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Theorem suplocexprlemex 7537
 Description: Lemma for suplocexpr 7540. The putative supremum is a positive real. (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
suplocexprlemex (𝜑𝐵P)
Distinct variable groups:   𝑢,𝐴,𝑤,𝑧   𝑥,𝐴,𝑢,𝑦,𝑧   𝑤,𝐵   𝜑,𝑢,𝑤,𝑧   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦,𝑧,𝑢)

Proof of Theorem suplocexprlemex
Dummy variables 𝑞 𝑟 𝑠 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 suplocexpr.m . . . . . 6 (𝜑 → ∃𝑥 𝑥𝐴)
2 suplocexpr.ub . . . . . 6 (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)
3 suplocexpr.loc . . . . . 6 (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))
41, 2, 3suplocexprlemss 7530 . . . . 5 (𝜑𝐴P)
5 suplocexpr.b . . . . . 6 𝐵 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}⟩
65suplocexprlem2b 7529 . . . . 5 (𝐴P → (2nd𝐵) = {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢})
74, 6syl 14 . . . 4 (𝜑 → (2nd𝐵) = {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢})
87opeq2d 3712 . . 3 (𝜑 → ⟨ (1st𝐴), (2nd𝐵)⟩ = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}⟩)
98, 5syl6reqr 2191 . 2 (𝜑𝐵 = ⟨ (1st𝐴), (2nd𝐵)⟩)
10 suplocexprlemell 7528 . . . . . . . . 9 (𝑠 (1st𝐴) ↔ ∃𝑡𝐴 𝑠 ∈ (1st𝑡))
1110biimpi 119 . . . . . . . 8 (𝑠 (1st𝐴) → ∃𝑡𝐴 𝑠 ∈ (1st𝑡))
1211adantl 275 . . . . . . 7 ((𝜑𝑠 (1st𝐴)) → ∃𝑡𝐴 𝑠 ∈ (1st𝑡))
134ad2antrr 479 . . . . . . . . . 10 (((𝜑𝑠 (1st𝐴)) ∧ (𝑡𝐴𝑠 ∈ (1st𝑡))) → 𝐴P)
14 simprl 520 . . . . . . . . . 10 (((𝜑𝑠 (1st𝐴)) ∧ (𝑡𝐴𝑠 ∈ (1st𝑡))) → 𝑡𝐴)
1513, 14sseldd 3098 . . . . . . . . 9 (((𝜑𝑠 (1st𝐴)) ∧ (𝑡𝐴𝑠 ∈ (1st𝑡))) → 𝑡P)
16 prop 7290 . . . . . . . . 9 (𝑡P → ⟨(1st𝑡), (2nd𝑡)⟩ ∈ P)
1715, 16syl 14 . . . . . . . 8 (((𝜑𝑠 (1st𝐴)) ∧ (𝑡𝐴𝑠 ∈ (1st𝑡))) → ⟨(1st𝑡), (2nd𝑡)⟩ ∈ P)
18 simprr 521 . . . . . . . 8 (((𝜑𝑠 (1st𝐴)) ∧ (𝑡𝐴𝑠 ∈ (1st𝑡))) → 𝑠 ∈ (1st𝑡))
19 elprnql 7296 . . . . . . . 8 ((⟨(1st𝑡), (2nd𝑡)⟩ ∈ P𝑠 ∈ (1st𝑡)) → 𝑠Q)
2017, 18, 19syl2anc 408 . . . . . . 7 (((𝜑𝑠 (1st𝐴)) ∧ (𝑡𝐴𝑠 ∈ (1st𝑡))) → 𝑠Q)
2112, 20rexlimddv 2554 . . . . . 6 ((𝜑𝑠 (1st𝐴)) → 𝑠Q)
2221ex 114 . . . . 5 (𝜑 → (𝑠 (1st𝐴) → 𝑠Q))
2322ssrdv 3103 . . . 4 (𝜑 (1st𝐴) ⊆ Q)
24 ssrab2 3182 . . . . 5 {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢} ⊆ Q
257, 24eqsstrdi 3149 . . . 4 (𝜑 → (2nd𝐵) ⊆ Q)
261, 2, 3suplocexprlemml 7531 . . . . 5 (𝜑 → ∃𝑞Q 𝑞 (1st𝐴))
271, 2, 3, 5suplocexprlemmu 7533 . . . . 5 (𝜑 → ∃𝑟Q 𝑟 ∈ (2nd𝐵))
2826, 27jca 304 . . . 4 (𝜑 → (∃𝑞Q 𝑞 (1st𝐴) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐵)))
2923, 25, 28jca31 307 . . 3 (𝜑 → (( (1st𝐴) ⊆ Q ∧ (2nd𝐵) ⊆ Q) ∧ (∃𝑞Q 𝑞 (1st𝐴) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐵))))
301, 2, 3suplocexprlemrl 7532 . . . . 5 (𝜑 → ∀𝑞Q (𝑞 (1st𝐴) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 (1st𝐴))))
311, 2, 3, 5suplocexprlemru 7534 . . . . 5 (𝜑 → ∀𝑟Q (𝑟 ∈ (2nd𝐵) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐵))))
3230, 31jca 304 . . . 4 (𝜑 → (∀𝑞Q (𝑞 (1st𝐴) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 (1st𝐴))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐵) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐵)))))
331, 2, 3, 5suplocexprlemdisj 7535 . . . 4 (𝜑 → ∀𝑞Q ¬ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵)))
341, 2, 3, 5suplocexprlemloc 7536 . . . 4 (𝜑 → ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 (1st𝐴) ∨ 𝑟 ∈ (2nd𝐵))))
3532, 33, 343jca 1161 . . 3 (𝜑 → ((∀𝑞Q (𝑞 (1st𝐴) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 (1st𝐴))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐵) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐵)))) ∧ ∀𝑞Q ¬ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵)) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 (1st𝐴) ∨ 𝑟 ∈ (2nd𝐵)))))
36 elinp 7289 . . 3 (⟨ (1st𝐴), (2nd𝐵)⟩ ∈ P ↔ ((( (1st𝐴) ⊆ Q ∧ (2nd𝐵) ⊆ Q) ∧ (∃𝑞Q 𝑞 (1st𝐴) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐵))) ∧ ((∀𝑞Q (𝑞 (1st𝐴) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 (1st𝐴))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐵) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐵)))) ∧ ∀𝑞Q ¬ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵)) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 (1st𝐴) ∨ 𝑟 ∈ (2nd𝐵))))))
3729, 35, 36sylanbrc 413 . 2 (𝜑 → ⟨ (1st𝐴), (2nd𝐵)⟩ ∈ P)
389, 37eqeltrd 2216 1 (𝜑𝐵P)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 697   ∧ w3a 962   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∀wral 2416  ∃wrex 2417  {crab 2420   ⊆ wss 3071  ⟨cop 3530  ∪ cuni 3736  ∩ cint 3771   class class class wbr 3929   “ cima 4542  ‘cfv 5123  1st c1st 6036  2nd c2nd 6037  Qcnq 7095
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