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Theorem aptiprlemu 7460
Description: Lemma for aptipr 7461. (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 7295 . . . . . 6 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
2 prnminu 7309 . . . . . 6 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑥 ∈ (2nd𝐵)) → ∃𝑠 ∈ (2nd𝐵)𝑠 <Q 𝑥)
31, 2sylan 281 . . . . 5 ((𝐵P𝑥 ∈ (2nd𝐵)) → ∃𝑠 ∈ (2nd𝐵)𝑠 <Q 𝑥)
433ad2antl2 1144 . . . 4 (((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) → ∃𝑠 ∈ (2nd𝐵)𝑠 <Q 𝑥)
5 simprr 521 . . . . . 6 ((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) → 𝑠 <Q 𝑥)
6 ltexnqi 7229 . . . . . 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 984 . . . . . . . 8 (((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) → 𝐴P)
98ad2antrr 479 . . . . . . 7 (((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) → 𝐴P)
10 simprl 520 . . . . . . 7 (((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) → 𝑡Q)
11 prop 7295 . . . . . . . 8 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
12 prarloc2 7324 . . . . . . . 8 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑡Q) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
1311, 12sylan 281 . . . . . . 7 ((𝐴P𝑡Q) → ∃𝑢 ∈ (1st𝐴)(𝑢 +Q 𝑡) ∈ (2nd𝐴))
149, 10, 13syl2anc 408 . . . . . 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 985 . . . . . . . . . 10 (((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) → 𝐵P)
1615ad3antrrr 483 . . . . . . . . 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 109 . . . . . . . . . 10 (((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) → 𝑥 ∈ (2nd𝐵))
1817ad3antrrr 483 . . . . . . . . 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 7302 . . . . . . . . . 10 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑥 ∈ (2nd𝐵)) → 𝑥Q)
201, 19sylan 281 . . . . . . . . 9 ((𝐵P𝑥 ∈ (2nd𝐵)) → 𝑥Q)
2116, 18, 20syl2anc 408 . . . . . . . 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 483 . . . . . . . . . 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 520 . . . . . . . . . 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 7301 . . . . . . . . . . 11 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑢 ∈ (1st𝐴)) → 𝑢Q)
2511, 24sylan 281 . . . . . . . . . 10 ((𝐴P𝑢 ∈ (1st𝐴)) → 𝑢Q)
2622, 23, 25syl2anc 408 . . . . . . . . 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 274 . . . . . . . . 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 7195 . . . . . . . . 9 ((𝑢Q𝑡Q) → (𝑢 +Q 𝑡) ∈ Q)
2926, 27, 28syl2anc 408 . . . . . . . 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 7216 . . . . . . . 8 ((𝑥Q ∧ (𝑢 +Q 𝑡) ∈ Q) → (𝑥 <Q (𝑢 +Q 𝑡) ∨ 𝑥 = (𝑢 +Q 𝑡) ∨ (𝑢 +Q 𝑡) <Q 𝑥))
3121, 29, 30syl2anc 408 . . . . . . 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 274 . . . . . . . . . . . . . 14 ((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) → 𝐵P)
33 simprl 520 . . . . . . . . . . . . . 14 ((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) → 𝑠 ∈ (2nd𝐵))
34 elprnqu 7302 . . . . . . . . . . . . . . 15 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑠 ∈ (2nd𝐵)) → 𝑠Q)
351, 34sylan 281 . . . . . . . . . . . . . 14 ((𝐵P𝑠 ∈ (2nd𝐵)) → 𝑠Q)
3632, 33, 35syl2anc 408 . . . . . . . . . . . . 13 ((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) → 𝑠Q)
3736ad3antrrr 483 . . . . . . . . . . . 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 483 . . . . . . . . . . . 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 525 . . . . . . . . . . . . . . . 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 3932 . . . . . . . . . . . . . . . . 17 ((𝑠 +Q 𝑡) = 𝑥 → ((𝑠 +Q 𝑡) <Q (𝑢 +Q 𝑡) ↔ 𝑥 <Q (𝑢 +Q 𝑡)))
4140biimprd 157 . . . . . . . . . . . . . . . 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 123 . . . . . . . . . . . . . 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 7220 . . . . . . . . . . . . . . . 16 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
4544adantl 275 . . . . . . . . . . . . . . 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 274 . . . . . . . . . . . . . . 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 274 . . . . . . . . . . . . . . 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 7201 . . . . . . . . . . . . . . . 16 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
4948adantl 275 . . . . . . . . . . . . . . 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 5940 . . . . . . . . . . . . . 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 166 . . . . . . . . . . . . 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 274 . . . . . . . . . . . . . 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 274 . . . . . . . . . . . . . 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 7304 . . . . . . . . . . . . . . 15 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑢 ∈ (1st𝐴)) → (𝑠 <Q 𝑢𝑠 ∈ (1st𝐴)))
5511, 54sylan 281 . . . . . . . . . . . . . 14 ((𝐴P𝑢 ∈ (1st𝐴)) → (𝑠 <Q 𝑢𝑠 ∈ (1st𝐴)))
5652, 53, 55syl2anc 408 . . . . . . . . . . . . 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 2481 . . . . . . . . . . . 12 ((𝑠Q ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝐴))) → ∃𝑠Q (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝐴)))
5937, 38, 57, 58syl12anc 1214 . . . . . . . . . . 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 274 . . . . . . . . . . . 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 7326 . . . . . . . . . . . 12 ((𝐵P𝐴P) → (𝐵<P 𝐴 ↔ ∃𝑠Q (𝑠 ∈ (2nd𝐵) ∧ 𝑠 ∈ (1st𝐴))))
6260, 52, 61syl2anc 408 . . . . . . . . . . 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 166 . . . . . . . . . 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 1022 . . . . . . . . . . 11 ((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) → ¬ 𝐵<P 𝐴)
6564ad3antrrr 483 . . . . . . . . . 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 609 . . . . . . . . 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 114 . . . . . . . 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 521 . . . . . . . . 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 2202 . . . . . . . . 9 (𝑥 = (𝑢 +Q 𝑡) → (𝑥 ∈ (2nd𝐴) ↔ (𝑢 +Q 𝑡) ∈ (2nd𝐴)))
7068, 69syl5ibrcom 156 . . . . . . . 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 7305 . . . . . . . . . 10 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴)) → ((𝑢 +Q 𝑡) <Q 𝑥𝑥 ∈ (2nd𝐴)))
7211, 71sylan 281 . . . . . . . . 9 ((𝐴P ∧ (𝑢 +Q 𝑡) ∈ (2nd𝐴)) → ((𝑢 +Q 𝑡) <Q 𝑥𝑥 ∈ (2nd𝐴)))
7322, 68, 72syl2anc 408 . . . . . . . 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 1282 . . . . . . 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 2554 . . . . 5 (((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) ∧ (𝑡Q ∧ (𝑠 +Q 𝑡) = 𝑥)) → 𝑥 ∈ (2nd𝐴))
777, 76rexlimddv 2554 . . . 4 ((((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) ∧ (𝑠 ∈ (2nd𝐵) ∧ 𝑠 <Q 𝑥)) → 𝑥 ∈ (2nd𝐴))
784, 77rexlimddv 2554 . . 3 (((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) ∧ 𝑥 ∈ (2nd𝐵)) → 𝑥 ∈ (2nd𝐴))
7978ex 114 . 2 ((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) → (𝑥 ∈ (2nd𝐵) → 𝑥 ∈ (2nd𝐴)))
8079ssrdv 3103 1 ((𝐴P𝐵P ∧ ¬ 𝐵<P 𝐴) → (2nd𝐵) ⊆ (2nd𝐴))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  w3o 961  w3a 962   = wceq 1331  wcel 1480  wrex 2417  wss 3071  cop 3530   class class class wbr 3929  cfv 5123  (class class class)co 5774  1st c1st 6036  2nd c2nd 6037  Qcnq 7100   +Q cplq 7102   <Q cltq 7105  Pcnp 7111  <P cltp 7115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7124  df-pli 7125  df-mi 7126  df-lti 7127  df-plpq 7164  df-mpq 7165  df-enq 7167  df-nqqs 7168  df-plqqs 7169  df-mqqs 7170  df-1nqqs 7171  df-rq 7172  df-ltnqqs 7173  df-enq0 7244  df-nq0 7245  df-0nq0 7246  df-plq0 7247  df-mq0 7248  df-inp 7286  df-iltp 7290
This theorem is referenced by:  aptipr  7461  suplocexprlemmu  7538
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