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Theorem suplocexprlemmu 7669
Description: Lemma for suplocexpr 7676. 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 7426 . . . . . . 7 (𝑥P → ⟨(1st𝑥), (2nd𝑥)⟩ ∈ P)
3 prmu 7429 . . . . . . 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 6135 . . . . . . . . . . . . 13 2nd :V–onto→V
7 fofun 5419 . . . . . . . . . . . . 13 (2nd :V–onto→V → Fun 2nd )
86, 7ax-mp 5 . . . . . . . . . . . 12 Fun 2nd
9 fvelima 5546 . . . . . . . . . . . 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 7666 . . . . . . . . . . . . . . 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 3990 . . . . . . . . . . . . . . 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 7547 . . . . . . . . . . . . . . . . 17 <P Or P
25 so2nr 4304 . . . . . . . . . . . . . . . . 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 7591 . . . . . . . . . . . . 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 3836 . . . . . . . 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 7317 . . . . . . . . 9 1QQ
49 addclnq 7326 . . . . . . . . 9 ((𝑠Q ∧ 1QQ) → (𝑠 +Q 1Q) ∈ Q)
5047, 48, 49sylancl 411 . . . . . . . 8 ((𝜑 ∧ (𝑠Q𝑠 (2nd𝐴))) → (𝑠 +Q 1Q) ∈ Q)
51 ltaddnq 7358 . . . . . . . . 9 ((𝑠Q ∧ 1QQ) → 𝑠 <Q (𝑠 +Q 1Q))
5247, 48, 51sylancl 411 . . . . . . . 8 ((𝜑 ∧ (𝑠Q𝑠 (2nd𝐴))) → 𝑠 <Q (𝑠 +Q 1Q))
53 breq2 3991 . . . . . . . . 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 3990 . . . . . . . . 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 7665 . . . . . . . . 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 3991 . . . . . . . . 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 3584   cuni 3794   cint 3829   class class class wbr 3987   Or wor 4278  cima 4612  Fun wfun 5190  ontowfo 5194  cfv 5196  (class class class)co 5851  1st c1st 6115  2nd c2nd 6116  Qcnq 7231  1Qc1q 7232   +Q cplq 7233   <Q cltq 7236  Pcnp 7242  <P cltp 7246
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 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570
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 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-eprel 4272  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-recs 6282  df-irdg 6347  df-1o 6393  df-2o 6394  df-oadd 6397  df-omul 6398  df-er 6510  df-ec 6512  df-qs 6516  df-ni 7255  df-pli 7256  df-mi 7257  df-lti 7258  df-plpq 7295  df-mpq 7296  df-enq 7298  df-nqqs 7299  df-plqqs 7300  df-mqqs 7301  df-1nqqs 7302  df-rq 7303  df-ltnqqs 7304  df-enq0 7375  df-nq0 7376  df-0nq0 7377  df-plq0 7378  df-mq0 7379  df-inp 7417  df-iltp 7421
This theorem is referenced by:  suplocexprlemex  7673
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