ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  suplocexprlemloc GIF version

Theorem suplocexprlemloc 7719
Description: Lemma for suplocexpr 7723. 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 7413 . . . . 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 7592 . . . . . . . . 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 4007 . . . . . . . . . 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 4007 . . . . . . . . . . . 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 4006 . . . . . . . . . . . 12 (𝑥 = ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩ → (𝑥<P 𝑦 ↔ ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑦))
18 breq1 4006 . . . . . . . . . . . . . 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 7545 . . . . . . . . . . 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 2846 . . . . . . . . 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 7545 . . . . . . . . . 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 2846 . . . . . . . 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 7713 . . . . . . . . . . . . . . 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 3156 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) ∧ 𝑧𝐴) ∧ ⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩<P 𝑧) → 𝑧P)
42 ltdfpr 7504 . . . . . . . . . . . . 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 2740 . . . . . . . . . . . . . 14 𝑤 ∈ V
46 breq2 4007 . . . . . . . . . . . . . 14 (𝑢 = 𝑤 → (𝑞 <Q 𝑢𝑞 <Q 𝑤))
47 ltnqex 7547 . . . . . . . . . . . . . . 15 {𝑙𝑙 <Q 𝑞} ∈ V
48 gtnqex 7548 . . . . . . . . . . . . . . 15 {𝑢𝑞 <Q 𝑢} ∈ V
4947, 48op2nd 6147 . . . . . . . . . . . . . 14 (2nd ‘⟨{𝑙𝑙 <Q 𝑞}, {𝑢𝑞 <Q 𝑢}⟩) = {𝑢𝑞 <Q 𝑢}
5045, 46, 49elab2 2885 . . . . . . . . . . . . 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 7473 . . . . . . . . . . . . . . . . 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 7482 . . . . . . . . . . . . . . . 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 7711 . . . . . . . . . . 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 6158 . . . . . . . . . . . . . . 15 2nd :V–onto→V
72 fofun 5439 . . . . . . . . . . . . . . 15 (2nd :V–onto→V → Fun 2nd )
7371, 72ax-mp 5 . . . . . . . . . . . . . 14 Fun 2nd
74 fvelima 5567 . . . . . . . . . . . . . 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 4006 . . . . . . . . . . . . . . 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 2846 . . . . . . . . . . . . . 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 3156 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑞Q𝑣Q)) ∧ 𝑞 <Q 𝑣) ∧ ∀𝑧𝐴 𝑧<P ⟨{𝑙𝑙 <Q 𝑣}, {𝑢𝑣 <Q 𝑢}⟩) ∧ 𝑠 ∈ (2nd𝐴)) ∧ (𝑡𝐴 ∧ (2nd𝑡) = 𝑠)) → 𝑡P)
84 nqpru 7550 . . . . . . . . . . . . . . 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 2740 . . . . . . . . . . 11 𝑣 ∈ V
9291elint2 3851 . . . . . . . . . 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 4007 . . . . . . . . . 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 4006 . . . . . . . . . . 11 (𝑤 = 𝑣 → (𝑤 <Q 𝑟𝑣 <Q 𝑟))
106105rspcev 2841 . . . . . . . . . 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 2895 . . . . . . . 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 7712 . . . . . . . . . . 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 2737  wss 3129  cop 3595   cuni 3809   cint 3844   class class class wbr 4003  cima 4629  Fun wfun 5210  ontowfo 5214  cfv 5216  1st c1st 6138  2nd c2nd 6139  Qcnq 7278   <Q cltq 7283  Pcnp 7289  <P cltp 7293
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 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-lti 7305  df-plpq 7342  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-mqqs 7348  df-1nqqs 7349  df-rq 7350  df-ltnqqs 7351  df-inp 7464  df-iltp 7468
This theorem is referenced by:  suplocexprlemex  7720
  Copyright terms: Public domain W3C validator