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Theorem aptiprlemu 7178
Description: Lemma for aptipr 7179. (Contributed by Jim Kingdon, 28-Jan-2020.)
Assertion
Ref Expression
aptiprlemu ((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) → (2nd𝐵) ⊆ (2nd𝐴))

Proof of Theorem aptiprlemu
Dummy variables 𝑓 𝑔 𝑠 𝑡 𝑢 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 7013 . . . . . 6 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
2 prnminu 7027 . . . . . 6 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑥 ∈ (2nd𝐵)) → ∃𝑠 ∈ (2nd𝐵)𝑠 <Q 𝑥)
31, 2sylan 277 . . . . 5 ((𝐵P𝑥 ∈ (2nd𝐵)) → ∃𝑠 ∈ (2nd𝐵)𝑠 <Q 𝑥)
433ad2antl2 1106 . . . 4 (((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) → ∃𝑠 ∈ (2nd𝐵)𝑠 <Q 𝑥)
5 simprr 499 . . . . . 6 ((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) → 𝑠 <Q 𝑥)
6 ltexnqi 6947 . . . . . 6 (𝑠 <Q 𝑥 → ∃𝑡Q (𝑠 +Q 𝑡) = 𝑥)
75, 6syl 14 . . . . 5 ((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) → ∃𝑡Q (𝑠 +Q 𝑡) = 𝑥)
8 simpl1 946 . . . . . . . 8 (((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) → 𝐴P)
98ad2antrr 472 . . . . . . 7 (((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) → 𝐴P)
10 simprl 498 . . . . . . 7 (((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) → 𝑡Q)
11 prop 7013 . . . . . . . 8 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
12 prarloc2 7042 . . . . . . . 8 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑡Q) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
1311, 12sylan 277 . . . . . . 7 ((𝐴P𝑡Q) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
149, 10, 13syl2anc 403 . . . . . 6 (((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
15 simpl2 947 . . . . . . . . . 10 (((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) → 𝐵P)
1615ad3antrrr 476 . . . . . . . . 9 ((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝐵P)
17 simpr 108 . . . . . . . . . 10 (((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) → 𝑥 ∈ (2nd𝐵))
1817ad3antrrr 476 . . . . . . . . 9 ((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝑥 ∈ (2nd𝐵))
19 elprnqu 7020 . . . . . . . . . 10 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑥 ∈ (2nd𝐵)) → 𝑥Q)
201, 19sylan 277 . . . . . . . . 9 ((𝐵P𝑥 ∈ (2nd𝐵)) → 𝑥Q)
2116, 18, 20syl2anc 403 . . . . . . . 8 ((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝑥Q)
228ad3antrrr 476 . . . . . . . . . 10 ((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝐴P)
23 simprl 498 . . . . . . . . . 10 ((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝑢 ∈ (1st𝐴))
24 elprnql 7019 . . . . . . . . . . 11 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑢 ∈ (1st𝐴)) → 𝑢Q)
2511, 24sylan 277 . . . . . . . . . 10 ((𝐴P𝑢 ∈ (1st𝐴)) → 𝑢Q)
2622, 23, 25syl2anc 403 . . . . . . . . 9 ((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝑢Q)
2710adantr 270 . . . . . . . . 9 ((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝑡Q)
28 addclnq 6913 . . . . . . . . 9 ((𝑢Q𝑡Q) → (𝑢 +Q 𝑡) ∈ Q)
2926, 27, 28syl2anc 403 . . . . . . . 8 ((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → (𝑢 +Q 𝑡) ∈ Q)
30 nqtri3or 6934 . . . . . . . 8 ((𝑥Q ∧ (𝑢 +Q 𝑡) ∈ Q) → (𝑥 <Q (𝑢 +Q 𝑡) ∨ 𝑥 = (𝑢 +Q 𝑡) ∨ (𝑢 +Q 𝑡) <Q 𝑥))
3121, 29, 30syl2anc 403 . . . . . . 7 ((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → (𝑥 <Q (𝑢 +Q 𝑡) ∨ 𝑥 = (𝑢 +Q 𝑡) ∨ (𝑢 +Q 𝑡) <Q 𝑥))
3215adantr 270 . . . . . . . . . . . . . 14 ((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) → 𝐵P)
33 simprl 498 . . . . . . . . . . . . . 14 ((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) → 𝑠 ∈ (2nd𝐵))
34 elprnqu 7020 . . . . . . . . . . . . . . 15 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑠 ∈ (2nd𝐵)) → 𝑠Q)
351, 34sylan 277 . . . . . . . . . . . . . 14 ((𝐵P𝑠 ∈ (2nd𝐵)) → 𝑠Q)
3632, 33, 35syl2anc 403 . . . . . . . . . . . . 13 ((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) → 𝑠Q)
3736ad3antrrr 476 . . . . . . . . . . . 12 (((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → 𝑠Q)
3833ad3antrrr 476 . . . . . . . . . . . 12 (((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → 𝑠 ∈ (2nd𝐵))
39 simplrr 503 . . . . . . . . . . . . . . . 16 ((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → (𝑠 +Q 𝑡) = 𝑥)
40 breq1 3840 . . . . . . . . . . . . . . . . 17 ((𝑠 +Q 𝑡) = 𝑥 → ((𝑠 +Q 𝑡) <Q (𝑢 +Q 𝑡) ↔ 𝑥 <Q (𝑢 +Q 𝑡)))
4140biimprd 156 . . . . . . . . . . . . . . . 16 ((𝑠 +Q 𝑡) = 𝑥 → (𝑥 <Q (𝑢 +Q 𝑡) → (𝑠 +Q 𝑡) <Q (𝑢 +Q 𝑡)))
4239, 41syl 14 . . . . . . . . . . . . . . 15 ((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → (𝑥 <Q (𝑢 +Q 𝑡) → (𝑠 +Q 𝑡) <Q (𝑢 +Q 𝑡)))
4342imp 122 . . . . . . . . . . . . . 14 (((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → (𝑠 +Q 𝑡) <Q (𝑢 +Q 𝑡))
44 ltanqg 6938 . . . . . . . . . . . . . . . 16 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
4544adantl 271 . . . . . . . . . . . . . . 15 ((((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) ∧ (𝑓Q𝑔QQ)) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
4626adantr 270 . . . . . . . . . . . . . . 15 (((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → 𝑢Q)
4727adantr 270 . . . . . . . . . . . . . . 15 (((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → 𝑡Q)
48 addcomnqg 6919 . . . . . . . . . . . . . . . 16 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
4948adantl 271 . . . . . . . . . . . . . . 15 ((((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) ∧ (𝑓Q𝑔Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
5045, 37, 46, 47, 49caovord2d 5796 . . . . . . . . . . . . . 14 (((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → (𝑠 <Q 𝑢 ↔ (𝑠 +Q 𝑡) <Q (𝑢 +Q 𝑡)))
5143, 50mpbird 165 . . . . . . . . . . . . 13 (((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → 𝑠 <Q 𝑢)
5222adantr 270 . . . . . . . . . . . . . 14 (((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → 𝐴P)
5323adantr 270 . . . . . . . . . . . . . 14 (((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → 𝑢 ∈ (1st𝐴))
54 prcdnql 7022 . . . . . . . . . . . . . . 15 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑢 ∈ (1st𝐴)) → (𝑠 <Q 𝑢𝑠 ∈ (1st𝐴)))
5511, 54sylan 277 . . . . . . . . . . . . . 14 ((𝐴P𝑢 ∈ (1st𝐴)) → (𝑠 <Q 𝑢𝑠 ∈ (1st𝐴)))
5652, 53, 55syl2anc 403 . . . . . . . . . . . . 13 (((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → (𝑠 <Q 𝑢𝑠 ∈ (1st𝐴)))
5751, 56mpd 13 . . . . . . . . . . . 12 (((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → 𝑠 ∈ (1st𝐴))
58 rspe 2424 . . . . . . . . . . . 12 ((𝑠Q ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝐴))) → ∃𝑠Q (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝐴)))
5937, 38, 57, 58syl12anc 1172 . . . . . . . . . . 11 (((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → ∃𝑠Q (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝐴)))
6016adantr 270 . . . . . . . . . . . 12 (((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → 𝐵P)
61 ltdfpr 7044 . . . . . . . . . . . 12 ((𝐵P𝐴P) → (𝐵<P 𝐴 ↔ ∃𝑠Q (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝐴))))
6260, 52, 61syl2anc 403 . . . . . . . . . . 11 (((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → (𝐵<P 𝐴 ↔ ∃𝑠Q (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝐴))))
6359, 62mpbird 165 . . . . . . . . . 10 (((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → 𝐵<P 𝐴)
64 simpll3 984 . . . . . . . . . . 11 ((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) → ¬ 𝐵<P 𝐴)
6564ad3antrrr 476 . . . . . . . . . 10 (((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → ¬ 𝐵<P 𝐴)
6663, 65pm2.21dd 585 . . . . . . . . 9 (((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) ∧ 𝑥 <Q (𝑢 +Q 𝑡)) → 𝑥 ∈ (2nd𝐴))
6766ex 113 . . . . . . . 8 ((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → (𝑥 <Q (𝑢 +Q 𝑡) → 𝑥 ∈ (2nd𝐴)))
68 simprr 499 . . . . . . . . 9 ((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → (𝑢 +Q 𝑡) ∈ (2nd𝐴))
69 eleq1 2150 . . . . . . . . 9 (𝑥 = (𝑢 +Q 𝑡) → (𝑥 ∈ (2nd𝐴) ↔ (𝑢 +Q 𝑡) ∈ (2nd𝐴)))
7068, 69syl5ibrcom 155 . . . . . . . 8 ((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → (𝑥 = (𝑢 +Q 𝑡) → 𝑥 ∈ (2nd𝐴)))
71 prcunqu 7023 . . . . . . . . . 10 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴)) → ((𝑢 +Q 𝑡) <Q 𝑥𝑥 ∈ (2nd𝐴)))
7211, 71sylan 277 . . . . . . . . 9 ((𝐴P ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴)) → ((𝑢 +Q 𝑡) <Q 𝑥𝑥 ∈ (2nd𝐴)))
7322, 68, 72syl2anc 403 . . . . . . . 8 ((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → ((𝑢 +Q 𝑡) <Q 𝑥𝑥 ∈ (2nd𝐴)))
7467, 70, 733jaod 1240 . . . . . . 7 ((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → ((𝑥 <Q (𝑢 +Q 𝑡) ∨ 𝑥 = (𝑢 +Q 𝑡) ∨ (𝑢 +Q 𝑡) <Q 𝑥) → 𝑥 ∈ (2nd𝐴)))
7531, 74mpd 13 . . . . . 6 ((((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) ∧ (𝑢 ∈ (1st𝐴) ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴))) → 𝑥 ∈ (2nd𝐴))
7614, 75rexlimddv 2493 . . . . 5 (((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) → 𝑥 ∈ (2nd𝐴))
777, 76rexlimddv 2493 . . . 4 ((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) → 𝑥 ∈ (2nd𝐴))
784, 77rexlimddv 2493 . . 3 (((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) → 𝑥 ∈ (2nd𝐴))
7978ex 113 . 2 ((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) → (𝑥 ∈ (2nd𝐵) → 𝑥 ∈ (2nd𝐴)))
8079ssrdv 3029 1 ((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) → (2nd𝐵) ⊆ (2nd𝐴))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  w3o 923  w3a 924   = wceq 1289  wcel 1438  wrex 2360  wss 2997  cop 3444   class class class wbr 3837  cfv 5002  (class class class)co 5634  1st c1st 5891  2nd c2nd 5892  Qcnq 6818   +Q cplq 6820   <Q cltq 6823  Pcnp 6829  <P cltp 6833
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-eprel 4107  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-1o 6163  df-2o 6164  df-oadd 6167  df-omul 6168  df-er 6272  df-ec 6274  df-qs 6278  df-ni 6842  df-pli 6843  df-mi 6844  df-lti 6845  df-plpq 6882  df-mpq 6883  df-enq 6885  df-nqqs 6886  df-plqqs 6887  df-mqqs 6888  df-1nqqs 6889  df-rq 6890  df-ltnqqs 6891  df-enq0 6962  df-nq0 6963  df-0nq0 6964  df-plq0 6965  df-mq0 6966  df-inp 7004  df-iltp 7008
This theorem is referenced by:  aptipr  7179
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