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Theorem addcanprlemu 7627
Description: Lemma for addcanprg 7628. (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 7487 . . . . . . 7 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
2 prnminu 7501 . . . . . . 7 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑣 ∈ (2nd𝐵)) → ∃𝑟 ∈ (2nd𝐵)𝑟 <Q 𝑣)
31, 2sylan 283 . . . . . 6 ((𝐵P𝑣 ∈ (2nd𝐵)) → ∃𝑟 ∈ (2nd𝐵)𝑟 <Q 𝑣)
433ad2antl2 1161 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ 𝑣 ∈ (2nd𝐵)) → ∃𝑟 ∈ (2nd𝐵)𝑟 <Q 𝑣)
54adantlr 477 . . . 4 ((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) → ∃𝑟 ∈ (2nd𝐵)𝑟 <Q 𝑣)
6 simprr 531 . . . . . 6 (((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) → 𝑟 <Q 𝑣)
7 ltexnqi 7421 . . . . . 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 529 . . . . . . 7 ((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) → 𝑤Q)
10 halfnqq 7422 . . . . . . 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 7487 . . . . . . . . . . . . . 14 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
13 prarloc2 7516 . . . . . . . . . . . . . 14 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑡Q) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
1412, 13sylan 283 . . . . . . . . . . . . 13 ((𝐴P𝑡Q) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
1514adantrr 479 . . . . . . . . . . . 12 ((𝐴P ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
16153ad2antl1 1160 . . . . . . . . . . 11 (((𝐴P𝐵P𝐶P) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
1716adantlr 477 . . . . . . . . . 10 ((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
1817adantlr 477 . . . . . . . . 9 (((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
1918adantlr 477 . . . . . . . 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 477 . . . . . . 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 533 . . . . . . . . . . . . . 14 (((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) → (𝐴P𝐵P𝐶P))
2221ad3antrrr 492 . . . . . . . . . . . . 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 1010 . . . . . . . . . . . 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 1011 . . . . . . . . . . . 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 7549 . . . . . . . . . . . 12 ((𝐴P𝐵P) → (𝐴 +P 𝐵) ∈ P)
2623, 24, 25syl2anc 411 . . . . . . . . . . 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 7487 . . . . . . . . . . 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 529 . . . . . . . . . . . . 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 7493 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑢 ∈ (1st𝐴)) → 𝑢Q)
3229, 30, 31syl2anc 411 . . . . . . . . . . . 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 535 . . . . . . . . . . . 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 7387 . . . . . . . . . . . 12 ((𝑢Q𝑡Q) → (𝑢 +Q 𝑡) ∈ Q)
3532, 33, 34syl2anc 411 . . . . . . . . . . 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 529 . . . . . . . . . . . . 13 (((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) → 𝑟 ∈ (2nd𝐵))
3837ad3antrrr 492 . . . . . . . . . . . 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 7494 . . . . . . . . . . . 12 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑟 ∈ (2nd𝐵)) → 𝑟Q)
4036, 38, 39syl2anc 411 . . . . . . . . . . 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 7387 . . . . . . . . . . 11 (((𝑢 +Q 𝑡) ∈ Q𝑟Q) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ Q)
4235, 40, 41syl2anc 411 . . . . . . . . . 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 7504 . . . . . . . . . 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 411 . . . . . . . . 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 7394 . . . . . . . . . . . . . . 15 ((𝑢Q𝑡Q𝑟Q) → ((𝑢 +Q 𝑡) +Q 𝑟) = (𝑢 +Q (𝑡 +Q 𝑟)))
4632, 33, 40, 45syl3anc 1248 . . . . . . . . . . . . . 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 7393 . . . . . . . . . . . . . . . 16 ((𝑡Q𝑟Q) → (𝑡 +Q 𝑟) = (𝑟 +Q 𝑡))
4847oveq2d 5904 . . . . . . . . . . . . . . 15 ((𝑡Q𝑟Q) → (𝑢 +Q (𝑡 +Q 𝑟)) = (𝑢 +Q (𝑟 +Q 𝑡)))
4933, 40, 48syl2anc 411 . . . . . . . . . . . . . 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 2220 . . . . . . . . . . . . 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 276 . . . . . . . . . . . 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 535 . . . . . . . . . . . . 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 110 . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . 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 1012 . . . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . 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 7480 . . . . . . . . . . . . . . 15 +P = (𝑞P, 𝑠P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑞) ∧ ∈ (1st𝑠) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑞) ∧ ∈ (2nd𝑠) ∧ 𝑓 = (𝑔 +Q ))}⟩)
58 addclnq 7387 . . . . . . . . . . . . . . 15 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
5957, 58genpprecll 7526 . . . . . . . . . . . . . 14 ((𝐴P𝐶P) → ((𝑢 ∈ (1st𝐴) ∧ (𝑟 +Q 𝑡) ∈ (1st𝐶)) → (𝑢 +Q (𝑟 +Q 𝑡)) ∈ (1st ‘(𝐴 +P 𝐶))))
6054, 56, 59syl2anc 411 . . . . . . . . . . . . 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 433 . . . . . . . . . . . 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 2264 . . . . . . . . . . 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 5527 . . . . . . . . . . . . 13 ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → (1st ‘(𝐴 +P 𝐵)) = (1st ‘(𝐴 +P 𝐶)))
6463eleq2d 2257 . . . . . . . . . . . 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 501 . . . . . . . . . . 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 167 . . . . . . . . . 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 7527 . . . . . . . . . . . . . . . . . . 19 ((𝐴P𝐵P) → (((𝑢 +Q 𝑡) ∈ (2nd𝐴) ∧ 𝑟 ∈ (2nd𝐵)) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (2nd ‘(𝐴 +P 𝐵))))
6867ancomsd 269 . . . . . . . . . . . . . . . . . 18 ((𝐴P𝐵P) → ((𝑟 ∈ (2nd𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴)) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (2nd ‘(𝐴 +P 𝐵))))
69683adant3 1018 . . . . . . . . . . . . . . . . 17 ((𝐴P𝐵P𝐶P) → ((𝑟 ∈ (2nd𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴)) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (2nd ‘(𝐴 +P 𝐵))))
7069ad2antrr 488 . . . . . . . . . . . . . . . 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 124 . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . 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 400 . . . . . . . . . . . . 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 511 . . . . . . . . . . . 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 477 . . . . . . . . . . 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 276 . . . . . . . . . 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 306 . . . . . . . . 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 666 . . . . . . . 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 7487 . . . . . . . . . . 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 7419 . . . . . . . . . . . . . 14 ((𝑡Q𝑡Q) → 𝑡 <Q (𝑡 +Q 𝑡))
8233, 33, 81syl2anc 411 . . . . . . . . . . . . 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 536 . . . . . . . . . . . . 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 4041 . . . . . . . . . . . 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 7414 . . . . . . . . . . . 12 ((𝑡 <Q 𝑤𝑟Q) → (𝑟 +Q 𝑡) <Q (𝑟 +Q 𝑤))
8684, 40, 85syl2anc 411 . . . . . . . . . . 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 531 . . . . . . . . . . . 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 488 . . . . . . . . . . 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 4041 . . . . . . . . . 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 7503 . . . . . . . . . 10 ((⟨(1st𝐶), (2nd𝐶)⟩ ∈ P ∧ (𝑟 +Q 𝑡) <Q 𝑣) → ((𝑟 +Q 𝑡) ∈ (1st𝐶) ∨ 𝑣 ∈ (2nd𝐶)))
9180, 89, 90syl2anc 411 . . . . . . . . 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 730 . . . . . . . 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 1359 . . . . . . 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 2609 . . . . . 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 2609 . . . . 5 ((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) → 𝑣 ∈ (2nd𝐶))
968, 95rexlimddv 2609 . . . 4 (((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) → 𝑣 ∈ (2nd𝐶))
975, 96rexlimddv 2609 . . 3 ((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) → 𝑣 ∈ (2nd𝐶))
9897ex 115 . 2 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (𝑣 ∈ (2nd𝐵) → 𝑣 ∈ (2nd𝐶)))
9998ssrdv 3173 1 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (2nd𝐵) ⊆ (2nd𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 709  w3a 979   = wceq 1363  wcel 2158  wrex 2466  wss 3141  cop 3607   class class class wbr 4015  cfv 5228  (class class class)co 5888  1st c1st 6152  2nd c2nd 6153  Qcnq 7292   +Q cplq 7294   <Q cltq 7297  Pcnp 7303   +P cpp 7305
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-eprel 4301  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-irdg 6384  df-1o 6430  df-2o 6431  df-oadd 6434  df-omul 6435  df-er 6548  df-ec 6550  df-qs 6554  df-ni 7316  df-pli 7317  df-mi 7318  df-lti 7319  df-plpq 7356  df-mpq 7357  df-enq 7359  df-nqqs 7360  df-plqqs 7361  df-mqqs 7362  df-1nqqs 7363  df-rq 7364  df-ltnqqs 7365  df-enq0 7436  df-nq0 7437  df-0nq0 7438  df-plq0 7439  df-mq0 7440  df-inp 7478  df-iplp 7480
This theorem is referenced by:  addcanprg  7628
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