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Theorem addcanprleml 7627
Description: Lemma for addcanprg 7629. (Contributed by Jim Kingdon, 25-Dec-2019.)
Assertion
Ref Expression
addcanprleml (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (1st𝐵) ⊆ (1st𝐶))

Proof of Theorem addcanprleml
Dummy variables 𝑓 𝑔 𝑟 𝑠 𝑡 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 7488 . . . . . . 7 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
2 prnmaddl 7503 . . . . . . 7 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑣 ∈ (1st𝐵)) → ∃𝑤Q (𝑣 +Q 𝑤) ∈ (1st𝐵))
31, 2sylan 283 . . . . . 6 ((𝐵P𝑣 ∈ (1st𝐵)) → ∃𝑤Q (𝑣 +Q 𝑤) ∈ (1st𝐵))
433ad2antl2 1161 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ 𝑣 ∈ (1st𝐵)) → ∃𝑤Q (𝑣 +Q 𝑤) ∈ (1st𝐵))
54adantlr 477 . . . 4 ((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) → ∃𝑤Q (𝑣 +Q 𝑤) ∈ (1st𝐵))
6 simprl 529 . . . . . 6 (((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) → 𝑤Q)
7 halfnqq 7423 . . . . . 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 533 . . . . . . . . . 10 (((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) → (𝐴P𝐵P𝐶P))
109adantr 276 . . . . . . . . 9 ((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → (𝐴P𝐵P𝐶P))
1110simp1d 1010 . . . . . . . 8 ((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → 𝐴P)
12 prop 7488 . . . . . . . 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 529 . . . . . . 7 ((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → 𝑡Q)
15 prarloc2 7517 . . . . . . 7 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑡Q) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
1613, 14, 15syl2anc 411 . . . . . 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 488 . . . . . . . . . . . 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 1010 . . . . . . . . . . 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 1011 . . . . . . . . . . 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 7550 . . . . . . . . . . 11 ((𝐴P𝐵P) → (𝐴 +P 𝐵) ∈ P)
2118, 19, 20syl2anc 411 . . . . . . . . . 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 7488 . . . . . . . . . 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 529 . . . . . . . . . . 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 7494 . . . . . . . . . . 11 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑢 ∈ (1st𝐴)) → 𝑢Q)
2724, 25, 26syl2anc 411 . . . . . . . . . 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 528 . . . . . . . . . . . . 13 (((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) → 𝑣 ∈ (1st𝐵))
3029ad2antrr 488 . . . . . . . . . . . 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 7494 . . . . . . . . . . . 12 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑣 ∈ (1st𝐵)) → 𝑣Q)
3228, 30, 31syl2anc 411 . . . . . . . . . . 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 535 . . . . . . . . . . . 12 ((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → 𝑤Q)
3433adantr 276 . . . . . . . . . . 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 7388 . . . . . . . . . . 11 ((𝑣Q𝑤Q) → (𝑣 +Q 𝑤) ∈ Q)
3632, 34, 35syl2anc 411 . . . . . . . . . 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 7388 . . . . . . . . . 10 ((𝑢Q ∧ (𝑣 +Q 𝑤) ∈ Q) → (𝑢 +Q (𝑣 +Q 𝑤)) ∈ Q)
3827, 36, 37syl2anc 411 . . . . . . . . 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 7505 . . . . . . . . 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 411 . . . . . . . 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 276 . . . . . . . . . 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 276 . . . . . . . . . 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 535 . . . . . . . . . 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 536 . . . . . . . . . . 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 488 . . . . . . . . . 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 7481 . . . . . . . . . . . 12 +P = (𝑟P, 𝑠P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑟) ∧ ∈ (1st𝑠) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑟) ∧ ∈ (2nd𝑠) ∧ 𝑓 = (𝑔 +Q ))}⟩)
47 addclnq 7388 . . . . . . . . . . . 12 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
4846, 47genpprecll 7527 . . . . . . . . . . 11 ((𝐴P𝐵P) → ((𝑢 ∈ (1st𝐴) ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵)) → (𝑢 +Q (𝑣 +Q 𝑤)) ∈ (1st ‘(𝐴 +P 𝐵))))
4948imp 124 . . . . . . . . . 10 (((𝐴P𝐵P) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) → (𝑢 +Q (𝑣 +Q 𝑤)) ∈ (1st ‘(𝐴 +P 𝐵)))
5041, 42, 43, 45, 49syl22anc 1249 . . . . . . . . 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 276 . . . . . . . . . . . . 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 488 . . . . . . . . . . . . 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 276 . . . . . . . . . . . . 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 7394 . . . . . . . . . . . . . 14 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
5554adantl 277 . . . . . . . . . . . . 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 7395 . . . . . . . . . . . . . 14 ((𝑓Q𝑔QQ) → ((𝑓 +Q 𝑔) +Q ) = (𝑓 +Q (𝑔 +Q )))
5756adantl 277 . . . . . . . . . . . . 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 7388 . . . . . . . . . . . . . 14 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) ∈ Q)
5958adantl 277 . . . . . . . . . . . . 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 6073 . . . . . . . . . . . 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 531 . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . 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 5904 . . . . . . . . . . . 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 488 . . . . . . . . . . . . 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 7395 . . . . . . . . . . . . 13 ((𝑢Q𝑣Q𝑤Q) → ((𝑢 +Q 𝑣) +Q 𝑤) = (𝑢 +Q (𝑣 +Q 𝑤)))
6651, 53, 64, 65syl3anc 1248 . . . . . . . . . . . 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 2224 . . . . . . . . . . 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 536 . . . . . . . . . . . 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 110 . . . . . . . . . . . 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 1012 . . . . . . . . . . . . . 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 276 . . . . . . . . . . . . 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 7528 . . . . . . . . . . . . 13 ((𝐴P𝐶P) → (((𝑢 +Q 𝑡) ∈ (2nd𝐴) ∧ (𝑣 +Q 𝑡) ∈ (2nd𝐶)) → ((𝑢 +Q 𝑡) +Q (𝑣 +Q 𝑡)) ∈ (2nd ‘(𝐴 +P 𝐶))))
7341, 71, 72syl2anc 411 . . . . . . . . . . . 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 433 . . . . . . . . . . 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 2265 . . . . . . . . . 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 110 . . . . . . . . . . . . 13 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (𝐴 +P 𝐵) = (𝐴 +P 𝐶))
7776ad3antrrr 492 . . . . . . . . . . . 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 488 . . . . . . . . . . 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 5527 . . . . . . . . . . . 12 ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → (2nd ‘(𝐴 +P 𝐵)) = (2nd ‘(𝐴 +P 𝐶)))
8079eleq2d 2257 . . . . . . . . . . 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 167 . . . . . . . . 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 306 . . . . . . . 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 666 . . . . . . 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 7488 . . . . . . . . 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 535 . . . . . . . . 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 7420 . . . . . . . . 9 ((𝑣Q𝑡Q) → 𝑣 <Q (𝑣 +Q 𝑡))
8932, 87, 88syl2anc 411 . . . . . . . 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 7504 . . . . . . . 8 ((⟨(1st𝐶), (2nd𝐶)⟩ ∈ P𝑣 <Q (𝑣 +Q 𝑡)) → (𝑣 ∈ (1st𝐶) ∨ (𝑣 +Q 𝑡) ∈ (2nd𝐶)))
9186, 89, 90syl2anc 411 . . . . . . 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 1359 . . . . . 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 2609 . . . . 5 ((((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) ∧ (𝑡Q ∧ (𝑡 +Q 𝑡) = 𝑤)) → 𝑣 ∈ (1st𝐶))
948, 93rexlimddv 2609 . . . 4 (((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) ∧ (𝑤Q ∧ (𝑣 +Q 𝑤) ∈ (1st𝐵))) → 𝑣 ∈ (1st𝐶))
955, 94rexlimddv 2609 . . 3 ((((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ∧ 𝑣 ∈ (1st𝐵)) → 𝑣 ∈ (1st𝐶))
9695ex 115 . 2 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (𝑣 ∈ (1st𝐵) → 𝑣 ∈ (1st𝐶)))
9796ssrdv 3173 1 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (1st𝐵) ⊆ (1st𝐶))
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 6153  2nd c2nd 6154  Qcnq 7293   +Q cplq 7295   <Q cltq 7298  Pcnp 7304   +P cpp 7306
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 6155  df-2nd 6156  df-recs 6320  df-irdg 6385  df-1o 6431  df-2o 6432  df-oadd 6435  df-omul 6436  df-er 6549  df-ec 6551  df-qs 6555  df-ni 7317  df-pli 7318  df-mi 7319  df-lti 7320  df-plpq 7357  df-mpq 7358  df-enq 7360  df-nqqs 7361  df-plqqs 7362  df-mqqs 7363  df-1nqqs 7364  df-rq 7365  df-ltnqqs 7366  df-enq0 7437  df-nq0 7438  df-0nq0 7439  df-plq0 7440  df-mq0 7441  df-inp 7479  df-iplp 7481
This theorem is referenced by:  addcanprg  7629
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