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Theorem suplocexprlemloc 7720
Description: Lemma for suplocexpr 7724. The putative supremum is located. (Contributed by Jim Kingdon, 9-Jan-2024.)
Hypotheses
Ref Expression
suplocexpr.m (𝜑 → ∃𝑥 𝑥𝐴)
suplocexpr.ub (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)
suplocexpr.loc (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))
suplocexpr.b 𝐵 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}⟩
Assertion
Ref Expression
suplocexprlemloc (𝜑 → ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 (1st𝐴) ∨ 𝑟 ∈ (2nd𝐵))))
Distinct variable groups:   𝑢,𝐴,𝑧,𝑤   𝑥,𝐴,𝑦,𝑢,𝑧   𝑢,𝑞,𝑧,𝑤   𝑥,𝑞,𝑦,𝜑   𝜑,𝑟,𝑤,𝑞   𝜑,𝑧,𝑥,𝑦   𝑢,𝑟
Allowed substitution hints:   𝜑(𝑢)   𝐴(𝑟,𝑞)   𝐵(𝑥,𝑦,𝑧,𝑤,𝑢,𝑟,𝑞)

Proof of Theorem suplocexprlemloc
Dummy variables 𝑠 𝑡 𝑣 𝑙 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 110 . . . . 5 (((𝜑 ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) → 𝑞 <Q 𝑟)
2 ltbtwnnqq 7414 . . . . 5 (𝑞 <Q 𝑟 ↔ ∃𝑣Q (𝑞 <Q 𝑣𝑣 <Q 𝑟))
31, 2sylib 122 . . . 4 (((𝜑 ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) → ∃𝑣Q (𝑞 <Q 𝑣𝑣 <Q 𝑟))
4 simplll 533 . . . . . . 7 ((((𝜑 ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ (𝑣Q ∧ (𝑞 <Q 𝑣𝑣 <Q 𝑟))) → 𝜑)
5 simprl 529 . . . . . . . 8 ((𝜑 ∧ (𝑞Q𝑟Q)) → 𝑞Q)
65ad2antrr 488 . . . . . . 7 ((((𝜑 ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ (𝑣Q ∧ (𝑞 <Q 𝑣𝑣 <Q 𝑟))) → 𝑞Q)
7 simprl 529 . . . . . . 7 ((((𝜑 ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ (𝑣Q ∧ (𝑞 <Q 𝑣𝑣 <Q 𝑟))) → 𝑣Q)
84, 6, 7jca32 310 . . . . . 6 ((((𝜑 ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ (𝑣Q ∧ (𝑞 <Q 𝑣𝑣 <Q 𝑟))) → (𝜑 ∧ (𝑞Q𝑣Q)))
9 simprrl 539 . . . . . 6 ((((𝜑 ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ (𝑣Q ∧ (𝑞 <Q 𝑣𝑣 <Q 𝑟))) → 𝑞 <Q 𝑣)
10 ltnqpri 7593 . . . . . . . . 9 (𝑞 <Q 𝑣 → ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩)
1110adantl 277 . . . . . . . 8 (((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) → ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩)
12 breq2 4008 . . . . . . . . . 10 (𝑦 = ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩ → (⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑦 ↔ ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩))
13 breq2 4008 . . . . . . . . . . . 12 (𝑦 = ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩ → (𝑧<P 𝑦𝑧<P ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩))
1413ralbidv 2477 . . . . . . . . . . 11 (𝑦 = ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩ → (∀𝑧𝐴 𝑧<P 𝑦 ↔ ∀𝑧𝐴 𝑧<P ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩))
1514orbi2d 790 . . . . . . . . . 10 (𝑦 = ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩ → ((∃𝑧𝐴 ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦) ↔ (∃𝑧𝐴 ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩)))
1612, 15imbi12d 234 . . . . . . . . 9 (𝑦 = ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩ → ((⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑦 → (∃𝑧𝐴 ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)) ↔ (⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩ → (∃𝑧𝐴 ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩))))
17 breq1 4007 . . . . . . . . . . . 12 (𝑥 = ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩ → (𝑥<P 𝑦 ↔ ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑦))
18 breq1 4007 . . . . . . . . . . . . . 14 (𝑥 = ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩ → (𝑥<P 𝑧 ↔ ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧))
1918rexbidv 2478 . . . . . . . . . . . . 13 (𝑥 = ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩ → (∃𝑧𝐴 𝑥<P 𝑧 ↔ ∃𝑧𝐴 ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧))
2019orbi1d 791 . . . . . . . . . . . 12 (𝑥 = ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩ → ((∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦) ↔ (∃𝑧𝐴 ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))
2117, 20imbi12d 234 . . . . . . . . . . 11 (𝑥 = ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩ → ((𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)) ↔ (⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑦 → (∃𝑧𝐴 ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦))))
2221ralbidv 2477 . . . . . . . . . 10 (𝑥 = ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩ → (∀𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)) ↔ ∀𝑦P (⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑦 → (∃𝑧𝐴 ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦))))
23 suplocexpr.loc . . . . . . . . . . 11 (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))
2423ad2antrr 488 . . . . . . . . . 10 (((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))
25 simplrl 535 . . . . . . . . . . 11 (((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) → 𝑞Q)
26 nqprlu 7546 . . . . . . . . . . 11 (𝑞Q → ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩ ∈ P)
2725, 26syl 14 . . . . . . . . . 10 (((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) → ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩ ∈ P)
2822, 24, 27rspcdva 2847 . . . . . . . . 9 (((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) → ∀𝑦P (⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑦 → (∃𝑧𝐴 ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))
29 simplrr 536 . . . . . . . . . 10 (((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) → 𝑣Q)
30 nqprlu 7546 . . . . . . . . . 10 (𝑣Q → ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩ ∈ P)
3129, 30syl 14 . . . . . . . . 9 (((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) → ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩ ∈ P)
3216, 28, 31rspcdva 2847 . . . . . . . 8 (((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) → (⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩ → (∃𝑧𝐴 ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩)))
3311, 32mpd 13 . . . . . . 7 (((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) → (∃𝑧𝐴 ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩))
34 simpr 110 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧𝐴) ∧ ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧) → ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧)
3527ad2antrr 488 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧𝐴) ∧ ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧) → ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩ ∈ P)
36 suplocexpr.m . . . . . . . . . . . . . . . 16 (𝜑 → ∃𝑥 𝑥𝐴)
37 suplocexpr.ub . . . . . . . . . . . . . . . 16 (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)
3836, 37, 23suplocexprlemss 7714 . . . . . . . . . . . . . . 15 (𝜑𝐴P)
3938ad4antr 494 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧𝐴) ∧ ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧) → 𝐴P)
40 simplr 528 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧𝐴) ∧ ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧) → 𝑧𝐴)
4139, 40sseldd 3157 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧𝐴) ∧ ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧) → 𝑧P)
42 ltdfpr 7505 . . . . . . . . . . . . 13 ((⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩ ∈ P𝑧P) → (⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧 ↔ ∃𝑤Q (𝑤 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩) ∧ 𝑤 ∈ (1st𝑧))))
4335, 41, 42syl2anc 411 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧𝐴) ∧ ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧) → (⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧 ↔ ∃𝑤Q (𝑤 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩) ∧ 𝑤 ∈ (1st𝑧))))
4434, 43mpbid 147 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧𝐴) ∧ ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧) → ∃𝑤Q (𝑤 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩) ∧ 𝑤 ∈ (1st𝑧)))
45 vex 2741 . . . . . . . . . . . . . 14 𝑤 ∈ V
46 breq2 4008 . . . . . . . . . . . . . 14 (𝑢 = 𝑤 → (𝑞 <Q 𝑢𝑞 <Q 𝑤))
47 ltnqex 7548 . . . . . . . . . . . . . . 15 {𝑙𝑙 <Q 𝑞} ∈ V
48 gtnqex 7549 . . . . . . . . . . . . . . 15 {𝑢𝑞 <Q 𝑢} ∈ V
4947, 48op2nd 6148 . . . . . . . . . . . . . 14 (2nd ‘⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩) = {𝑢𝑞 <Q 𝑢}
5045, 46, 49elab2 2886 . . . . . . . . . . . . 13 (𝑤 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩) ↔ 𝑞 <Q 𝑤)
5150anbi1i 458 . . . . . . . . . . . 12 ((𝑤 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩) ∧ 𝑤 ∈ (1st𝑧)) ↔ (𝑞 <Q 𝑤𝑤 ∈ (1st𝑧)))
5251rexbii 2484 . . . . . . . . . . 11 (∃𝑤Q (𝑤 ∈ (2nd ‘⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩) ∧ 𝑤 ∈ (1st𝑧)) ↔ ∃𝑤Q (𝑞 <Q 𝑤𝑤 ∈ (1st𝑧)))
5344, 52sylib 122 . . . . . . . . . 10 (((((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧𝐴) ∧ ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧) → ∃𝑤Q (𝑞 <Q 𝑤𝑤 ∈ (1st𝑧)))
54 simpllr 534 . . . . . . . . . . . . . 14 ((((((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧𝐴) ∧ ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧) ∧ (𝑤Q ∧ (𝑞 <Q 𝑤𝑤 ∈ (1st𝑧)))) → 𝑧𝐴)
55 simprrl 539 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧𝐴) ∧ ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧) ∧ (𝑤Q ∧ (𝑞 <Q 𝑤𝑤 ∈ (1st𝑧)))) → 𝑞 <Q 𝑤)
5641adantr 276 . . . . . . . . . . . . . . . . 17 ((((((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧𝐴) ∧ ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧) ∧ (𝑤Q ∧ (𝑞 <Q 𝑤𝑤 ∈ (1st𝑧)))) → 𝑧P)
57 prop 7474 . . . . . . . . . . . . . . . . 17 (𝑧P → ⟨(1st𝑧), (2nd𝑧)⟩ ∈ P)
5856, 57syl 14 . . . . . . . . . . . . . . . 16 ((((((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧𝐴) ∧ ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧) ∧ (𝑤Q ∧ (𝑞 <Q 𝑤𝑤 ∈ (1st𝑧)))) → ⟨(1st𝑧), (2nd𝑧)⟩ ∈ P)
59 simprrr 540 . . . . . . . . . . . . . . . 16 ((((((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧𝐴) ∧ ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧) ∧ (𝑤Q ∧ (𝑞 <Q 𝑤𝑤 ∈ (1st𝑧)))) → 𝑤 ∈ (1st𝑧))
60 prcdnql 7483 . . . . . . . . . . . . . . . 16 ((⟨(1st𝑧), (2nd𝑧)⟩ ∈ P𝑤 ∈ (1st𝑧)) → (𝑞 <Q 𝑤𝑞 ∈ (1st𝑧)))
6158, 59, 60syl2anc 411 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧𝐴) ∧ ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧) ∧ (𝑤Q ∧ (𝑞 <Q 𝑤𝑤 ∈ (1st𝑧)))) → (𝑞 <Q 𝑤𝑞 ∈ (1st𝑧)))
6255, 61mpd 13 . . . . . . . . . . . . . 14 ((((((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧𝐴) ∧ ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧) ∧ (𝑤Q ∧ (𝑞 <Q 𝑤𝑤 ∈ (1st𝑧)))) → 𝑞 ∈ (1st𝑧))
6354, 62jca 306 . . . . . . . . . . . . 13 ((((((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧𝐴) ∧ ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧) ∧ (𝑤Q ∧ (𝑞 <Q 𝑤𝑤 ∈ (1st𝑧)))) → (𝑧𝐴𝑞 ∈ (1st𝑧)))
646319.8ad 1591 . . . . . . . . . . . 12 ((((((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧𝐴) ∧ ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧) ∧ (𝑤Q ∧ (𝑞 <Q 𝑤𝑤 ∈ (1st𝑧)))) → ∃𝑧(𝑧𝐴𝑞 ∈ (1st𝑧)))
65 df-rex 2461 . . . . . . . . . . . 12 (∃𝑧𝐴 𝑞 ∈ (1st𝑧) ↔ ∃𝑧(𝑧𝐴𝑞 ∈ (1st𝑧)))
6664, 65sylibr 134 . . . . . . . . . . 11 ((((((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧𝐴) ∧ ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧) ∧ (𝑤Q ∧ (𝑞 <Q 𝑤𝑤 ∈ (1st𝑧)))) → ∃𝑧𝐴 𝑞 ∈ (1st𝑧))
67 suplocexprlemell 7712 . . . . . . . . . . 11 (𝑞 (1st𝐴) ↔ ∃𝑧𝐴 𝑞 ∈ (1st𝑧))
6866, 67sylibr 134 . . . . . . . . . 10 ((((((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧𝐴) ∧ ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧) ∧ (𝑤Q ∧ (𝑞 <Q 𝑤𝑤 ∈ (1st𝑧)))) → 𝑞 (1st𝐴))
6953, 68rexlimddv 2599 . . . . . . . . 9 (((((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧𝐴) ∧ ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧) → 𝑞 (1st𝐴))
7069rexlimdva2 2597 . . . . . . . 8 (((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) → (∃𝑧𝐴 ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧𝑞 (1st𝐴)))
71 fo2nd 6159 . . . . . . . . . . . . . . 15 2nd :V–onto→V
72 fofun 5440 . . . . . . . . . . . . . . 15 (2nd :V–onto→V → Fun 2nd )
7371, 72ax-mp 5 . . . . . . . . . . . . . 14 Fun 2nd
74 fvelima 5568 . . . . . . . . . . . . . 14 ((Fun 2nd𝑠 ∈ (2nd𝐴)) → ∃𝑡𝐴 (2nd𝑡) = 𝑠)
7573, 74mpan 424 . . . . . . . . . . . . 13 (𝑠 ∈ (2nd𝐴) → ∃𝑡𝐴 (2nd𝑡) = 𝑠)
7675adantl 277 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) ∧ ∀𝑧𝐴 𝑧<P ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩) ∧ 𝑠 ∈ (2nd𝐴)) → ∃𝑡𝐴 (2nd𝑡) = 𝑠)
77 breq1 4007 . . . . . . . . . . . . . . 15 (𝑧 = 𝑡 → (𝑧<P ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩ ↔ 𝑡<P ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩))
78 simpllr 534 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) ∧ ∀𝑧𝐴 𝑧<P ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩) ∧ 𝑠 ∈ (2nd𝐴)) ∧ (𝑡𝐴 ∧ (2nd𝑡) = 𝑠)) → ∀𝑧𝐴 𝑧<P ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩)
79 simprl 529 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) ∧ ∀𝑧𝐴 𝑧<P ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩) ∧ 𝑠 ∈ (2nd𝐴)) ∧ (𝑡𝐴 ∧ (2nd𝑡) = 𝑠)) → 𝑡𝐴)
8077, 78, 79rspcdva 2847 . . . . . . . . . . . . . 14 ((((((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) ∧ ∀𝑧𝐴 𝑧<P ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩) ∧ 𝑠 ∈ (2nd𝐴)) ∧ (𝑡𝐴 ∧ (2nd𝑡) = 𝑠)) → 𝑡<P ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩)
8129ad3antrrr 492 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) ∧ ∀𝑧𝐴 𝑧<P ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩) ∧ 𝑠 ∈ (2nd𝐴)) ∧ (𝑡𝐴 ∧ (2nd𝑡) = 𝑠)) → 𝑣Q)
8238ad5antr 496 . . . . . . . . . . . . . . . 16 ((((((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) ∧ ∀𝑧𝐴 𝑧<P ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩) ∧ 𝑠 ∈ (2nd𝐴)) ∧ (𝑡𝐴 ∧ (2nd𝑡) = 𝑠)) → 𝐴P)
8382, 79sseldd 3157 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) ∧ ∀𝑧𝐴 𝑧<P ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩) ∧ 𝑠 ∈ (2nd𝐴)) ∧ (𝑡𝐴 ∧ (2nd𝑡) = 𝑠)) → 𝑡P)
84 nqpru 7551 . . . . . . . . . . . . . . 15 ((𝑣Q𝑡P) → (𝑣 ∈ (2nd𝑡) ↔ 𝑡<P ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩))
8581, 83, 84syl2anc 411 . . . . . . . . . . . . . 14 ((((((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) ∧ ∀𝑧𝐴 𝑧<P ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩) ∧ 𝑠 ∈ (2nd𝐴)) ∧ (𝑡𝐴 ∧ (2nd𝑡) = 𝑠)) → (𝑣 ∈ (2nd𝑡) ↔ 𝑡<P ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩))
8680, 85mpbird 167 . . . . . . . . . . . . 13 ((((((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) ∧ ∀𝑧𝐴 𝑧<P ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩) ∧ 𝑠 ∈ (2nd𝐴)) ∧ (𝑡𝐴 ∧ (2nd𝑡) = 𝑠)) → 𝑣 ∈ (2nd𝑡))
87 simprr 531 . . . . . . . . . . . . 13 ((((((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) ∧ ∀𝑧𝐴 𝑧<P ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩) ∧ 𝑠 ∈ (2nd𝐴)) ∧ (𝑡𝐴 ∧ (2nd𝑡) = 𝑠)) → (2nd𝑡) = 𝑠)
8886, 87eleqtrd 2256 . . . . . . . . . . . 12 ((((((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) ∧ ∀𝑧𝐴 𝑧<P ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩) ∧ 𝑠 ∈ (2nd𝐴)) ∧ (𝑡𝐴 ∧ (2nd𝑡) = 𝑠)) → 𝑣𝑠)
8976, 88rexlimddv 2599 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) ∧ ∀𝑧𝐴 𝑧<P ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩) ∧ 𝑠 ∈ (2nd𝐴)) → 𝑣𝑠)
9089ralrimiva 2550 . . . . . . . . . 10 ((((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) ∧ ∀𝑧𝐴 𝑧<P ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩) → ∀𝑠 ∈ (2nd𝐴)𝑣𝑠)
91 vex 2741 . . . . . . . . . . 11 𝑣 ∈ V
9291elint2 3852 . . . . . . . . . 10 (𝑣 (2nd𝐴) ↔ ∀𝑠 ∈ (2nd𝐴)𝑣𝑠)
9390, 92sylibr 134 . . . . . . . . 9 ((((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) ∧ ∀𝑧𝐴 𝑧<P ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩) → 𝑣 (2nd𝐴))
9493ex 115 . . . . . . . 8 (((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) → (∀𝑧𝐴 𝑧<P ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩ → 𝑣 (2nd𝐴)))
9570, 94orim12d 786 . . . . . . 7 (((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) → ((∃𝑧𝐴 ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩) → (𝑞 (1st𝐴) ∨ 𝑣 (2nd𝐴))))
9633, 95mpd 13 . . . . . 6 (((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) → (𝑞 (1st𝐴) ∨ 𝑣 (2nd𝐴)))
978, 9, 96syl2anc 411 . . . . 5 ((((𝜑 ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ (𝑣Q ∧ (𝑞 <Q 𝑣𝑣 <Q 𝑟))) → (𝑞 (1st𝐴) ∨ 𝑣 (2nd𝐴)))
98 breq2 4008 . . . . . . . . . 10 (𝑢 = 𝑟 → (𝑤 <Q 𝑢𝑤 <Q 𝑟))
9998rexbidv 2478 . . . . . . . . 9 (𝑢 = 𝑟 → (∃𝑤 (2nd𝐴)𝑤 <Q 𝑢 ↔ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑟))
100 simprr 531 . . . . . . . . . 10 ((𝜑 ∧ (𝑞Q𝑟Q)) → 𝑟Q)
101100ad3antrrr 492 . . . . . . . . 9 (((((𝜑 ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ (𝑣Q ∧ (𝑞 <Q 𝑣𝑣 <Q 𝑟))) ∧ 𝑣 (2nd𝐴)) → 𝑟Q)
102 simpr 110 . . . . . . . . . 10 (((((𝜑 ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ (𝑣Q ∧ (𝑞 <Q 𝑣𝑣 <Q 𝑟))) ∧ 𝑣 (2nd𝐴)) → 𝑣 (2nd𝐴))
103 simprrr 540 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ (𝑣Q ∧ (𝑞 <Q 𝑣𝑣 <Q 𝑟))) → 𝑣 <Q 𝑟)
104103adantr 276 . . . . . . . . . 10 (((((𝜑 ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ (𝑣Q ∧ (𝑞 <Q 𝑣𝑣 <Q 𝑟))) ∧ 𝑣 (2nd𝐴)) → 𝑣 <Q 𝑟)
105 breq1 4007 . . . . . . . . . . 11 (𝑤 = 𝑣 → (𝑤 <Q 𝑟𝑣 <Q 𝑟))
106105rspcev 2842 . . . . . . . . . 10 ((𝑣 (2nd𝐴) ∧ 𝑣 <Q 𝑟) → ∃𝑤 (2nd𝐴)𝑤 <Q 𝑟)
107102, 104, 106syl2anc 411 . . . . . . . . 9 (((((𝜑 ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ (𝑣Q ∧ (𝑞 <Q 𝑣𝑣 <Q 𝑟))) ∧ 𝑣 (2nd𝐴)) → ∃𝑤 (2nd𝐴)𝑤 <Q 𝑟)
10899, 101, 107elrabd 2896 . . . . . . . 8 (((((𝜑 ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ (𝑣Q ∧ (𝑞 <Q 𝑣𝑣 <Q 𝑟))) ∧ 𝑣 (2nd𝐴)) → 𝑟 ∈ {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢})
109 suplocexpr.b . . . . . . . . . . . 12 𝐵 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}⟩
110109suplocexprlem2b 7713 . . . . . . . . . . 11 (𝐴P → (2nd𝐵) = {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢})
11138, 110syl 14 . . . . . . . . . 10 (𝜑 → (2nd𝐵) = {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢})
112111eleq2d 2247 . . . . . . . . 9 (𝜑 → (𝑟 ∈ (2nd𝐵) ↔ 𝑟 ∈ {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}))
113112ad4antr 494 . . . . . . . 8 (((((𝜑 ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ (𝑣Q ∧ (𝑞 <Q 𝑣𝑣 <Q 𝑟))) ∧ 𝑣 (2nd𝐴)) → (𝑟 ∈ (2nd𝐵) ↔ 𝑟 ∈ {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}))
114108, 113mpbird 167 . . . . . . 7 (((((𝜑 ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ (𝑣Q ∧ (𝑞 <Q 𝑣𝑣 <Q 𝑟))) ∧ 𝑣 (2nd𝐴)) → 𝑟 ∈ (2nd𝐵))
115114ex 115 . . . . . 6 ((((𝜑 ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ (𝑣Q ∧ (𝑞 <Q 𝑣𝑣 <Q 𝑟))) → (𝑣 (2nd𝐴) → 𝑟 ∈ (2nd𝐵)))
116115orim2d 788 . . . . 5 ((((𝜑 ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ (𝑣Q ∧ (𝑞 <Q 𝑣𝑣 <Q 𝑟))) → ((𝑞 (1st𝐴) ∨ 𝑣 (2nd𝐴)) → (𝑞 (1st𝐴) ∨ 𝑟 ∈ (2nd𝐵))))
11797, 116mpd 13 . . . 4 ((((𝜑 ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) ∧ (𝑣Q ∧ (𝑞 <Q 𝑣𝑣 <Q 𝑟))) → (𝑞 (1st𝐴) ∨ 𝑟 ∈ (2nd𝐵)))
1183, 117rexlimddv 2599 . . 3 (((𝜑 ∧ (𝑞Q𝑟Q)) ∧ 𝑞 <Q 𝑟) → (𝑞 (1st𝐴) ∨ 𝑟 ∈ (2nd𝐵)))
119118ex 115 . 2 ((𝜑 ∧ (𝑞Q𝑟Q)) → (𝑞 <Q 𝑟 → (𝑞 (1st𝐴) ∨ 𝑟 ∈ (2nd𝐵))))
120119ralrimivva 2559 1 (𝜑 → ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 (1st𝐴) ∨ 𝑟 ∈ (2nd𝐵))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wo 708   = wceq 1353  wex 1492  wcel 2148  {cab 2163  wral 2455  wrex 2456  {crab 2459  Vcvv 2738  wss 3130  cop 3596   cuni 3810   cint 3845   class class class wbr 4004  cima 4630  Fun wfun 5211  ontowfo 5215  cfv 5217  1st c1st 6139  2nd c2nd 6140  Qcnq 7279   <Q cltq 7284  Pcnp 7290  <P cltp 7294
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-eprel 4290  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-1o 6417  df-oadd 6421  df-omul 6422  df-er 6535  df-ec 6537  df-qs 6541  df-ni 7303  df-pli 7304  df-mi 7305  df-lti 7306  df-plpq 7343  df-mpq 7344  df-enq 7346  df-nqqs 7347  df-plqqs 7348  df-mqqs 7349  df-1nqqs 7350  df-rq 7351  df-ltnqqs 7352  df-inp 7465  df-iltp 7469
This theorem is referenced by:  suplocexprlemex  7721
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