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Theorem addcanprlemu 7391
Description: Lemma for addcanprg 7392. (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 7251 . . . . . . 7 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
2 prnminu 7265 . . . . . . 7 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑣 ∈ (2nd𝐵)) → ∃𝑟 ∈ (2nd𝐵)𝑟 <Q 𝑣)
31, 2sylan 281 . . . . . 6 ((𝐵P𝑣 ∈ (2nd𝐵)) → ∃𝑟 ∈ (2nd𝐵)𝑟 <Q 𝑣)
433ad2antl2 1129 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ 𝑣 ∈ (2nd𝐵)) → ∃𝑟 ∈ (2nd𝐵)𝑟 <Q 𝑣)
54adantlr 468 . . . 4 ((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) → ∃𝑟 ∈ (2nd𝐵)𝑟 <Q 𝑣)
6 simprr 506 . . . . . 6 (((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) → 𝑟 <Q 𝑣)
7 ltexnqi 7185 . . . . . 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 505 . . . . . . 7 ((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) → 𝑤Q)
10 halfnqq 7186 . . . . . . 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 7251 . . . . . . . . . . . . . 14 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
13 prarloc2 7280 . . . . . . . . . . . . . 14 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑡Q) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
1412, 13sylan 281 . . . . . . . . . . . . 13 ((𝐴P𝑡Q) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
1514adantrr 470 . . . . . . . . . . . 12 ((𝐴P ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
16153ad2antl1 1128 . . . . . . . . . . 11 (((𝐴P𝐵P𝐶P) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
1716adantlr 468 . . . . . . . . . 10 ((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
1817adantlr 468 . . . . . . . . 9 (((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
1918adantlr 468 . . . . . . . 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 468 . . . . . . 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 507 . . . . . . . . . . . . . 14 (((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) → (𝐴P𝐵P𝐶P))
2221ad3antrrr 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))) → (𝐴P𝐵P𝐶P))
2322simp1d 978 . . . . . . . . . . . 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 979 . . . . . . . . . . . 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 7313 . . . . . . . . . . . 12 ((𝐴P𝐵P) → (𝐴 +P 𝐵) ∈ P)
2623, 24, 25syl2anc 408 . . . . . . . . . . 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 7251 . . . . . . . . . . 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 505 . . . . . . . . . . . . 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 7257 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑢 ∈ (1st𝐴)) → 𝑢Q)
3229, 30, 31syl2anc 408 . . . . . . . . . . . 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 509 . . . . . . . . . . . 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 7151 . . . . . . . . . . . 12 ((𝑢Q𝑡Q) → (𝑢 +Q 𝑡) ∈ Q)
3532, 33, 34syl2anc 408 . . . . . . . . . . 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 505 . . . . . . . . . . . . 13 (((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) → 𝑟 ∈ (2nd𝐵))
3837ad3antrrr 483 . . . . . . . . . . . 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 7258 . . . . . . . . . . . 12 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑟 ∈ (2nd𝐵)) → 𝑟Q)
4036, 38, 39syl2anc 408 . . . . . . . . . . 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 7151 . . . . . . . . . . 11 (((𝑢 +Q 𝑡) ∈ Q𝑟Q) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ Q)
4235, 40, 41syl2anc 408 . . . . . . . . . 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 7268 . . . . . . . . . 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 408 . . . . . . . . 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 7158 . . . . . . . . . . . . . . 15 ((𝑢Q𝑡Q𝑟Q) → ((𝑢 +Q 𝑡) +Q 𝑟) = (𝑢 +Q (𝑡 +Q 𝑟)))
4632, 33, 40, 45syl3anc 1201 . . . . . . . . . . . . . 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 7157 . . . . . . . . . . . . . . . 16 ((𝑡Q𝑟Q) → (𝑡 +Q 𝑟) = (𝑟 +Q 𝑡))
4847oveq2d 5758 . . . . . . . . . . . . . . 15 ((𝑡Q𝑟Q) → (𝑢 +Q (𝑡 +Q 𝑟)) = (𝑢 +Q (𝑟 +Q 𝑡)))
4933, 40, 48syl2anc 408 . . . . . . . . . . . . . 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 2150 . . . . . . . . . . . . 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 274 . . . . . . . . . . . 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 509 . . . . . . . . . . . . 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 109 . . . . . . . . . . . . 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 274 . . . . . . . . . . . . . 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 980 . . . . . . . . . . . . . . 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 274 . . . . . . . . . . . . . 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 7244 . . . . . . . . . . . . . . 15 +P = (𝑞P, 𝑠P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑞) ∧ ∈ (1st𝑠) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑞) ∧ ∈ (2nd𝑠) ∧ 𝑓 = (𝑔 +Q ))}⟩)
58 addclnq 7151 . . . . . . . . . . . . . . 15 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
5957, 58genpprecll 7290 . . . . . . . . . . . . . 14 ((𝐴P𝐶P) → ((𝑢 ∈ (1st𝐴) ∧ (𝑟 +Q 𝑡) ∈ (1st𝐶)) → (𝑢 +Q (𝑟 +Q 𝑡)) ∈ (1st ‘(𝐴 +P 𝐶))))
6054, 56, 59syl2anc 408 . . . . . . . . . . . . 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 429 . . . . . . . . . . . 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 2194 . . . . . . . . . . 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 5389 . . . . . . . . . . . . 13 ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → (1st ‘(𝐴 +P 𝐵)) = (1st ‘(𝐴 +P 𝐶)))
6463eleq2d 2187 . . . . . . . . . . . 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 492 . . . . . . . . . . 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 166 . . . . . . . . . 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 7291 . . . . . . . . . . . . . . . . . . 19 ((𝐴P𝐵P) → (((𝑢 +Q 𝑡) ∈ (2nd𝐴) ∧ 𝑟 ∈ (2nd𝐵)) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (2nd ‘(𝐴 +P 𝐵))))
6867ancomsd 267 . . . . . . . . . . . . . . . . . 18 ((𝐴P𝐵P) → ((𝑟 ∈ (2nd𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴)) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (2nd ‘(𝐴 +P 𝐵))))
69683adant3 986 . . . . . . . . . . . . . . . . 17 ((𝐴P𝐵P𝐶P) → ((𝑟 ∈ (2nd𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴)) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (2nd ‘(𝐴 +P 𝐵))))
7069ad2antrr 479 . . . . . . . . . . . . . . . 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 123 . . . . . . . . . . . . . . 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 476 . . . . . . . . . . . . . 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 397 . . . . . . . . . . . . 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 502 . . . . . . . . . . . 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 468 . . . . . . . . . . 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 274 . . . . . . . . . 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 304 . . . . . . . . 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 639 . . . . . . . 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 7251 . . . . . . . . . . 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 7183 . . . . . . . . . . . . . 14 ((𝑡Q𝑡Q) → 𝑡 <Q (𝑡 +Q 𝑡))
8233, 33, 81syl2anc 408 . . . . . . . . . . . . 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 510 . . . . . . . . . . . . 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 3924 . . . . . . . . . . . 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 7178 . . . . . . . . . . . 12 ((𝑡 <Q 𝑤𝑟Q) → (𝑟 +Q 𝑡) <Q (𝑟 +Q 𝑤))
8684, 40, 85syl2anc 408 . . . . . . . . . . 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 506 . . . . . . . . . . . 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 479 . . . . . . . . . . 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 3924 . . . . . . . . . 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 7267 . . . . . . . . . 10 ((⟨(1st𝐶), (2nd𝐶)⟩ ∈ P ∧ (𝑟 +Q 𝑡) <Q 𝑣) → ((𝑟 +Q 𝑡) ∈ (1st𝐶) ∨ 𝑣 ∈ (2nd𝐶)))
9180, 89, 90syl2anc 408 . . . . . . . . 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 703 . . . . . . . 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 1312 . . . . . . 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 2531 . . . . . 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 2531 . . . . 5 ((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) → 𝑣 ∈ (2nd𝐶))
968, 95rexlimddv 2531 . . . 4 (((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) → 𝑣 ∈ (2nd𝐶))
975, 96rexlimddv 2531 . . 3 ((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) → 𝑣 ∈ (2nd𝐶))
9897ex 114 . 2 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (𝑣 ∈ (2nd𝐵) → 𝑣 ∈ (2nd𝐶)))
9998ssrdv 3073 1 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (2nd𝐵) ⊆ (2nd𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 682  w3a 947   = wceq 1316  wcel 1465  wrex 2394  wss 3041  cop 3500   class class class wbr 3899  cfv 5093  (class class class)co 5742  1st c1st 6004  2nd c2nd 6005  Qcnq 7056   +Q cplq 7058   <Q cltq 7061  Pcnp 7067   +P cpp 7069
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-2o 6282  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-plpq 7120  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-mqqs 7126  df-1nqqs 7127  df-rq 7128  df-ltnqqs 7129  df-enq0 7200  df-nq0 7201  df-0nq0 7202  df-plq0 7203  df-mq0 7204  df-inp 7242  df-iplp 7244
This theorem is referenced by:  addcanprg  7392
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