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Theorem suplocexprlemmu 7708
Description: Lemma for suplocexpr 7715. The upper cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-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
suplocexprlemmu (𝜑 → ∃𝑠Q 𝑠 ∈ (2nd𝐵))
Distinct variable groups:   𝐴,𝑠,𝑢,𝑤   𝑥,𝐴,𝑦,𝑠,𝑢   𝐵,𝑠   𝜑,𝑠,𝑢,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑧,𝑤)   𝐴(𝑧)   𝐵(𝑥,𝑦,𝑧,𝑤,𝑢)

Proof of Theorem suplocexprlemmu
Dummy variables 𝑗 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 suplocexpr.ub . . . 4 (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)
2 prop 7465 . . . . . . 7 (𝑥P → ⟨(1st𝑥), (2nd𝑥)⟩ ∈ P)
3 prmu 7468 . . . . . . 7 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ P → ∃𝑠Q 𝑠 ∈ (2nd𝑥))
42, 3syl 14 . . . . . 6 (𝑥P → ∃𝑠Q 𝑠 ∈ (2nd𝑥))
54ad2antrl 490 . . . . 5 ((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) → ∃𝑠Q 𝑠 ∈ (2nd𝑥))
6 fo2nd 6153 . . . . . . . . . . . . 13 2nd :V–onto→V
7 fofun 5435 . . . . . . . . . . . . 13 (2nd :V–onto→V → Fun 2nd )
86, 7ax-mp 5 . . . . . . . . . . . 12 Fun 2nd
9 fvelima 5563 . . . . . . . . . . . 12 ((Fun 2nd𝑡 ∈ (2nd𝐴)) → ∃𝑢𝐴 (2nd𝑢) = 𝑡)
108, 9mpan 424 . . . . . . . . . . 11 (𝑡 ∈ (2nd𝐴) → ∃𝑢𝐴 (2nd𝑢) = 𝑡)
1110adantl 277 . . . . . . . . . 10 (((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) → ∃𝑢𝐴 (2nd𝑢) = 𝑡)
12 suplocexpr.m . . . . . . . . . . . . . . . 16 (𝜑 → ∃𝑥 𝑥𝐴)
13 suplocexpr.loc . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))
1412, 1, 13suplocexprlemss 7705 . . . . . . . . . . . . . . 15 (𝜑𝐴P)
1514ad5antr 496 . . . . . . . . . . . . . 14 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → 𝐴P)
16 simprl 529 . . . . . . . . . . . . . 14 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → 𝑢𝐴)
1715, 16sseldd 3156 . . . . . . . . . . . . 13 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → 𝑢P)
18 simprl 529 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) → 𝑥P)
1918ad4antr 494 . . . . . . . . . . . . 13 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → 𝑥P)
20 breq1 4003 . . . . . . . . . . . . . . 15 (𝑦 = 𝑢 → (𝑦<P 𝑥𝑢<P 𝑥))
21 simprr 531 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) → ∀𝑦𝐴 𝑦<P 𝑥)
2221ad4antr 494 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → ∀𝑦𝐴 𝑦<P 𝑥)
2320, 22, 16rspcdva 2846 . . . . . . . . . . . . . 14 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → 𝑢<P 𝑥)
24 ltsopr 7586 . . . . . . . . . . . . . . . . 17 <P Or P
25 so2nr 4318 . . . . . . . . . . . . . . . . 17 ((<P Or P ∧ (𝑢P𝑥P)) → ¬ (𝑢<P 𝑥𝑥<P 𝑢))
2624, 25mpan 424 . . . . . . . . . . . . . . . 16 ((𝑢P𝑥P) → ¬ (𝑢<P 𝑥𝑥<P 𝑢))
2717, 19, 26syl2anc 411 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → ¬ (𝑢<P 𝑥𝑥<P 𝑢))
28 imnan 690 . . . . . . . . . . . . . . 15 ((𝑢<P 𝑥 → ¬ 𝑥<P 𝑢) ↔ ¬ (𝑢<P 𝑥𝑥<P 𝑢))
2927, 28sylibr 134 . . . . . . . . . . . . . 14 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → (𝑢<P 𝑥 → ¬ 𝑥<P 𝑢))
3023, 29mpd 13 . . . . . . . . . . . . 13 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → ¬ 𝑥<P 𝑢)
31 aptiprlemu 7630 . . . . . . . . . . . . 13 ((𝑢P𝑥P ∧ ¬ 𝑥<P 𝑢) → (2nd𝑥) ⊆ (2nd𝑢))
3217, 19, 30, 31syl3anc 1238 . . . . . . . . . . . 12 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → (2nd𝑥) ⊆ (2nd𝑢))
33 simpllr 534 . . . . . . . . . . . 12 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → 𝑠 ∈ (2nd𝑥))
3432, 33sseldd 3156 . . . . . . . . . . 11 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → 𝑠 ∈ (2nd𝑢))
35 simprr 531 . . . . . . . . . . 11 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → (2nd𝑢) = 𝑡)
3634, 35eleqtrd 2256 . . . . . . . . . 10 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → 𝑠𝑡)
3711, 36rexlimddv 2599 . . . . . . . . 9 (((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) → 𝑠𝑡)
3837ralrimiva 2550 . . . . . . . 8 ((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) → ∀𝑡 ∈ (2nd𝐴)𝑠𝑡)
39 vex 2740 . . . . . . . . 9 𝑠 ∈ V
4039elint2 3849 . . . . . . . 8 (𝑠 (2nd𝐴) ↔ ∀𝑡 ∈ (2nd𝐴)𝑠𝑡)
4138, 40sylibr 134 . . . . . . 7 ((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) → 𝑠 (2nd𝐴))
4241ex 115 . . . . . 6 (((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) → (𝑠 ∈ (2nd𝑥) → 𝑠 (2nd𝐴)))
4342reximdva 2579 . . . . 5 ((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) → (∃𝑠Q 𝑠 ∈ (2nd𝑥) → ∃𝑠Q 𝑠 (2nd𝐴)))
445, 43mpd 13 . . . 4 ((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) → ∃𝑠Q 𝑠 (2nd𝐴))
451, 44rexlimddv 2599 . . 3 (𝜑 → ∃𝑠Q 𝑠 (2nd𝐴))
46 simprr 531 . . . . . . 7 ((𝜑 ∧ (𝑠Q𝑠 (2nd𝐴))) → 𝑠 (2nd𝐴))
47 simprl 529 . . . . . . . . 9 ((𝜑 ∧ (𝑠Q𝑠 (2nd𝐴))) → 𝑠Q)
48 1nq 7356 . . . . . . . . 9 1QQ
49 addclnq 7365 . . . . . . . . 9 ((𝑠Q ∧ 1QQ) → (𝑠 +Q 1Q) ∈ Q)
5047, 48, 49sylancl 413 . . . . . . . 8 ((𝜑 ∧ (𝑠Q𝑠 (2nd𝐴))) → (𝑠 +Q 1Q) ∈ Q)
51 ltaddnq 7397 . . . . . . . . 9 ((𝑠Q ∧ 1QQ) → 𝑠 <Q (𝑠 +Q 1Q))
5247, 48, 51sylancl 413 . . . . . . . 8 ((𝜑 ∧ (𝑠Q𝑠 (2nd𝐴))) → 𝑠 <Q (𝑠 +Q 1Q))
53 breq2 4004 . . . . . . . . 9 (𝑗 = (𝑠 +Q 1Q) → (𝑠 <Q 𝑗𝑠 <Q (𝑠 +Q 1Q)))
5453rspcev 2841 . . . . . . . 8 (((𝑠 +Q 1Q) ∈ Q𝑠 <Q (𝑠 +Q 1Q)) → ∃𝑗Q 𝑠 <Q 𝑗)
5550, 52, 54syl2anc 411 . . . . . . 7 ((𝜑 ∧ (𝑠Q𝑠 (2nd𝐴))) → ∃𝑗Q 𝑠 <Q 𝑗)
56 breq1 4003 . . . . . . . . 9 (𝑤 = 𝑠 → (𝑤 <Q 𝑗𝑠 <Q 𝑗))
5756rexbidv 2478 . . . . . . . 8 (𝑤 = 𝑠 → (∃𝑗Q 𝑤 <Q 𝑗 ↔ ∃𝑗Q 𝑠 <Q 𝑗))
5857rspcev 2841 . . . . . . 7 ((𝑠 (2nd𝐴) ∧ ∃𝑗Q 𝑠 <Q 𝑗) → ∃𝑤 (2nd𝐴)∃𝑗Q 𝑤 <Q 𝑗)
5946, 55, 58syl2anc 411 . . . . . 6 ((𝜑 ∧ (𝑠Q𝑠 (2nd𝐴))) → ∃𝑤 (2nd𝐴)∃𝑗Q 𝑤 <Q 𝑗)
60 rexcom 2641 . . . . . 6 (∃𝑤 (2nd𝐴)∃𝑗Q 𝑤 <Q 𝑗 ↔ ∃𝑗Q𝑤 (2nd𝐴)𝑤 <Q 𝑗)
6159, 60sylib 122 . . . . 5 ((𝜑 ∧ (𝑠Q𝑠 (2nd𝐴))) → ∃𝑗Q𝑤 (2nd𝐴)𝑤 <Q 𝑗)
62 ssid 3175 . . . . . 6 QQ
63 rexss 3222 . . . . . 6 (QQ → (∃𝑗Q𝑤 (2nd𝐴)𝑤 <Q 𝑗 ↔ ∃𝑗Q (𝑗Q ∧ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑗)))
6462, 63ax-mp 5 . . . . 5 (∃𝑗Q𝑤 (2nd𝐴)𝑤 <Q 𝑗 ↔ ∃𝑗Q (𝑗Q ∧ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑗))
6561, 64sylib 122 . . . 4 ((𝜑 ∧ (𝑠Q𝑠 (2nd𝐴))) → ∃𝑗Q (𝑗Q ∧ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑗))
66 suplocexpr.b . . . . . . . . . 10 𝐵 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}⟩
6766suplocexprlem2b 7704 . . . . . . . . 9 (𝐴P → (2nd𝐵) = {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢})
6814, 67syl 14 . . . . . . . 8 (𝜑 → (2nd𝐵) = {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢})
6968eleq2d 2247 . . . . . . 7 (𝜑 → (𝑗 ∈ (2nd𝐵) ↔ 𝑗 ∈ {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}))
70 breq2 4004 . . . . . . . . 9 (𝑢 = 𝑗 → (𝑤 <Q 𝑢𝑤 <Q 𝑗))
7170rexbidv 2478 . . . . . . . 8 (𝑢 = 𝑗 → (∃𝑤 (2nd𝐴)𝑤 <Q 𝑢 ↔ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑗))
7271elrab 2893 . . . . . . 7 (𝑗 ∈ {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢} ↔ (𝑗Q ∧ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑗))
7369, 72bitrdi 196 . . . . . 6 (𝜑 → (𝑗 ∈ (2nd𝐵) ↔ (𝑗Q ∧ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑗)))
7473rexbidv 2478 . . . . 5 (𝜑 → (∃𝑗Q 𝑗 ∈ (2nd𝐵) ↔ ∃𝑗Q (𝑗Q ∧ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑗)))
7574adantr 276 . . . 4 ((𝜑 ∧ (𝑠Q𝑠 (2nd𝐴))) → (∃𝑗Q 𝑗 ∈ (2nd𝐵) ↔ ∃𝑗Q (𝑗Q ∧ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑗)))
7665, 75mpbird 167 . . 3 ((𝜑 ∧ (𝑠Q𝑠 (2nd𝐴))) → ∃𝑗Q 𝑗 ∈ (2nd𝐵))
7745, 76rexlimddv 2599 . 2 (𝜑 → ∃𝑗Q 𝑗 ∈ (2nd𝐵))
78 eleq1w 2238 . . 3 (𝑗 = 𝑠 → (𝑗 ∈ (2nd𝐵) ↔ 𝑠 ∈ (2nd𝐵)))
7978cbvrexv 2704 . 2 (∃𝑗Q 𝑗 ∈ (2nd𝐵) ↔ ∃𝑠Q 𝑠 ∈ (2nd𝐵))
8077, 79sylib 122 1 (𝜑 → ∃𝑠Q 𝑠 ∈ (2nd𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708   = wceq 1353  wex 1492  wcel 2148  wral 2455  wrex 2456  {crab 2459  Vcvv 2737  wss 3129  cop 3594   cuni 3807   cint 3842   class class class wbr 4000   Or wor 4292  cima 4626  Fun wfun 5206  ontowfo 5210  cfv 5212  (class class class)co 5869  1st c1st 6133  2nd c2nd 6134  Qcnq 7270  1Qc1q 7271   +Q cplq 7272   <Q cltq 7275  Pcnp 7281  <P cltp 7285
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 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
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 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-2o 6412  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343  df-enq0 7414  df-nq0 7415  df-0nq0 7416  df-plq0 7417  df-mq0 7418  df-inp 7456  df-iltp 7460
This theorem is referenced by:  suplocexprlemex  7712
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