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Theorem suplocsr 7629
 Description: An inhabited, bounded, located set of signed reals has a supremum. (Contributed by Jim Kingdon, 22-Jan-2024.)
Hypotheses
Ref Expression
suplocsr.m (𝜑 → ∃𝑥 𝑥𝐴)
suplocsr.ub (𝜑 → ∃𝑥R𝑦𝐴 𝑦 <R 𝑥)
suplocsr.loc (𝜑 → ∀𝑥R𝑦R (𝑥 <R 𝑦 → (∃𝑧𝐴 𝑥 <R 𝑧 ∨ ∀𝑧𝐴 𝑧 <R 𝑦)))
Assertion
Ref Expression
suplocsr (𝜑 → ∃𝑥R (∀𝑦𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦R (𝑦 <R 𝑥 → ∃𝑧𝐴 𝑦 <R 𝑧)))
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧

Proof of Theorem suplocsr
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 suplocsr.m . . 3 (𝜑 → ∃𝑥 𝑥𝐴)
2 eleq1w 2200 . . . 4 (𝑥 = 𝑎 → (𝑥𝐴𝑎𝐴))
32cbvexv 1890 . . 3 (∃𝑥 𝑥𝐴 ↔ ∃𝑎 𝑎𝐴)
41, 3sylib 121 . 2 (𝜑 → ∃𝑎 𝑎𝐴)
5 opeq1 3705 . . . . . . 7 (𝑏 = 𝑐 → ⟨𝑏, 1P⟩ = ⟨𝑐, 1P⟩)
65eceq1d 6465 . . . . . 6 (𝑏 = 𝑐 → [⟨𝑏, 1P⟩] ~R = [⟨𝑐, 1P⟩] ~R )
76oveq2d 5790 . . . . 5 (𝑏 = 𝑐 → (𝑎 +R [⟨𝑏, 1P⟩] ~R ) = (𝑎 +R [⟨𝑐, 1P⟩] ~R ))
87eleq1d 2208 . . . 4 (𝑏 = 𝑐 → ((𝑎 +R [⟨𝑏, 1P⟩] ~R ) ∈ 𝐴 ↔ (𝑎 +R [⟨𝑐, 1P⟩] ~R ) ∈ 𝐴))
98cbvrabv 2685 . . 3 {𝑏P ∣ (𝑎 +R [⟨𝑏, 1P⟩] ~R ) ∈ 𝐴} = {𝑐P ∣ (𝑎 +R [⟨𝑐, 1P⟩] ~R ) ∈ 𝐴}
10 suplocsr.ub . . . . 5 (𝜑 → ∃𝑥R𝑦𝐴 𝑦 <R 𝑥)
11 ltrelsr 7558 . . . . . . . . . 10 <R ⊆ (R × R)
1211brel 4591 . . . . . . . . 9 (𝑦 <R 𝑥 → (𝑦R𝑥R))
1312simpld 111 . . . . . . . 8 (𝑦 <R 𝑥𝑦R)
1413ralimi 2495 . . . . . . 7 (∀𝑦𝐴 𝑦 <R 𝑥 → ∀𝑦𝐴 𝑦R)
15 dfss3 3087 . . . . . . 7 (𝐴R ↔ ∀𝑦𝐴 𝑦R)
1614, 15sylibr 133 . . . . . 6 (∀𝑦𝐴 𝑦 <R 𝑥𝐴R)
1716rexlimivw 2545 . . . . 5 (∃𝑥R𝑦𝐴 𝑦 <R 𝑥𝐴R)
1810, 17syl 14 . . . 4 (𝜑𝐴R)
1918adantr 274 . . 3 ((𝜑𝑎𝐴) → 𝐴R)
20 simpr 109 . . 3 ((𝜑𝑎𝐴) → 𝑎𝐴)
2110adantr 274 . . 3 ((𝜑𝑎𝐴) → ∃𝑥R𝑦𝐴 𝑦 <R 𝑥)
22 suplocsr.loc . . . 4 (𝜑 → ∀𝑥R𝑦R (𝑥 <R 𝑦 → (∃𝑧𝐴 𝑥 <R 𝑧 ∨ ∀𝑧𝐴 𝑧 <R 𝑦)))
2322adantr 274 . . 3 ((𝜑𝑎𝐴) → ∀𝑥R𝑦R (𝑥 <R 𝑦 → (∃𝑧𝐴 𝑥 <R 𝑧 ∨ ∀𝑧𝐴 𝑧 <R 𝑦)))
249, 19, 20, 21, 23suplocsrlem 7628 . 2 ((𝜑𝑎𝐴) → ∃𝑥R (∀𝑦𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦R (𝑦 <R 𝑥 → ∃𝑧𝐴 𝑦 <R 𝑧)))
254, 24exlimddv 1870 1 (𝜑 → ∃𝑥R (∀𝑦𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦R (𝑦 <R 𝑥 → ∃𝑧𝐴 𝑦 <R 𝑧)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 697  ∃wex 1468   ∈ wcel 1480  ∀wral 2416  ∃wrex 2417  {crab 2420   ⊆ wss 3071  ⟨cop 3530   class class class wbr 3929  (class class class)co 5774  [cec 6427  Pcnp 7111  1Pc1p 7112   ~R cer 7116  Rcnr 7117   +R cplr 7121
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