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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   <Q cltq 7100  Pcnp 7106  <P cltp 7110
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 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
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 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7119  df-pli 7120  df-mi 7121  df-lti 7122  df-plpq 7159  df-mpq 7160  df-enq 7162  df-nqqs 7163  df-plqqs 7164  df-mqqs 7165  df-1nqqs 7166  df-rq 7167  df-ltnqqs 7168  df-enq0 7239  df-nq0 7240  df-0nq0 7241  df-plq0 7242  df-mq0 7243  df-inp 7281  df-iltp 7285
This theorem is referenced by:  suplocexprlemub  7538  suplocexpr  7540
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