Theorem addcanprleml 7981

Description: Lemma for addcanprg 7983. (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 7842	. . . . . . 7 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
2		prnmaddl 7857	. . . . . . 7 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑣 ∈ (1st ‘𝐵)) → ∃𝑤 ∈ Q (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))
3	1, 2	sylan 283	. . . . . 6 ⊢ ((𝐵 ∈ P ∧ 𝑣 ∈ (1st ‘𝐵)) → ∃𝑤 ∈ Q (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))
4	3	3ad2antl2 1191	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ 𝑣 ∈ (1st ‘𝐵)) → ∃𝑤 ∈ Q (𝑣 +Q 𝑤) ∈ (1st ‘𝐵))
5	4	adantlr 481	. . . 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 535	. . . . . 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 7777	. . . . . 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 539	. . . . . . . . . 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 276	. . . . . . . . 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 1040	. . . . . . . 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 7842	. . . . . . . 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 535	. . . . . . 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 7871	. . . . . . 7 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑡 ∈ Q) → ∃𝑢 ∈ (1st ‘𝐴)(𝑢 +Q 𝑡) ∈ (2nd ‘𝐴))
16	13, 14, 15	syl2anc 415	. . . . . 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 492	. . . . . . . . . . . 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 1040	. . . . . . . . . . 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 1041	. . . . . . . . . . 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 7904	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +P 𝐵) ∈ P)
21	18, 19, 20	syl2anc 415	. . . . . . . . . 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 7842	. . . . . . . . . 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 535	. . . . . . . . . . 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 7848	. . . . . . . . . . 11 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑢 ∈ (1st ‘𝐴)) → 𝑢 ∈ Q)
27	24, 25, 26	syl2anc 415	. . . . . . . . . 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 533	. . . . . . . . . . . . 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 492	. . . . . . . . . . . 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 7848	. . . . . . . . . . . 12 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑣 ∈ (1st ‘𝐵)) → 𝑣 ∈ Q)
32	28, 30, 31	syl2anc 415	. . . . . . . . . . 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 541	. . . . . . . . . . . 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 276	. . . . . . . . . . 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 7742	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ Q ∧ 𝑤 ∈ Q) → (𝑣 +Q 𝑤) ∈ Q)
36	32, 34, 35	syl2anc 415	. . . . . . . . . 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 7742	. . . . . . . . . 10 ⊢ ((𝑢 ∈ Q ∧ (𝑣 +Q 𝑤) ∈ Q) → (𝑢 +Q (𝑣 +Q 𝑤)) ∈ Q)
38	27, 36, 37	syl2anc 415	. . . . . . . . 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 7859	. . . . . . . . 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 415	. . . . . . . 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 276	. . . . . . . . . 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 276	. . . . . . . . . 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 541	. . . . . . . . . 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 542	. . . . . . . . . . 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 492	. . . . . . . . . 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 7835	. . . . . . . . . . . 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 7742	. . . . . . . . . . . 12 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +Q ℎ) ∈ Q)
48	46, 47	genpprecll 7881	. . . . . . . . . . 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 124	. . . . . . . . . 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 1279	. . . . . . . . 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 276	. . . . . . . . . . . . 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 492	. . . . . . . . . . . . 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 276	. . . . . . . . . . . . 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 7748	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
55	54	adantl 277	. . . . . . . . . . . . 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 7749	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑓 +Q 𝑔) +Q ℎ) = (𝑓 +Q (𝑔 +Q ℎ)))
57	56	adantl 277	. . . . . . . . . . . . 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 7742	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +Q 𝑔) ∈ Q)
59	58	adantl 277	. . . . . . . . . . . . 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 6274	. . . . . . . . . . . 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 537	. . . . . . . . . . . . . 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 492	. . . . . . . . . . . . 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 6101	. . . . . . . . . . . 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 492	. . . . . . . . . . . . 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 7749	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ Q ∧ 𝑣 ∈ Q ∧ 𝑤 ∈ Q) → ((𝑢 +Q 𝑣) +Q 𝑤) = (𝑢 +Q (𝑣 +Q 𝑤)))
66	51, 53, 64, 65	syl3anc 1278	. . . . . . . . . . . 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 2275	. . . . . . . . . . 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 542	. . . . . . . . . . . 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 110	. . . . . . . . . . . 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 1042	. . . . . . . . . . . . . 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 276	. . . . . . . . . . . . 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 7882	. . . . . . . . . . . . 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 415	. . . . . . . . . . . 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 437	. . . . . . . . . . 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 2316	. . . . . . . . . 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 110	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (𝐴 +P 𝐵) = (𝐴 +P 𝐶))
77	76	ad3antrrr 496	. . . . . . . . . . . 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 492	. . . . . . . . . . 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 5695	. . . . . . . . . . . 12 ⊢ ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → (2nd ‘(𝐴 +P 𝐵)) = (2nd ‘(𝐴 +P 𝐶)))
80	79	eleq2d 2308	. . . . . . . . . . 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 167	. . . . . . . . 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 306	. . . . . . . 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 675	. . . . . . 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 7842	. . . . . . . . 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 541	. . . . . . . . 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 7774	. . . . . . . . 9 ⊢ ((𝑣 ∈ Q ∧ 𝑡 ∈ Q) → 𝑣 <Q (𝑣 +Q 𝑡))
89	32, 87, 88	syl2anc 415	. . . . . . . 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 7858	. . . . . . . 8 ⊢ ((⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ∈ P ∧ 𝑣 <Q (𝑣 +Q 𝑡)) → (𝑣 ∈ (1st ‘𝐶) ∨ (𝑣 +Q 𝑡) ∈ (2nd ‘𝐶)))
91	86, 89, 90	syl2anc 415	. . . . . . 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 1390	. . . . . 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 2673	. . . . 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 2673	. . . 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 2673	. . 3 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st ‘𝐵)) → 𝑣 ∈ (1st ‘𝐶))
96	95	ex 115	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (𝑣 ∈ (1st ‘𝐵) → 𝑣 ∈ (1st ‘𝐶)))
97	96	ssrdv 3254	1 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (1st ‘𝐵) ⊆ (1st ‘𝐶))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ∃wrex 2529   ⊆ wss 3220  ⟨cop 3712   class class class wbr 4130  ‘cfv 5377  (class class class)co 6085  1^st c1st 6372  2^nd c2nd 6373  Qcnq 7647   +_Q cplq 7649   <_Q cltq 7652  Pcnp 7658   +_P cpp 7660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-eprel 4434  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-1o 6687  df-2o 6688  df-oadd 6691  df-omul 6692  df-er 6807  df-ec 6809  df-qs 6813  df-ni 7671  df-pli 7672  df-mi 7673  df-lti 7674  df-plpq 7711  df-mpq 7712  df-enq 7714  df-nqqs 7715  df-plqqs 7716  df-mqqs 7717  df-1nqqs 7718  df-rq 7719  df-ltnqqs 7720  df-enq0 7791  df-nq0 7792  df-0nq0 7793  df-plq0 7794  df-mq0 7795  df-inp 7833  df-iplp 7835

This theorem is used by:  addcanprg  7983