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Theorem addcanprlemu 6711
Description: Lemma for addcanprg 6712. (Contributed by Jim Kingdon, 25-Dec-2019.)
Assertion
Ref Expression
addcanprlemu (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (2nd𝐵) ⊆ (2nd𝐶))

Proof of Theorem addcanprlemu
Dummy variables 𝑓 𝑔 𝑞 𝑟 𝑠 𝑡 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 6571 . . . . . . 7 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
2 prnminu 6585 . . . . . . 7 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑣 ∈ (2nd𝐵)) → ∃𝑟 ∈ (2nd𝐵)𝑟 <Q 𝑣)
31, 2sylan 267 . . . . . 6 ((𝐵P𝑣 ∈ (2nd𝐵)) → ∃𝑟 ∈ (2nd𝐵)𝑟 <Q 𝑣)
433ad2antl2 1067 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ 𝑣 ∈ (2nd𝐵)) → ∃𝑟 ∈ (2nd𝐵)𝑟 <Q 𝑣)
54adantlr 446 . . . 4 ((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) → ∃𝑟 ∈ (2nd𝐵)𝑟 <Q 𝑣)
6 simprr 484 . . . . . 6 (((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) → 𝑟 <Q 𝑣)
7 ltexnqi 6505 . . . . . 6 (𝑟 <Q 𝑣 → ∃𝑤Q (𝑟 +Q 𝑤) = 𝑣)
86, 7syl 14 . . . . 5 (((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) → ∃𝑤Q (𝑟 +Q 𝑤) = 𝑣)
9 simprl 483 . . . . . . 7 ((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) → 𝑤Q)
10 halfnqq 6506 . . . . . . 7 (𝑤Q → ∃𝑡Q (𝑡 +Q 𝑡) = 𝑤)
119, 10syl 14 . . . . . 6 ((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) → ∃𝑡Q (𝑡 +Q 𝑡) = 𝑤)
12 prop 6571 . . . . . . . . . . . . . 14 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
13 prarloc2 6600 . . . . . . . . . . . . . 14 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑡Q) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
1412, 13sylan 267 . . . . . . . . . . . . 13 ((𝐴P𝑡Q) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
1514adantrr 448 . . . . . . . . . . . 12 ((𝐴P ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
16153ad2antl1 1066 . . . . . . . . . . 11 (((𝐴P𝐵P𝐶P) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
1716adantlr 446 . . . . . . . . . 10 ((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
1817adantlr 446 . . . . . . . . 9 (((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
1918adantlr 446 . . . . . . . 8 ((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
2019adantlr 446 . . . . . . 7 (((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
21 simplll 485 . . . . . . . . . . . . . 14 (((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) → (𝐴P𝐵P𝐶P))
2221ad3antrrr 461 . . . . . . . . . . . . 13 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → (𝐴P𝐵P𝐶P))
2322simp1d 916 . . . . . . . . . . . 12 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝐴P)
2422simp2d 917 . . . . . . . . . . . 12 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝐵P)
25 addclpr 6633 . . . . . . . . . . . 12 ((𝐴P𝐵P) → (𝐴 +P 𝐵) ∈ P)
2623, 24, 25syl2anc 391 . . . . . . . . . . 11 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → (𝐴 +P 𝐵) ∈ P)
27 prop 6571 . . . . . . . . . . 11 ((𝐴 +P 𝐵) ∈ P → ⟨(1st ‘(𝐴 +P 𝐵)), (2nd ‘(𝐴 +P 𝐵))⟩ ∈ P)
2826, 27syl 14 . . . . . . . . . 10 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → ⟨(1st ‘(𝐴 +P 𝐵)), (2nd ‘(𝐴 +P 𝐵))⟩ ∈ P)
2923, 12syl 14 . . . . . . . . . . . . 13 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
30 simprl 483 . . . . . . . . . . . . 13 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝑢 ∈ (1st𝐴))
31 elprnql 6577 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑢 ∈ (1st𝐴)) → 𝑢Q)
3229, 30, 31syl2anc 391 . . . . . . . . . . . 12 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝑢Q)
33 simplrl 487 . . . . . . . . . . . 12 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝑡Q)
34 addclnq 6471 . . . . . . . . . . . 12 ((𝑢Q𝑡Q) → (𝑢 +Q 𝑡) ∈ Q)
3532, 33, 34syl2anc 391 . . . . . . . . . . 11 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → (𝑢 +Q 𝑡) ∈ Q)
3624, 1syl 14 . . . . . . . . . . . 12 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
37 simprl 483 . . . . . . . . . . . . 13 (((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) → 𝑟 ∈ (2nd𝐵))
3837ad3antrrr 461 . . . . . . . . . . . 12 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝑟 ∈ (2nd𝐵))
39 elprnqu 6578 . . . . . . . . . . . 12 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑟 ∈ (2nd𝐵)) → 𝑟Q)
4036, 38, 39syl2anc 391 . . . . . . . . . . 11 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝑟Q)
41 addclnq 6471 . . . . . . . . . . 11 (((𝑢 +Q 𝑡) ∈ Q𝑟Q) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ Q)
4235, 40, 41syl2anc 391 . . . . . . . . . 10 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ Q)
43 prdisj 6588 . . . . . . . . . 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))))
4428, 42, 43syl2anc 391 . . . . . . . . 9 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → ¬ (((𝑢 +Q 𝑡) +Q 𝑟) ∈ (1st ‘(𝐴 +P 𝐵)) ∧ ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (2nd ‘(𝐴 +P 𝐵))))
45 addassnqg 6478 . . . . . . . . . . . . . . 15 ((𝑢Q𝑡Q𝑟Q) → ((𝑢 +Q 𝑡) +Q 𝑟) = (𝑢 +Q (𝑡 +Q 𝑟)))
4632, 33, 40, 45syl3anc 1135 . . . . . . . . . . . . . 14 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → ((𝑢 +Q 𝑡) +Q 𝑟) = (𝑢 +Q (𝑡 +Q 𝑟)))
47 addcomnqg 6477 . . . . . . . . . . . . . . . 16 ((𝑡Q𝑟Q) → (𝑡 +Q 𝑟) = (𝑟 +Q 𝑡))
4847oveq2d 5528 . . . . . . . . . . . . . . 15 ((𝑡Q𝑟Q) → (𝑢 +Q (𝑡 +Q 𝑟)) = (𝑢 +Q (𝑟 +Q 𝑡)))
4933, 40, 48syl2anc 391 . . . . . . . . . . . . . 14 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → (𝑢 +Q (𝑡 +Q 𝑟)) = (𝑢 +Q (𝑟 +Q 𝑡)))
5046, 49eqtrd 2072 . . . . . . . . . . . . 13 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → ((𝑢 +Q 𝑡) +Q 𝑟) = (𝑢 +Q (𝑟 +Q 𝑡)))
5150adantr 261 . . . . . . . . . . . 12 (((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑟 +Q 𝑡) ∈ (1st𝐶)) → ((𝑢 +Q 𝑡) +Q 𝑟) = (𝑢 +Q (𝑟 +Q 𝑡)))
52 simplrl 487 . . . . . . . . . . . . 13 (((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑟 +Q 𝑡) ∈ (1st𝐶)) → 𝑢 ∈ (1st𝐴))
53 simpr 103 . . . . . . . . . . . . 13 (((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑟 +Q 𝑡) ∈ (1st𝐶)) → (𝑟 +Q 𝑡) ∈ (1st𝐶))
5423adantr 261 . . . . . . . . . . . . . 14 (((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑟 +Q 𝑡) ∈ (1st𝐶)) → 𝐴P)
5522simp3d 918 . . . . . . . . . . . . . . 15 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝐶P)
5655adantr 261 . . . . . . . . . . . . . 14 (((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑟 +Q 𝑡) ∈ (1st𝐶)) → 𝐶P)
57 df-iplp 6564 . . . . . . . . . . . . . . 15 +P = (𝑞P, 𝑠P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑞) ∧ ∈ (1st𝑠) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑞) ∧ ∈ (2nd𝑠) ∧ 𝑓 = (𝑔 +Q ))}⟩)
58 addclnq 6471 . . . . . . . . . . . . . . 15 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
5957, 58genpprecll 6610 . . . . . . . . . . . . . 14 ((𝐴P𝐶P) → ((𝑢 ∈ (1st𝐴) ∧ (𝑟 +Q 𝑡) ∈ (1st𝐶)) → (𝑢 +Q (𝑟 +Q 𝑡)) ∈ (1st ‘(𝐴 +P 𝐶))))
6054, 56, 59syl2anc 391 . . . . . . . . . . . . 13 (((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑟 +Q 𝑡) ∈ (1st𝐶)) → ((𝑢 ∈ (1st𝐴) ∧ (𝑟 +Q 𝑡) ∈ (1st𝐶)) → (𝑢 +Q (𝑟 +Q 𝑡)) ∈ (1st ‘(𝐴 +P 𝐶))))
6152, 53, 60mp2and 409 . . . . . . . . . . . 12 (((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑟 +Q 𝑡) ∈ (1st𝐶)) → (𝑢 +Q (𝑟 +Q 𝑡)) ∈ (1st ‘(𝐴 +P 𝐶)))
6251, 61eqeltrd 2114 . . . . . . . . . . 11 (((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑟 +Q 𝑡) ∈ (1st𝐶)) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (1st ‘(𝐴 +P 𝐶)))
63 fveq2 5178 . . . . . . . . . . . . 13 ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → (1st ‘(𝐴 +P 𝐵)) = (1st ‘(𝐴 +P 𝐶)))
6463eleq2d 2107 . . . . . . . . . . . 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))))
6564ad7antlr 470 . . . . . . . . . . 11 (((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑟 +Q 𝑡) ∈ (1st𝐶)) → (((𝑢 +Q 𝑡) +Q 𝑟) ∈ (1st ‘(𝐴 +P 𝐵)) ↔ ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (1st ‘(𝐴 +P 𝐶))))
6662, 65mpbird 156 . . . . . . . . . 10 (((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑟 +Q 𝑡) ∈ (1st𝐶)) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (1st ‘(𝐴 +P 𝐵)))
6757, 58genppreclu 6611 . . . . . . . . . . . . . . . . . . 19 ((𝐴P𝐵P) → (((𝑢 +Q 𝑡) ∈ (2nd𝐴) ∧ 𝑟 ∈ (2nd𝐵)) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (2nd ‘(𝐴 +P 𝐵))))
6867ancomsd 256 . . . . . . . . . . . . . . . . . 18 ((𝐴P𝐵P) → ((𝑟 ∈ (2nd𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴)) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (2nd ‘(𝐴 +P 𝐵))))
69683adant3 924 . . . . . . . . . . . . . . . . 17 ((𝐴P𝐵P𝐶P) → ((𝑟 ∈ (2nd𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴)) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (2nd ‘(𝐴 +P 𝐵))))
7069ad2antrr 457 . . . . . . . . . . . . . . . 16 ((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) → ((𝑟 ∈ (2nd𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴)) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (2nd ‘(𝐴 +P 𝐵))))
7170imp 115 . . . . . . . . . . . . . . 15 (((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (2nd ‘(𝐴 +P 𝐵)))
7271adantrlr 454 . . . . . . . . . . . . . 14 (((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ ((𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (2nd ‘(𝐴 +P 𝐵)))
7372anassrs 380 . . . . . . . . . . . . 13 ((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴)) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (2nd ‘(𝐴 +P 𝐵)))
7473ad2ant2rl 480 . . . . . . . . . . . 12 (((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (2nd ‘(𝐴 +P 𝐵)))
7574adantlr 446 . . . . . . . . . . 11 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (2nd ‘(𝐴 +P 𝐵)))
7675adantr 261 . . . . . . . . . 10 (((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑟 +Q 𝑡) ∈ (1st𝐶)) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (2nd ‘(𝐴 +P 𝐵)))
7766, 76jca 290 . . . . . . . . 9 (((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑟 +Q 𝑡) ∈ (1st𝐶)) → (((𝑢 +Q 𝑡) +Q 𝑟) ∈ (1st ‘(𝐴 +P 𝐵)) ∧ ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (2nd ‘(𝐴 +P 𝐵))))
7844, 77mtand 591 . . . . . . . 8 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → ¬ (𝑟 +Q 𝑡) ∈ (1st𝐶))
79 prop 6571 . . . . . . . . . . 11 (𝐶P → ⟨(1st𝐶), (2nd𝐶)⟩ ∈ P)
8055, 79syl 14 . . . . . . . . . 10 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → ⟨(1st𝐶), (2nd𝐶)⟩ ∈ P)
81 ltaddnq 6503 . . . . . . . . . . . . . 14 ((𝑡Q𝑡Q) → 𝑡 <Q (𝑡 +Q 𝑡))
8233, 33, 81syl2anc 391 . . . . . . . . . . . . 13 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝑡 <Q (𝑡 +Q 𝑡))
83 simplrr 488 . . . . . . . . . . . . 13 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → (𝑡 +Q 𝑡) = 𝑤)
8482, 83breqtrd 3788 . . . . . . . . . . . 12 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝑡 <Q 𝑤)
85 ltanqi 6498 . . . . . . . . . . . 12 ((𝑡 <Q 𝑤𝑟Q) → (𝑟 +Q 𝑡) <Q (𝑟 +Q 𝑤))
8684, 40, 85syl2anc 391 . . . . . . . . . . 11 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → (𝑟 +Q 𝑡) <Q (𝑟 +Q 𝑤))
87 simprr 484 . . . . . . . . . . . 12 ((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) → (𝑟 +Q 𝑤) = 𝑣)
8887ad2antrr 457 . . . . . . . . . . 11 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → (𝑟 +Q 𝑤) = 𝑣)
8986, 88breqtrd 3788 . . . . . . . . . 10 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → (𝑟 +Q 𝑡) <Q 𝑣)
90 prloc 6587 . . . . . . . . . 10 ((⟨(1st𝐶), (2nd𝐶)⟩ ∈ P ∧ (𝑟 +Q 𝑡) <Q 𝑣) → ((𝑟 +Q 𝑡) ∈ (1st𝐶) ∨ 𝑣 ∈ (2nd𝐶)))
9180, 89, 90syl2anc 391 . . . . . . . . 9 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → ((𝑟 +Q 𝑡) ∈ (1st𝐶) ∨ 𝑣 ∈ (2nd𝐶)))
9291orcomd 648 . . . . . . . 8 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → (𝑣 ∈ (2nd𝐶) ∨ (𝑟 +Q 𝑡) ∈ (1st𝐶)))
9378, 92ecased 1239 . . . . . . 7 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝑣 ∈ (2nd𝐶))
9420, 93rexlimddv 2437 . . . . . 6 (((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → 𝑣 ∈ (2nd𝐶))
9511, 94rexlimddv 2437 . . . . 5 ((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) → 𝑣 ∈ (2nd𝐶))
968, 95rexlimddv 2437 . . . 4 (((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) → 𝑣 ∈ (2nd𝐶))
975, 96rexlimddv 2437 . . 3 ((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) → 𝑣 ∈ (2nd𝐶))
9897ex 108 . 2 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (𝑣 ∈ (2nd𝐵) → 𝑣 ∈ (2nd𝐶)))
9998ssrdv 2951 1 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (2nd𝐵) ⊆ (2nd𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 97  wb 98  wo 629  w3a 885   = wceq 1243  wcel 1393  wrex 2307  wss 2917  cop 3378   class class class wbr 3764  cfv 4902  (class class class)co 5512  1st c1st 5765  2nd c2nd 5766  Qcnq 6376   +Q cplq 6378   <Q cltq 6381  Pcnp 6387   +P cpp 6389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6400  df-pli 6401  df-mi 6402  df-lti 6403  df-plpq 6440  df-mpq 6441  df-enq 6443  df-nqqs 6444  df-plqqs 6445  df-mqqs 6446  df-1nqqs 6447  df-rq 6448  df-ltnqqs 6449  df-enq0 6520  df-nq0 6521  df-0nq0 6522  df-plq0 6523  df-mq0 6524  df-inp 6562  df-iplp 6564
This theorem is referenced by:  addcanprg  6712
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