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Theorem suplocexprlemmu 7519
Description: Lemma for suplocexpr 7526. 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 7276 . . . . . . 7 (𝑥P → ⟨(1st𝑥), (2nd𝑥)⟩ ∈ P)
3 prmu 7279 . . . . . . 7 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ P → ∃𝑠Q 𝑠 ∈ (2nd𝑥))
42, 3syl 14 . . . . . 6 (𝑥P → ∃𝑠Q 𝑠 ∈ (2nd𝑥))
54ad2antrl 481 . . . . 5 ((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) → ∃𝑠Q 𝑠 ∈ (2nd𝑥))
6 fo2nd 6049 . . . . . . . . . . . . 13 2nd :V–onto→V
7 fofun 5341 . . . . . . . . . . . . 13 (2nd :V–onto→V → Fun 2nd )
86, 7ax-mp 5 . . . . . . . . . . . 12 Fun 2nd
9 fvelima 5466 . . . . . . . . . . . 12 ((Fun 2nd𝑡 ∈ (2nd𝐴)) → ∃𝑢𝐴 (2nd𝑢) = 𝑡)
108, 9mpan 420 . . . . . . . . . . 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 7516 . . . . . . . . . . . . . . 15 (𝜑𝐴P)
1514ad5antr 487 . . . . . . . . . . . . . 14 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → 𝐴P)
16 simprl 520 . . . . . . . . . . . . . 14 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → 𝑢𝐴)
1715, 16sseldd 3093 . . . . . . . . . . . . 13 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → 𝑢P)
18 simprl 520 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) → 𝑥P)
1918ad4antr 485 . . . . . . . . . . . . 13 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → 𝑥P)
20 breq1 3927 . . . . . . . . . . . . . . 15 (𝑦 = 𝑢 → (𝑦<P 𝑥𝑢<P 𝑥))
21 simprr 521 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) → ∀𝑦𝐴 𝑦<P 𝑥)
2221ad4antr 485 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → ∀𝑦𝐴 𝑦<P 𝑥)
2320, 22, 16rspcdva 2789 . . . . . . . . . . . . . 14 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → 𝑢<P 𝑥)
24 ltsopr 7397 . . . . . . . . . . . . . . . . 17 <P Or P
25 so2nr 4238 . . . . . . . . . . . . . . . . 17 ((<P Or P ∧ (𝑢P𝑥P)) → ¬ (𝑢<P 𝑥𝑥<P 𝑢))
2624, 25mpan 420 . . . . . . . . . . . . . . . 16 ((𝑢P𝑥P) → ¬ (𝑢<P 𝑥𝑥<P 𝑢))
2717, 19, 26syl2anc 408 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → ¬ (𝑢<P 𝑥𝑥<P 𝑢))
28 imnan 679 . . . . . . . . . . . . . . 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 7441 . . . . . . . . . . . . 13 ((𝑢P𝑥P ∧ ¬ 𝑥<P 𝑢) → (2nd𝑥) ⊆ (2nd𝑢))
3217, 19, 30, 31syl3anc 1216 . . . . . . . . . . . 12 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → (2nd𝑥) ⊆ (2nd𝑢))
33 simpllr 523 . . . . . . . . . . . 12 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → 𝑠 ∈ (2nd𝑥))
3432, 33sseldd 3093 . . . . . . . . . . 11 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → 𝑠 ∈ (2nd𝑢))
35 simprr 521 . . . . . . . . . . 11 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → (2nd𝑢) = 𝑡)
3634, 35eleqtrd 2216 . . . . . . . . . 10 ((((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) ∧ (𝑢𝐴 ∧ (2nd𝑢) = 𝑡)) → 𝑠𝑡)
3711, 36rexlimddv 2552 . . . . . . . . 9 (((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) ∧ 𝑡 ∈ (2nd𝐴)) → 𝑠𝑡)
3837ralrimiva 2503 . . . . . . . 8 ((((𝜑 ∧ (𝑥P ∧ ∀𝑦𝐴 𝑦<P 𝑥)) ∧ 𝑠Q) ∧ 𝑠 ∈ (2nd𝑥)) → ∀𝑡 ∈ (2nd𝐴)𝑠𝑡)
39 vex 2684 . . . . . . . . 9 𝑠 ∈ V
4039elint2 3773 . . . . . . . 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 2532 . . . . 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 2552 . . 3 (𝜑 → ∃𝑠Q 𝑠 (2nd𝐴))
46 simprr 521 . . . . . . 7 ((𝜑 ∧ (𝑠Q𝑠 (2nd𝐴))) → 𝑠 (2nd𝐴))
47 simprl 520 . . . . . . . . 9 ((𝜑 ∧ (𝑠Q𝑠 (2nd𝐴))) → 𝑠Q)
48 1nq 7167 . . . . . . . . 9 1QQ
49 addclnq 7176 . . . . . . . . 9 ((𝑠Q ∧ 1QQ) → (𝑠 +Q 1Q) ∈ Q)
5047, 48, 49sylancl 409 . . . . . . . 8 ((𝜑 ∧ (𝑠Q𝑠 (2nd𝐴))) → (𝑠 +Q 1Q) ∈ Q)
51 ltaddnq 7208 . . . . . . . . 9 ((𝑠Q ∧ 1QQ) → 𝑠 <Q (𝑠 +Q 1Q))
5247, 48, 51sylancl 409 . . . . . . . 8 ((𝜑 ∧ (𝑠Q𝑠 (2nd𝐴))) → 𝑠 <Q (𝑠 +Q 1Q))
53 breq2 3928 . . . . . . . . 9 (𝑗 = (𝑠 +Q 1Q) → (𝑠 <Q 𝑗𝑠 <Q (𝑠 +Q 1Q)))
5453rspcev 2784 . . . . . . . 8 (((𝑠 +Q 1Q) ∈ Q𝑠 <Q (𝑠 +Q 1Q)) → ∃𝑗Q 𝑠 <Q 𝑗)
5550, 52, 54syl2anc 408 . . . . . . 7 ((𝜑 ∧ (𝑠Q𝑠 (2nd𝐴))) → ∃𝑗Q 𝑠 <Q 𝑗)
56 breq1 3927 . . . . . . . . 9 (𝑤 = 𝑠 → (𝑤 <Q 𝑗𝑠 <Q 𝑗))
5756rexbidv 2436 . . . . . . . 8 (𝑤 = 𝑠 → (∃𝑗Q 𝑤 <Q 𝑗 ↔ ∃𝑗Q 𝑠 <Q 𝑗))
5857rspcev 2784 . . . . . . 7 ((𝑠 (2nd𝐴) ∧ ∃𝑗Q 𝑠 <Q 𝑗) → ∃𝑤 (2nd𝐴)∃𝑗Q 𝑤 <Q 𝑗)
5946, 55, 58syl2anc 408 . . . . . 6 ((𝜑 ∧ (𝑠Q𝑠 (2nd𝐴))) → ∃𝑤 (2nd𝐴)∃𝑗Q 𝑤 <Q 𝑗)
60 rexcom 2593 . . . . . 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 3112 . . . . . 6 QQ
63 rexss 3159 . . . . . 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 7515 . . . . . . . . 9 (𝐴P → (2nd𝐵) = {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢})
6814, 67syl 14 . . . . . . . 8 (𝜑 → (2nd𝐵) = {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢})
6968eleq2d 2207 . . . . . . 7 (𝜑 → (𝑗 ∈ (2nd𝐵) ↔ 𝑗 ∈ {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}))
70 breq2 3928 . . . . . . . . 9 (𝑢 = 𝑗 → (𝑤 <Q 𝑢𝑤 <Q 𝑗))
7170rexbidv 2436 . . . . . . . 8 (𝑢 = 𝑗 → (∃𝑤 (2nd𝐴)𝑤 <Q 𝑢 ↔ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑗))
7271elrab 2835 . . . . . . 7 (𝑗 ∈ {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢} ↔ (𝑗Q ∧ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑗))
7369, 72syl6bb 195 . . . . . 6 (𝜑 → (𝑗 ∈ (2nd𝐵) ↔ (𝑗Q ∧ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑗)))
7473rexbidv 2436 . . . . 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 2552 . 2 (𝜑 → ∃𝑗Q 𝑗 ∈ (2nd𝐵))
78 eleq1w 2198 . . 3 (𝑗 = 𝑠 → (𝑗 ∈ (2nd𝐵) ↔ 𝑠 ∈ (2nd𝐵)))
7978cbvrexv 2653 . 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 697   = wceq 1331  wex 1468  wcel 1480  wral 2414  wrex 2415  {crab 2418  Vcvv 2681  wss 3066  cop 3525   cuni 3731   cint 3766   class class class wbr 3924   Or wor 4212  cima 4537  Fun wfun 5112  ontowfo 5116  cfv 5118  (class class class)co 5767  1st c1st 6029  2nd c2nd 6030  Qcnq 7081  1Qc1q 7082   +Q cplq 7083   <Q cltq 7086  Pcnp 7092  <P cltp 7096
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-2o 6307  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-pli 7106  df-mi 7107  df-lti 7108  df-plpq 7145  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-plqqs 7150  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154  df-enq0 7225  df-nq0 7226  df-0nq0 7227  df-plq0 7228  df-mq0 7229  df-inp 7267  df-iltp 7271
This theorem is referenced by:  suplocexprlemex  7523
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