓 +Q (𝑔 +Q ℎ)))
57	56	adantl 271	. . . . . . . . . . . . 13 ⊢ (((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd ‘𝐶)) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q)) → ((𝑓 +Q 𝑔) +Q ℎ) = (𝑓 +Q (𝑔 +Q ℎ)))
58		addclnq 6565	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +Q 𝑔) ∈ Q)
59	58	adantl 271	. . . . . . . . . . . . 13 ⊢ (((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd ‘𝐶)) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q)) → (𝑓 +Q 𝑔) ∈ Q)
60	51, 52, 53, 55, 57, 52, 59	caov4d 5705	. . . . . . . . . . . 12 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd ‘𝐶)) → ((𝑢 +Q 𝑡) +Q (𝑣 +Q 𝑡)) = ((𝑢 +Q 𝑣) +Q (𝑡 +Q 𝑡)))
61		simprr 498	. . . . . . . . . . . . . 14 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → (𝑡 +Q 𝑡) = 𝑤)
62	61	ad2antrr 471	. . . . . . . . . . . . 13 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd ‘𝐶)) → (𝑡 +Q 𝑡) = 𝑤)
63	62	oveq2d 5548	. . . . . . . . . . . 12 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd ‘𝐶)) → ((𝑢 +Q 𝑣) +Q (𝑡 +Q 𝑡)) = ((𝑢 +Q 𝑣) +Q 𝑤))
64	33	ad2antrr 471	. . . . . . . . . . . . 13 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd ‘𝐶)) → 𝑤 ∈ Q)
65		addassnqg 6572	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ Q ∧ 𝑣 ∈ Q ∧ 𝑤 ∈ Q) → ((𝑢 +Q 𝑣) +Q 𝑤) = (𝑢 +Q (𝑣 +Q 𝑤)))
66	51, 53, 64, 65	syl3anc 1169	. . . . . . . . . . . 12 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd ‘𝐶)) → ((𝑢 +Q 𝑣) +Q 𝑤) = (𝑢 +Q (𝑣 +Q 𝑤)))
67	60, 63, 66	3eqtrd 2117	. . . . . . . . . . 11 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd ‘𝐶)) → ((𝑢 +Q 𝑡) +Q (𝑣 +Q 𝑡)) = (𝑢 +Q (𝑣 +Q 𝑤)))
68		simplrr 502	. . . . . . . . . . . 12 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd ‘𝐶)) → (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))
69		simpr 108	. . . . . . . . . . . 12 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd ‘𝐶)) → (𝑣 +Q 𝑡) ∈ (2nd ‘𝐶))
70	17	simp3d 952	. . . . . . . . . . . . . 14 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → 𝐶 ∈ P)
71	70	adantr 270	. . . . . . . . . . . . 13 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd ‘𝐶)) → 𝐶 ∈ P)
72	46, 47	genppreclu 6705	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (((𝑢 +Q 𝑡) ∈ (2nd ‘𝐴) ∧ (𝑣 +Q 𝑡) ∈ (2nd ‘𝐶)) → ((𝑢 +Q 𝑡) +Q (𝑣 +Q 𝑡)) ∈ (2nd ‘(𝐴 +P 𝐶))))
73	41, 71, 72	syl2anc 403	. . . . . . . . . . . 12 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd ‘𝐶)) → (((𝑢 +Q 𝑡) ∈ (2nd ‘𝐴) ∧ (𝑣 +Q 𝑡) ∈ (2nd ‘𝐶)) → ((𝑢 +Q 𝑡) +Q (𝑣 +Q 𝑡)) ∈ (2nd ‘(𝐴 +P 𝐶))))
74	68, 69, 73	mp2and 423	. . . . . . . . . . 11 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd ‘𝐶)) → ((𝑢 +Q 𝑡) +Q (𝑣 +Q 𝑡)) ∈ (2nd ‘(𝐴 +P 𝐶)))
75	67, 74	eqeltrrd 2156	. . . . . . . . . 10 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd ‘𝐶)) → (𝑢 +Q (𝑣 +Q 𝑤)) ∈ (2nd ‘(𝐴 +P 𝐶)))
76		simpr 108	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (𝐴 +P 𝐵) = (𝐴 +P 𝐶))
77	76	ad3antrrr 475	. . . . . . . . . . . 12 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → (𝐴 +P 𝐵) = (𝐴 +P 𝐶))
78	77	ad2antrr 471	. . . . . . . . . . 11 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd ‘𝐶)) → (𝐴 +P 𝐵) = (𝐴 +P 𝐶))
79		fveq2 5198	. . . . . . . . . . . 12 ⊢ ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → (2nd ‘(𝐴 +P 𝐵)) = (2nd ‘(𝐴 +P 𝐶)))
80	79	eleq2d 2148	. . . . . . . . . . 11 ⊢ ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → ((𝑢 +Q (𝑣 +Q 𝑤)) ∈ (2nd ‘(𝐴 +P 𝐵)) ↔ (𝑢 +Q (𝑣 +Q 𝑤)) ∈ (2nd ‘(𝐴 +P 𝐶))))
81	78, 80	syl 14	. . . . . . . . . 10 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd ‘𝐶)) → ((𝑢 +Q (𝑣 +Q 𝑤)) ∈ (2nd ‘(𝐴 +P 𝐵)) ↔ (𝑢 +Q (𝑣 +Q 𝑤)) ∈ (2nd ‘(𝐴 +P 𝐶))))
82	75, 81	mpbird 165	. . . . . . . . 9 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd ‘𝐶)) → (𝑢 +Q (𝑣 +Q 𝑤)) ∈ (2nd ‘(𝐴 +P 𝐵)))
83	50, 82	jca 300	. . . . . . . 8 ⊢ ((((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd ‘𝐶)) → ((𝑢 +Q (𝑣 +Q 𝑤)) ∈ (1st ‘(𝐴 +P 𝐵)) ∧ (𝑢 +Q (𝑣 +Q 𝑤)) ∈ (2nd ‘(𝐴 +P 𝐵))))
84	40, 83	mtand 623	. . . . . . 7 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → ¬ (𝑣 +Q 𝑡) ∈ (2nd ‘𝐶))
85		prop 6665	. . . . . . . . 9 ⊢ (𝐶 ∈ P → ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ∈ P)
86	70, 85	syl 14	. . . . . . . 8 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ∈ P)
87		simplrl 501	. . . . . . . . 9 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → 𝑡 ∈ Q)
88		ltaddnq 6597	. . . . . . . . 9 ⊢ ((𝑣 ∈ Q ∧ 𝑡 ∈ Q) → 𝑣 <Q (𝑣 +Q 𝑡))
89	32, 87, 88	syl2anc 403	. . . . . . . 8 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → 𝑣 <Q (𝑣 +Q 𝑡))
90		prloc 6681	. . . . . . . 8 ⊢ ((⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ∈ P ∧ 𝑣 <Q (𝑣 +Q 𝑡)) → (𝑣 ∈ (1st ‘𝐶) ∨ (𝑣 +Q 𝑡) ∈ (2nd ‘𝐶)))
91	86, 89, 90	syl2anc 403	. . . . . . 7 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → (𝑣 ∈ (1st ‘𝐶) ∨ (𝑣 +Q 𝑡) ∈ (2nd ‘𝐶)))
92	84, 91	ecased 1280	. . . . . 6 ⊢ (((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st ‘𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))) → 𝑣 ∈ (1st ‘𝐶))
93	16, 92	rexlimddv 2481	. . . . 5 ⊢ ((((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) ∧ (𝑡 ∈ Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → 𝑣 ∈ (1st ‘𝐶))
94	8, 93	rexlimddv 2481	. . . 4 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) ∧ (𝑤 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))) → 𝑣 ∈ (1st ‘𝐶))
95	5, 94	rexlimddv 2481	. . 3 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) → 𝑣 ∈ (1st ‘𝐶))
96	95	ex 113	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (𝑣 ∈ (1st ‘𝐵) → 𝑣 ∈ (1st ‘𝐶)))
97	96	ssrdv 3005	1 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (1st ‘𝐵) ⊆ (1st ‘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