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Theorem suplocexprlemmu 7680
Description: Lemma for suplocexpr 7687. 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 7437 . . . . . . 7 (𝑥P → ⟨(1st𝑥), (2nd𝑥)⟩ ∈ P)
3 prmu 7440 . . . . . . 7 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ P → ∃𝑠Q 𝑠 ∈ (2nd𝑥))
42, 3syl 14 . . . . . 6 (𝑥P → ∃𝑠Q 𝑠 ∈ (2nd𝑥))
54ad2antrl 487 . . . . 5 ((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) → ∃𝑠Q 𝑠 ∈ (2nd𝑥))
6 fo2nd 6137 . . . . . . . . . . . . 13 2nd :V–onto→V
7 fofun 5421 . . . . . . . . . . . . 13 (2nd :V–onto→V → Fun 2nd )
86, 7ax-mp 5 . . . . . . . . . . . 12 Fun 2nd
9 fvelima 5548 . . . . . . . . . . . 12 ((Fun 2nd𝑡 ∈ (2nd𝐴)) → ∃𝑢𝐴 (2nd𝑢) = 𝑡)
108, 9mpan 422 . . . . . . . . . . 11 (𝑡 ∈ (2nd𝐴) → ∃𝑢𝐴 (2nd𝑢) = 𝑡)
1110adantl 275 . . . . . . . . . 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 7677 . . . . . . . . . . . . . . 15 (𝜑𝐴P)
1514ad5antr 493 . . . . . . . . . . . . . 14 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → 𝐴P)
16 simprl 526 . . . . . . . . . . . . . 14 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → 𝑢𝐴)
1715, 16sseldd 3148 . . . . . . . . . . . . 13 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → 𝑢P)
18 simprl 526 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) → 𝑥P)
1918ad4antr 491 . . . . . . . . . . . . 13 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → 𝑥P)
20 breq1 3992 . . . . . . . . . . . . . . 15 (𝑦 = 𝑢 → (𝑦<P 𝑥𝑢<P 𝑥))
21 simprr 527 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) → ∀𝑦𝐴 𝑦<P 𝑥)
2221ad4antr 491 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → ∀𝑦𝐴 𝑦<P 𝑥)
2320, 22, 16rspcdva 2839 . . . . . . . . . . . . . 14 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → 𝑢<P 𝑥)
24 ltsopr 7558 . . . . . . . . . . . . . . . . 17 <P Or P
25 so2nr 4306 . . . . . . . . . . . . . . . . 17 ((<P Or P ∧ (𝑢P𝑥P)) → ¬ (𝑢<P 𝑥𝑥<P 𝑢))
2624, 25mpan 422 . . . . . . . . . . . . . . . 16 ((𝑢P𝑥P) → ¬ (𝑢<P 𝑥𝑥<P 𝑢))
2717, 19, 26syl2anc 409 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → ¬ (𝑢<P 𝑥𝑥<P 𝑢))
28 imnan 685 . . . . . . . . . . . . . . 15 ((𝑢<P 𝑥 → ¬ 𝑥<P 𝑢) ↔ ¬ (𝑢<P 𝑥𝑥<P 𝑢))
2927, 28sylibr 133 . . . . . . . . . . . . . 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 7602 . . . . . . . . . . . . 13 ((𝑢P𝑥P ∧ ¬ 𝑥<P 𝑢) → (2nd𝑥) ⊆ (2nd𝑢))
3217, 19, 30, 31syl3anc 1233 . . . . . . . . . . . 12 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → (2nd𝑥) ⊆ (2nd𝑢))
33 simpllr 529 . . . . . . . . . . . 12 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → 𝑠 ∈ (2nd𝑥))
3432, 33sseldd 3148 . . . . . . . . . . 11 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → 𝑠 ∈ (2nd𝑢))
35 simprr 527 . . . . . . . . . . 11 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → (2nd𝑢) = 𝑡)
3634, 35eleqtrd 2249 . . . . . . . . . 10 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → 𝑠𝑡)
3711, 36rexlimddv 2592 . . . . . . . . 9 (((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) → 𝑠𝑡)
3837ralrimiva 2543 . . . . . . . 8 ((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) → ∀𝑡 ∈ (2nd𝐴)𝑠𝑡)
39 vex 2733 . . . . . . . . 9 𝑠 ∈ V
4039elint2 3838 . . . . . . . 8 (𝑠 (2nd𝐴) ↔ ∀𝑡 ∈ (2nd𝐴)𝑠𝑡)
4138, 40sylibr 133 . . . . . . 7 ((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) → 𝑠 (2nd𝐴))
4241ex 114 . . . . . 6 (((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) → (𝑠 ∈ (2nd𝑥) → 𝑠 (2nd𝐴)))
4342reximdva 2572 . . . . 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 2592 . . 3 (𝜑 → ∃𝑠Q 𝑠 (2nd𝐴))
46 simprr 527 . . . . . . 7 ((𝜑 ∧ (𝑠Q𝑠 (2nd𝐴))) → 𝑠 (2nd𝐴))
47 simprl 526 . . . . . . . . 9 ((𝜑 ∧ (𝑠Q𝑠 (2nd𝐴))) → 𝑠Q)
48 1nq 7328 . . . . . . . . 9 1QQ
49 addclnq 7337 . . . . . . . . 9 ((𝑠Q ∧ 1QQ) → (𝑠 +Q 1Q) ∈ Q)
5047, 48, 49sylancl 411 . . . . . . . 8 ((𝜑 ∧ (𝑠Q𝑠 (2nd𝐴))) → (𝑠 +Q 1Q) ∈ Q)
51 ltaddnq 7369 . . . . . . . . 9 ((𝑠Q ∧ 1QQ) → 𝑠 <Q (𝑠 +Q 1Q))
5247, 48, 51sylancl 411 . . . . . . . 8 ((𝜑 ∧ (𝑠Q𝑠 (2nd𝐴))) → 𝑠 <Q (𝑠 +Q 1Q))
53 breq2 3993 . . . . . . . . 9 (𝑗 = (𝑠 +Q 1Q) → (𝑠 <Q 𝑗𝑠 <Q (𝑠 +Q 1Q)))
5453rspcev 2834 . . . . . . . 8 (((𝑠 +Q 1Q) ∈ Q𝑠 <Q (𝑠 +Q 1Q)) → ∃𝑗Q 𝑠 <Q 𝑗)
5550, 52, 54syl2anc 409 . . . . . . 7 ((𝜑 ∧ (𝑠Q𝑠 (2nd𝐴))) → ∃𝑗Q 𝑠 <Q 𝑗)
56 breq1 3992 . . . . . . . . 9 (𝑤 = 𝑠 → (𝑤 <Q 𝑗𝑠 <Q 𝑗))
5756rexbidv 2471 . . . . . . . 8 (𝑤 = 𝑠 → (∃𝑗Q 𝑤 <Q 𝑗 ↔ ∃𝑗Q 𝑠 <Q 𝑗))
5857rspcev 2834 . . . . . . 7 ((𝑠 (2nd𝐴) ∧ ∃𝑗Q 𝑠 <Q 𝑗) → ∃𝑤 (2nd𝐴)∃𝑗Q 𝑤 <Q 𝑗)
5946, 55, 58syl2anc 409 . . . . . 6 ((𝜑 ∧ (𝑠Q𝑠 (2nd𝐴))) → ∃𝑤 (2nd𝐴)∃𝑗Q 𝑤 <Q 𝑗)
60 rexcom 2634 . . . . . 6 (∃𝑤 (2nd𝐴)∃𝑗Q 𝑤 <Q 𝑗 ↔ ∃𝑗Q𝑤 (2nd𝐴)𝑤 <Q 𝑗)
6159, 60sylib 121 . . . . 5 ((𝜑 ∧ (𝑠Q𝑠 (2nd𝐴))) → ∃𝑗Q𝑤 (2nd𝐴)𝑤 <Q 𝑗)
62 ssid 3167 . . . . . 6 QQ
63 rexss 3214 . . . . . 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 121 . . . 4 ((𝜑 ∧ (𝑠Q𝑠 (2nd𝐴))) → ∃𝑗Q (𝑗Q ∧ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑗))
66 suplocexpr.b . . . . . . . . . 10 𝐵 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}⟩
6766suplocexprlem2b 7676 . . . . . . . . 9 (𝐴P → (2nd𝐵) = {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢})
6814, 67syl 14 . . . . . . . 8 (𝜑 → (2nd𝐵) = {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢})
6968eleq2d 2240 . . . . . . 7 (𝜑 → (𝑗 ∈ (2nd𝐵) ↔ 𝑗 ∈ {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}))
70 breq2 3993 . . . . . . . . 9 (𝑢 = 𝑗 → (𝑤 <Q 𝑢𝑤 <Q 𝑗))
7170rexbidv 2471 . . . . . . . 8 (𝑢 = 𝑗 → (∃𝑤 (2nd𝐴)𝑤 <Q 𝑢 ↔ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑗))
7271elrab 2886 . . . . . . 7 (𝑗 ∈ {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢} ↔ (𝑗Q ∧ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑗))
7369, 72bitrdi 195 . . . . . 6 (𝜑 → (𝑗 ∈ (2nd𝐵) ↔ (𝑗Q ∧ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑗)))
7473rexbidv 2471 . . . . 5 (𝜑 → (∃𝑗Q 𝑗 ∈ (2nd𝐵) ↔ ∃𝑗Q (𝑗Q ∧ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑗)))
7574adantr 274 . . . 4 ((𝜑 ∧ (𝑠Q𝑠 (2nd𝐴))) → (∃𝑗Q 𝑗 ∈ (2nd𝐵) ↔ ∃𝑗Q (𝑗Q ∧ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑗)))
7665, 75mpbird 166 . . 3 ((𝜑 ∧ (𝑠Q𝑠 (2nd𝐴))) → ∃𝑗Q 𝑗 ∈ (2nd𝐵))
7745, 76rexlimddv 2592 . 2 (𝜑 → ∃𝑗Q 𝑗 ∈ (2nd𝐵))
78 eleq1w 2231 . . 3 (𝑗 = 𝑠 → (𝑗 ∈ (2nd𝐵) ↔ 𝑠 ∈ (2nd𝐵)))
7978cbvrexv 2697 . 2 (∃𝑗Q 𝑗 ∈ (2nd𝐵) ↔ ∃𝑠Q 𝑠 ∈ (2nd𝐵))
8077, 79sylib 121 1 (𝜑 → ∃𝑠Q 𝑠 ∈ (2nd𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 703   = wceq 1348  wex 1485  wcel 2141  wral 2448  wrex 2449  {crab 2452  Vcvv 2730  wss 3121  cop 3586   cuni 3796   cint 3831   class class class wbr 3989   Or wor 4280  cima 4614  Fun wfun 5192  ontowfo 5196  cfv 5198  (class class class)co 5853  1st c1st 6117  2nd c2nd 6118  Qcnq 7242  1Qc1q 7243   +Q cplq 7244   <Q cltq 7247  Pcnp 7253  <P cltp 7257
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-iltp 7432
This theorem is referenced by:  suplocexprlemex  7684
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