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

 Description: Lemma for addcanprg 7236. (Contributed by Jim Kingdon, 25-Dec-2019.)
Assertion
Ref Expression
addcanprleml (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (1st𝐵) ⊆ (1st𝐶))

Dummy variables 𝑓 𝑔 𝑟 𝑠 𝑡 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 7095 . . . . . . 7 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
2 prnmaddl 7110 . . . . . . 7 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑣 ∈ (1st𝐵)) → ∃𝑤Q (𝑣 +Q 𝑤) ∈ (1st𝐵))
31, 2sylan 278 . . . . . 6 ((𝐵P𝑣 ∈ (1st𝐵)) → ∃𝑤Q (𝑣 +Q 𝑤) ∈ (1st𝐵))
433ad2antl2 1107 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ 𝑣 ∈ (1st𝐵)) → ∃𝑤Q (𝑣 +Q 𝑤) ∈ (1st𝐵))
54adantlr 462 . . . 4 ((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) → ∃𝑤Q (𝑣 +Q 𝑤) ∈ (1st𝐵))
6 simprl 499 . . . . . 6 (((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) → 𝑤Q)
7 halfnqq 7030 . . . . . 6 (𝑤Q → ∃𝑡Q (𝑡 +Q 𝑡) = 𝑤)
86, 7syl 14 . . . . 5 (((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) → ∃𝑡Q (𝑡 +Q 𝑡) = 𝑤)
9 simplll 501 . . . . . . . . . 10 (((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) → (𝐴P𝐵P𝐶P))
109adantr 271 . . . . . . . . 9 ((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → (𝐴P𝐵P𝐶P))
1110simp1d 956 . . . . . . . 8 ((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → 𝐴P)
12 prop 7095 . . . . . . . 8 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
1311, 12syl 14 . . . . . . 7 ((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
14 simprl 499 . . . . . . 7 ((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → 𝑡Q)
15 prarloc2 7124 . . . . . . 7 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑡Q) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
1613, 14, 15syl2anc 404 . . . . . 6 ((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
179ad2antrr 473 . . . . . . . . . . . 12 (((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → (𝐴P𝐵P𝐶P))
1817simp1d 956 . . . . . . . . . . 11 (((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝐴P)
1917simp2d 957 . . . . . . . . . . 11 (((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝐵P)
20 addclpr 7157 . . . . . . . . . . 11 ((𝐴P𝐵P) → (𝐴 +P 𝐵) ∈ P)
2118, 19, 20syl2anc 404 . . . . . . . . . 10 (((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → (𝐴 +P 𝐵) ∈ P)
22 prop 7095 . . . . . . . . . 10 ((𝐴 +P 𝐵) ∈ P → ⟨(1st ‘(𝐴 +P 𝐵)), (2nd ‘(𝐴 +P 𝐵))⟩ ∈ P)
2321, 22syl 14 . . . . . . . . 9 (((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → ⟨(1st ‘(𝐴 +P 𝐵)), (2nd ‘(𝐴 +P 𝐵))⟩ ∈ P)
2418, 12syl 14 . . . . . . . . . . 11 (((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
25 simprl 499 . . . . . . . . . . 11 (((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝑢 ∈ (1st𝐴))
26 elprnql 7101 . . . . . . . . . . 11 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑢 ∈ (1st𝐴)) → 𝑢Q)
2724, 25, 26syl2anc 404 . . . . . . . . . 10 (((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝑢Q)
2819, 1syl 14 . . . . . . . . . . . 12 (((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
29 simplr 498 . . . . . . . . . . . . 13 (((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) → 𝑣 ∈ (1st𝐵))
3029ad2antrr 473 . . . . . . . . . . . 12 (((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝑣 ∈ (1st𝐵))
31 elprnql 7101 . . . . . . . . . . . 12 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑣 ∈ (1st𝐵)) → 𝑣Q)
3228, 30, 31syl2anc 404 . . . . . . . . . . 11 (((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝑣Q)
33 simplrl 503 . . . . . . . . . . . 12 ((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → 𝑤Q)
3433adantr 271 . . . . . . . . . . 11 (((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝑤Q)
35 addclnq 6995 . . . . . . . . . . 11 ((𝑣Q𝑤Q) → (𝑣 +Q 𝑤) ∈ Q)
3632, 34, 35syl2anc 404 . . . . . . . . . 10 (((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → (𝑣 +Q 𝑤) ∈ Q)
37 addclnq 6995 . . . . . . . . . 10 ((𝑢Q ∧ (𝑣 +Q 𝑤) ∈ Q) → (𝑢 +Q (𝑣 +Q 𝑤)) ∈ Q)
3827, 36, 37syl2anc 404 . . . . . . . . 9 (((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → (𝑢 +Q (𝑣 +Q 𝑤)) ∈ Q)
39 prdisj 7112 . . . . . . . . 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))))
4023, 38, 39syl2anc 404 . . . . . . . 8 (((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → ¬ ((𝑢 +Q (𝑣 +Q 𝑤)) ∈ (1st ‘(𝐴 +P 𝐵)) ∧ (𝑢 +Q (𝑣 +Q 𝑤)) ∈ (2nd ‘(𝐴 +P 𝐵))))
4118adantr 271 . . . . . . . . . 10 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd𝐶)) → 𝐴P)
4219adantr 271 . . . . . . . . . 10 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd𝐶)) → 𝐵P)
43 simplrl 503 . . . . . . . . . 10 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd𝐶)) → 𝑢 ∈ (1st𝐴))
44 simplrr 504 . . . . . . . . . . 11 ((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → (𝑣 +Q 𝑤) ∈ (1st𝐵))
4544ad2antrr 473 . . . . . . . . . 10 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd𝐶)) → (𝑣 +Q 𝑤) ∈ (1st𝐵))
46 df-iplp 7088 . . . . . . . . . . . 12 +P = (𝑟P, 𝑠P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑟) ∧ ∈ (1st𝑠) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑟) ∧ ∈ (2nd𝑠) ∧ 𝑓 = (𝑔 +Q ))}⟩)
47 addclnq 6995 . . . . . . . . . . . 12 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
4846, 47genpprecll 7134 . . . . . . . . . . 11 ((𝐴P𝐵P) → ((𝑢 ∈ (1st𝐴) ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵)) → (𝑢 +Q (𝑣 +Q 𝑤)) ∈ (1st ‘(𝐴 +P 𝐵))))
4948imp 123 . . . . . . . . . 10 (((𝐴P𝐵P) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) → (𝑢 +Q (𝑣 +Q 𝑤)) ∈ (1st ‘(𝐴 +P 𝐵)))
5041, 42, 43, 45, 49syl22anc 1176 . . . . . . . . 9 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd𝐶)) → (𝑢 +Q (𝑣 +Q 𝑤)) ∈ (1st ‘(𝐴 +P 𝐵)))
5127adantr 271 . . . . . . . . . . . . 13 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd𝐶)) → 𝑢Q)
5214ad2antrr 473 . . . . . . . . . . . . 13 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd𝐶)) → 𝑡Q)
5332adantr 271 . . . . . . . . . . . . 13 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd𝐶)) → 𝑣Q)
54 addcomnqg 7001 . . . . . . . . . . . . . 14 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
5554adantl 272 . . . . . . . . . . . . 13 (((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd𝐶)) ∧ (𝑓Q𝑔Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
56 addassnqg 7002 . . . . . . . . . . . . . 14 ((𝑓Q𝑔QQ) → ((𝑓 +Q 𝑔) +Q ) = (𝑓 +Q (𝑔 +Q )))
5756adantl 272 . . . . . . . . . . . . 13 (((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd𝐶)) ∧ (𝑓Q𝑔QQ)) → ((𝑓 +Q 𝑔) +Q ) = (𝑓 +Q (𝑔 +Q )))
58 addclnq 6995 . . . . . . . . . . . . . 14 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) ∈ Q)
5958adantl 272 . . . . . . . . . . . . 13 (((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd𝐶)) ∧ (𝑓Q𝑔Q)) → (𝑓 +Q 𝑔) ∈ Q)
6051, 52, 53, 55, 57, 52, 59caov4d 5843 . . . . . . . . . . . 12 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd𝐶)) → ((𝑢 +Q 𝑡) +Q (𝑣 +Q 𝑡)) = ((𝑢 +Q 𝑣) +Q (𝑡 +Q 𝑡)))
61 simprr 500 . . . . . . . . . . . . . 14 ((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → (𝑡 +Q 𝑡) = 𝑤)
6261ad2antrr 473 . . . . . . . . . . . . 13 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd𝐶)) → (𝑡 +Q 𝑡) = 𝑤)
6362oveq2d 5682 . . . . . . . . . . . 12 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd𝐶)) → ((𝑢 +Q 𝑣) +Q (𝑡 +Q 𝑡)) = ((𝑢 +Q 𝑣) +Q 𝑤))
6433ad2antrr 473 . . . . . . . . . . . . 13 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd𝐶)) → 𝑤Q)
65 addassnqg 7002 . . . . . . . . . . . . 13 ((𝑢Q𝑣Q𝑤Q) → ((𝑢 +Q 𝑣) +Q 𝑤) = (𝑢 +Q (𝑣 +Q 𝑤)))
6651, 53, 64, 65syl3anc 1175 . . . . . . . . . . . 12 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd𝐶)) → ((𝑢 +Q 𝑣) +Q 𝑤) = (𝑢 +Q (𝑣 +Q 𝑤)))
6760, 63, 663eqtrd 2125 . . . . . . . . . . 11 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd𝐶)) → ((𝑢 +Q 𝑡) +Q (𝑣 +Q 𝑡)) = (𝑢 +Q (𝑣 +Q 𝑤)))
68 simplrr 504 . . . . . . . . . . . 12 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd𝐶)) → (𝑢 +Q 𝑡) ∈ (2nd𝐴))
69 simpr 109 . . . . . . . . . . . 12 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd𝐶)) → (𝑣 +Q 𝑡) ∈ (2nd𝐶))
7017simp3d 958 . . . . . . . . . . . . . 14 (((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝐶P)
7170adantr 271 . . . . . . . . . . . . 13 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd𝐶)) → 𝐶P)
7246, 47genppreclu 7135 . . . . . . . . . . . . 13 ((𝐴P𝐶P) → (((𝑢 +Q 𝑡) ∈ (2nd𝐴) ∧ (𝑣 +Q 𝑡) ∈ (2nd𝐶)) → ((𝑢 +Q 𝑡) +Q (𝑣 +Q 𝑡)) ∈ (2nd ‘(𝐴 +P 𝐶))))
7341, 71, 72syl2anc 404 . . . . . . . . . . . 12 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd𝐶)) → (((𝑢 +Q 𝑡) ∈ (2nd𝐴) ∧ (𝑣 +Q 𝑡) ∈ (2nd𝐶)) → ((𝑢 +Q 𝑡) +Q (𝑣 +Q 𝑡)) ∈ (2nd ‘(𝐴 +P 𝐶))))
7468, 69, 73mp2and 425 . . . . . . . . . . 11 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd𝐶)) → ((𝑢 +Q 𝑡) +Q (𝑣 +Q 𝑡)) ∈ (2nd ‘(𝐴 +P 𝐶)))
7567, 74eqeltrrd 2166 . . . . . . . . . 10 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd𝐶)) → (𝑢 +Q (𝑣 +Q 𝑤)) ∈ (2nd ‘(𝐴 +P 𝐶)))
76 simpr 109 . . . . . . . . . . . . 13 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (𝐴 +P 𝐵) = (𝐴 +P 𝐶))
7776ad3antrrr 477 . . . . . . . . . . . 12 ((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → (𝐴 +P 𝐵) = (𝐴 +P 𝐶))
7877ad2antrr 473 . . . . . . . . . . 11 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd𝐶)) → (𝐴 +P 𝐵) = (𝐴 +P 𝐶))
79 fveq2 5318 . . . . . . . . . . . 12 ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → (2nd ‘(𝐴 +P 𝐵)) = (2nd ‘(𝐴 +P 𝐶)))
8079eleq2d 2158 . . . . . . . . . . 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))))
8178, 80syl 14 . . . . . . . . . 10 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd𝐶)) → ((𝑢 +Q (𝑣 +Q 𝑤)) ∈ (2nd ‘(𝐴 +P 𝐵)) ↔ (𝑢 +Q (𝑣 +Q 𝑤)) ∈ (2nd ‘(𝐴 +P 𝐶))))
8275, 81mpbird 166 . . . . . . . . 9 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd𝐶)) → (𝑢 +Q (𝑣 +Q 𝑤)) ∈ (2nd ‘(𝐴 +P 𝐵)))
8350, 82jca 301 . . . . . . . 8 ((((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ (𝑣 +Q 𝑡) ∈ (2nd𝐶)) → ((𝑢 +Q (𝑣 +Q 𝑤)) ∈ (1st ‘(𝐴 +P 𝐵)) ∧ (𝑢 +Q (𝑣 +Q 𝑤)) ∈ (2nd ‘(𝐴 +P 𝐵))))
8440, 83mtand 627 . . . . . . 7 (((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → ¬ (𝑣 +Q 𝑡) ∈ (2nd𝐶))
85 prop 7095 . . . . . . . . 9 (𝐶P → ⟨(1st𝐶), (2nd𝐶)⟩ ∈ P)
8670, 85syl 14 . . . . . . . 8 (((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → ⟨(1st𝐶), (2nd𝐶)⟩ ∈ P)
87 simplrl 503 . . . . . . . . 9 (((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝑡Q)
88 ltaddnq 7027 . . . . . . . . 9 ((𝑣Q𝑡Q) → 𝑣 <Q (𝑣 +Q 𝑡))
8932, 87, 88syl2anc 404 . . . . . . . 8 (((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝑣 <Q (𝑣 +Q 𝑡))
90 prloc 7111 . . . . . . . 8 ((⟨(1st𝐶), (2nd𝐶)⟩ ∈ P𝑣 <Q (𝑣 +Q 𝑡)) → (𝑣 ∈ (1st𝐶) ∨ (𝑣 +Q 𝑡) ∈ (2nd𝐶)))
9186, 89, 90syl2anc 404 . . . . . . 7 (((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → (𝑣 ∈ (1st𝐶) ∨ (𝑣 +Q 𝑡) ∈ (2nd𝐶)))
9284, 91ecased 1286 . . . . . 6 (((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝑣 ∈ (1st𝐶))
9316, 92rexlimddv 2494 . . . . 5 ((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → 𝑣 ∈ (1st𝐶))
948, 93rexlimddv 2494 . . . 4 (((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) → 𝑣 ∈ (1st𝐶))
955, 94rexlimddv 2494 . . 3 ((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) → 𝑣 ∈ (1st𝐶))
9695ex 114 . 2 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (𝑣 ∈ (1st𝐵) → 𝑣 ∈ (1st𝐶)))
9796ssrdv 3032 1 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (1st𝐵) ⊆ (1st𝐶))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 665   ∧ w3a 925   = wceq 1290   ∈ wcel 1439  ∃wrex 2361   ⊆ wss 3000  ⟨cop 3453   class class class wbr 3851  ‘cfv 5028  (class class class)co 5666  1st c1st 5923  2nd c2nd 5924  Qcnq 6900   +Q cplq 6902
 Copyright terms: Public domain W3C validator