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Theorem aptiprlemu 7658
Description: Lemma for aptipr 7659. (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 7493 . . . . . 6 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
2 prnminu 7507 . . . . . 6 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑥 ∈ (2nd𝐵)) → ∃𝑠 ∈ (2nd𝐵)𝑠 <Q 𝑥)
31, 2sylan 283 . . . . 5 ((𝐵P𝑥 ∈ (2nd𝐵)) → ∃𝑠 ∈ (2nd𝐵)𝑠 <Q 𝑥)
433ad2antl2 1162 . . . 4 (((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) → ∃𝑠 ∈ (2nd𝐵)𝑠 <Q 𝑥)
5 simprr 531 . . . . . 6 ((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) → 𝑠 <Q 𝑥)
6 ltexnqi 7427 . . . . . 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 1002 . . . . . . . 8 (((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) → 𝐴P)
98ad2antrr 488 . . . . . . 7 (((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) → 𝐴P)
10 simprl 529 . . . . . . 7 (((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) → 𝑡Q)
11 prop 7493 . . . . . . . 8 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
12 prarloc2 7522 . . . . . . . 8 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑡Q) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
1311, 12sylan 283 . . . . . . 7 ((𝐴P𝑡Q) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
149, 10, 13syl2anc 411 . . . . . 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 1003 . . . . . . . . . 10 (((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) → 𝐵P)
1615ad3antrrr 492 . . . . . . . . 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 110 . . . . . . . . . 10 (((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) → 𝑥 ∈ (2nd𝐵))
1817ad3antrrr 492 . . . . . . . . 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 7500 . . . . . . . . . 10 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑥 ∈ (2nd𝐵)) → 𝑥Q)
201, 19sylan 283 . . . . . . . . 9 ((𝐵P𝑥 ∈ (2nd𝐵)) → 𝑥Q)
2116, 18, 20syl2anc 411 . . . . . . . 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 492 . . . . . . . . . 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 529 . . . . . . . . . 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 7499 . . . . . . . . . . 11 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑢 ∈ (1st𝐴)) → 𝑢Q)
2511, 24sylan 283 . . . . . . . . . 10 ((𝐴P𝑢 ∈ (1st𝐴)) → 𝑢Q)
2622, 23, 25syl2anc 411 . . . . . . . . 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 276 . . . . . . . . 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 7393 . . . . . . . . 9 ((𝑢Q𝑡Q) → (𝑢 +Q 𝑡) ∈ Q)
2926, 27, 28syl2anc 411 . . . . . . . 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 7414 . . . . . . . 8 ((𝑥Q ∧ (𝑢 +Q 𝑡) ∈ Q) → (𝑥 <Q (𝑢 +Q 𝑡) ∨ 𝑥 = (𝑢 +Q 𝑡) ∨ (𝑢 +Q 𝑡) <Q 𝑥))
3121, 29, 30syl2anc 411 . . . . . . 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 276 . . . . . . . . . . . . . 14 ((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) → 𝐵P)
33 simprl 529 . . . . . . . . . . . . . 14 ((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) → 𝑠 ∈ (2nd𝐵))
34 elprnqu 7500 . . . . . . . . . . . . . . 15 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑠 ∈ (2nd𝐵)) → 𝑠Q)
351, 34sylan 283 . . . . . . . . . . . . . 14 ((𝐵P𝑠 ∈ (2nd𝐵)) → 𝑠Q)
3632, 33, 35syl2anc 411 . . . . . . . . . . . . 13 ((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) → 𝑠Q)
3736ad3antrrr 492 . . . . . . . . . . . 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 492 . . . . . . . . . . . 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 536 . . . . . . . . . . . . . . . 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 4021 . . . . . . . . . . . . . . . . 17 ((𝑠 +Q 𝑡) = 𝑥 → ((𝑠 +Q 𝑡) <Q (𝑢 +Q 𝑡) ↔ 𝑥 <Q (𝑢 +Q 𝑡)))
4140biimprd 158 . . . . . . . . . . . . . . . 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 124 . . . . . . . . . . . . . 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 7418 . . . . . . . . . . . . . . . 16 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
4544adantl 277 . . . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . . 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 7399 . . . . . . . . . . . . . . . 16 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
4948adantl 277 . . . . . . . . . . . . . . 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 6061 . . . . . . . . . . . . . 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 167 . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . 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 7502 . . . . . . . . . . . . . . 15 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑢 ∈ (1st𝐴)) → (𝑠 <Q 𝑢𝑠 ∈ (1st𝐴)))
5511, 54sylan 283 . . . . . . . . . . . . . 14 ((𝐴P𝑢 ∈ (1st𝐴)) → (𝑠 <Q 𝑢𝑠 ∈ (1st𝐴)))
5652, 53, 55syl2anc 411 . . . . . . . . . . . . 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 2539 . . . . . . . . . . . 12 ((𝑠Q ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝐴))) → ∃𝑠Q (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝐴)))
5937, 38, 57, 58syl12anc 1247 . . . . . . . . . . 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 276 . . . . . . . . . . . 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 7524 . . . . . . . . . . . 12 ((𝐵P𝐴P) → (𝐵<P 𝐴 ↔ ∃𝑠Q (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝐴))))
6260, 52, 61syl2anc 411 . . . . . . . . . . 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 167 . . . . . . . . . 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 1040 . . . . . . . . . . 11 ((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) → ¬ 𝐵<P 𝐴)
6564ad3antrrr 492 . . . . . . . . . 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 621 . . . . . . . . 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 115 . . . . . . . 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 531 . . . . . . . . 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 2252 . . . . . . . . 9 (𝑥 = (𝑢 +Q 𝑡) → (𝑥 ∈ (2nd𝐴) ↔ (𝑢 +Q 𝑡) ∈ (2nd𝐴)))
7068, 69syl5ibrcom 157 . . . . . . . 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 7503 . . . . . . . . . 10 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴)) → ((𝑢 +Q 𝑡) <Q 𝑥𝑥 ∈ (2nd𝐴)))
7211, 71sylan 283 . . . . . . . . 9 ((𝐴P ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴)) → ((𝑢 +Q 𝑡) <Q 𝑥𝑥 ∈ (2nd𝐴)))
7322, 68, 72syl2anc 411 . . . . . . . 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 1315 . . . . . . 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 2612 . . . . 5 (((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) → 𝑥 ∈ (2nd𝐴))
777, 76rexlimddv 2612 . . . 4 ((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) → 𝑥 ∈ (2nd𝐴))
784, 77rexlimddv 2612 . . 3 (((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) → 𝑥 ∈ (2nd𝐴))
7978ex 115 . 2 ((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) → (𝑥 ∈ (2nd𝐵) → 𝑥 ∈ (2nd𝐴)))
8079ssrdv 3176 1 ((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) → (2nd𝐵) ⊆ (2nd𝐴))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  w3o 979  w3a 980   = wceq 1364  wcel 2160  wrex 2469  wss 3144  cop 3610   class class class wbr 4018  cfv 5231  (class class class)co 5891  1st c1st 6157  2nd c2nd 6158  Qcnq 7298   +Q cplq 7300   <Q cltq 7303  Pcnp 7309  <P cltp 7313
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4304  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-irdg 6389  df-1o 6435  df-2o 6436  df-oadd 6439  df-omul 6440  df-er 6553  df-ec 6555  df-qs 6559  df-ni 7322  df-pli 7323  df-mi 7324  df-lti 7325  df-plpq 7362  df-mpq 7363  df-enq 7365  df-nqqs 7366  df-plqqs 7367  df-mqqs 7368  df-1nqqs 7369  df-rq 7370  df-ltnqqs 7371  df-enq0 7442  df-nq0 7443  df-0nq0 7444  df-plq0 7445  df-mq0 7446  df-inp 7484  df-iltp 7488
This theorem is referenced by:  aptipr  7659  suplocexprlemmu  7736
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