Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  addcanprlemu GIF version

Theorem addcanprlemu 6867
 Description: Lemma for addcanprg 6868. (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 6727 . . . . . . 7 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
2 prnminu 6741 . . . . . . 7 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑣 ∈ (2nd𝐵)) → ∃𝑟 ∈ (2nd𝐵)𝑟 <Q 𝑣)
31, 2sylan 277 . . . . . 6 ((𝐵P𝑣 ∈ (2nd𝐵)) → ∃𝑟 ∈ (2nd𝐵)𝑟 <Q 𝑣)
433ad2antl2 1102 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ 𝑣 ∈ (2nd𝐵)) → ∃𝑟 ∈ (2nd𝐵)𝑟 <Q 𝑣)
54adantlr 461 . . . 4 ((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) → ∃𝑟 ∈ (2nd𝐵)𝑟 <Q 𝑣)
6 simprr 499 . . . . . 6 (((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) → 𝑟 <Q 𝑣)
7 ltexnqi 6661 . . . . . 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 498 . . . . . . 7 ((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) → 𝑤Q)
10 halfnqq 6662 . . . . . . 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 6727 . . . . . . . . . . . . . 14 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
13 prarloc2 6756 . . . . . . . . . . . . . 14 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑡Q) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
1412, 13sylan 277 . . . . . . . . . . . . 13 ((𝐴P𝑡Q) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
1514adantrr 463 . . . . . . . . . . . 12 ((𝐴P ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
16153ad2antl1 1101 . . . . . . . . . . 11 (((𝐴P𝐵P𝐶P) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
1716adantlr 461 . . . . . . . . . 10 ((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
1817adantlr 461 . . . . . . . . 9 (((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
1918adantlr 461 . . . . . . . 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 461 . . . . . . 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 500 . . . . . . . . . . . . . 14 (((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) → (𝐴P𝐵P𝐶P))
2221ad3antrrr 476 . . . . . . . . . . . . 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 951 . . . . . . . . . . . 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 952 . . . . . . . . . . . 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 6789 . . . . . . . . . . . 12 ((𝐴P𝐵P) → (𝐴 +P 𝐵) ∈ P)
2623, 24, 25syl2anc 403 . . . . . . . . . . 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 6727 . . . . . . . . . . 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 498 . . . . . . . . . . . . 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 6733 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑢 ∈ (1st𝐴)) → 𝑢Q)
3229, 30, 31syl2anc 403 . . . . . . . . . . . 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 502 . . . . . . . . . . . 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 6627 . . . . . . . . . . . 12 ((𝑢Q𝑡Q) → (𝑢 +Q 𝑡) ∈ Q)
3532, 33, 34syl2anc 403 . . . . . . . . . . 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 498 . . . . . . . . . . . . 13 (((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) → 𝑟 ∈ (2nd𝐵))
3837ad3antrrr 476 . . . . . . . . . . . 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 6734 . . . . . . . . . . . 12 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑟 ∈ (2nd𝐵)) → 𝑟Q)
4036, 38, 39syl2anc 403 . . . . . . . . . . 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 6627 . . . . . . . . . . 11 (((𝑢 +Q 𝑡) ∈ Q𝑟Q) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ Q)
4235, 40, 41syl2anc 403 . . . . . . . . . 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 6744 . . . . . . . . . 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 403 . . . . . . . . 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 6634 . . . . . . . . . . . . . . 15 ((𝑢Q𝑡Q𝑟Q) → ((𝑢 +Q 𝑡) +Q 𝑟) = (𝑢 +Q (𝑡 +Q 𝑟)))
4632, 33, 40, 45syl3anc 1170 . . . . . . . . . . . . . 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 6633 . . . . . . . . . . . . . . . 16 ((𝑡Q𝑟Q) → (𝑡 +Q 𝑟) = (𝑟 +Q 𝑡))
4847oveq2d 5559 . . . . . . . . . . . . . . 15 ((𝑡Q𝑟Q) → (𝑢 +Q (𝑡 +Q 𝑟)) = (𝑢 +Q (𝑟 +Q 𝑡)))
4933, 40, 48syl2anc 403 . . . . . . . . . . . . . 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 2114 . . . . . . . . . . . . 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 270 . . . . . . . . . . . 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 502 . . . . . . . . . . . . 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 108 . . . . . . . . . . . . 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 270 . . . . . . . . . . . . . 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 953 . . . . . . . . . . . . . . 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 270 . . . . . . . . . . . . . 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 6720 . . . . . . . . . . . . . . 15 +P = (𝑞P, 𝑠P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑞) ∧ ∈ (1st𝑠) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑞) ∧ ∈ (2nd𝑠) ∧ 𝑓 = (𝑔 +Q ))}⟩)
58 addclnq 6627 . . . . . . . . . . . . . . 15 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
5957, 58genpprecll 6766 . . . . . . . . . . . . . 14 ((𝐴P𝐶P) → ((𝑢 ∈ (1st𝐴) ∧ (𝑟 +Q 𝑡) ∈ (1st𝐶)) → (𝑢 +Q (𝑟 +Q 𝑡)) ∈ (1st ‘(𝐴 +P 𝐶))))
6054, 56, 59syl2anc 403 . . . . . . . . . . . . 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 424 . . . . . . . . . . . 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 2156 . . . . . . . . . . 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 5209 . . . . . . . . . . . . 13 ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → (1st ‘(𝐴 +P 𝐵)) = (1st ‘(𝐴 +P 𝐶)))
6463eleq2d 2149 . . . . . . . . . . . 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 485 . . . . . . . . . . 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 165 . . . . . . . . . 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 6767 . . . . . . . . . . . . . . . . . . 19 ((𝐴P𝐵P) → (((𝑢 +Q 𝑡) ∈ (2nd𝐴) ∧ 𝑟 ∈ (2nd𝐵)) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (2nd ‘(𝐴 +P 𝐵))))
6867ancomsd 265 . . . . . . . . . . . . . . . . . 18 ((𝐴P𝐵P) → ((𝑟 ∈ (2nd𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴)) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (2nd ‘(𝐴 +P 𝐵))))
69683adant3 959 . . . . . . . . . . . . . . . . 17 ((𝐴P𝐵P𝐶P) → ((𝑟 ∈ (2nd𝐵) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴)) → ((𝑢 +Q 𝑡) +Q 𝑟) ∈ (2nd ‘(𝐴 +P 𝐵))))
7069ad2antrr 472 . . . . . . . . . . . . . . . 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 122 . . . . . . . . . . . . . . 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 469 . . . . . . . . . . . . . 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 392 . . . . . . . . . . . . 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 495 . . . . . . . . . . . 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 461 . . . . . . . . . . 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 270 . . . . . . . . . 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 300 . . . . . . . . 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 624 . . . . . . . 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 6727 . . . . . . . . . . 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 6659 . . . . . . . . . . . . . 14 ((𝑡Q𝑡Q) → 𝑡 <Q (𝑡 +Q 𝑡))
8233, 33, 81syl2anc 403 . . . . . . . . . . . . 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 503 . . . . . . . . . . . . 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 3817 . . . . . . . . . . . 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 6654 . . . . . . . . . . . 12 ((𝑡 <Q 𝑤𝑟Q) → (𝑟 +Q 𝑡) <Q (𝑟 +Q 𝑤))
8684, 40, 85syl2anc 403 . . . . . . . . . . 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 499 . . . . . . . . . . . 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 472 . . . . . . . . . . 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 3817 . . . . . . . . . 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 6743 . . . . . . . . . 10 ((⟨(1st𝐶), (2nd𝐶)⟩ ∈ P ∧ (𝑟 +Q 𝑡) <Q 𝑣) → ((𝑟 +Q 𝑡) ∈ (1st𝐶) ∨ 𝑣 ∈ (2nd𝐶)))
9180, 89, 90syl2anc 403 . . . . . . . . 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 681 . . . . . . . 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 1281 . . . . . . 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 2482 . . . . . 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 2482 . . . . 5 ((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) ∧ (𝑤Q ∧ (𝑟 +Q 𝑤) = 𝑣)) → 𝑣 ∈ (2nd𝐶))
968, 95rexlimddv 2482 . . . 4 (((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) ∧ (𝑟 ∈ (2nd𝐵) ∧ 𝑟 <Q 𝑣)) → 𝑣 ∈ (2nd𝐶))
975, 96rexlimddv 2482 . . 3 ((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (2nd𝐵)) → 𝑣 ∈ (2nd𝐶))
9897ex 113 . 2 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (𝑣 ∈ (2nd𝐵) → 𝑣 ∈ (2nd𝐶)))
9998ssrdv 3006 1 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (2nd𝐵) ⊆ (2nd𝐶))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103   ∨ wo 662   ∧ w3a 920   = wceq 1285   ∈ wcel 1434  ∃wrex 2350   ⊆ wss 2974  ⟨cop 3409   class class class wbr 3793  ‘cfv 4932  (class class class)co 5543  1st c1st 5796  2nd c2nd 5797  Qcnq 6532   +Q cplq 6534
 Copyright terms: Public domain W3C validator