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Theorem suplocexprlemex 7671
Description: Lemma for suplocexpr 7674. 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.b . . 3 𝐵 = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}⟩
2 suplocexpr.m . . . . . 6 (𝜑 → ∃𝑥 𝑥𝐴)
3 suplocexpr.ub . . . . . 6 (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)
4 suplocexpr.loc . . . . . 6 (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))
52, 3, 4suplocexprlemss 7664 . . . . 5 (𝜑𝐴P)
61suplocexprlem2b 7663 . . . . 5 (𝐴P → (2nd𝐵) = {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢})
75, 6syl 14 . . . 4 (𝜑 → (2nd𝐵) = {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢})
87opeq2d 3770 . . 3 (𝜑 → ⟨ (1st𝐴), (2nd𝐵)⟩ = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}⟩)
91, 8eqtr4id 2222 . 2 (𝜑𝐵 = ⟨ (1st𝐴), (2nd𝐵)⟩)
10 suplocexprlemell 7662 . . . . . . . . 9 (𝑠 (1st𝐴) ↔ ∃𝑡𝐴 𝑠 ∈ (1st𝑡))
1110biimpi 119 . . . . . . . 8 (𝑠 (1st𝐴) → ∃𝑡𝐴 𝑠 ∈ (1st𝑡))
1211adantl 275 . . . . . . 7 ((𝜑𝑠 (1st𝐴)) → ∃𝑡𝐴 𝑠 ∈ (1st𝑡))
135ad2antrr 485 . . . . . . . . . 10 (((𝜑𝑠 (1st𝐴)) ∧ (𝑡𝐴𝑠 ∈ (1st𝑡))) → 𝐴P)
14 simprl 526 . . . . . . . . . 10 (((𝜑𝑠 (1st𝐴)) ∧ (𝑡𝐴𝑠 ∈ (1st𝑡))) → 𝑡𝐴)
1513, 14sseldd 3148 . . . . . . . . 9 (((𝜑𝑠 (1st𝐴)) ∧ (𝑡𝐴𝑠 ∈ (1st𝑡))) → 𝑡P)
16 prop 7424 . . . . . . . . 9 (𝑡P → ⟨(1st𝑡), (2nd𝑡)⟩ ∈ P)
1715, 16syl 14 . . . . . . . 8 (((𝜑𝑠 (1st𝐴)) ∧ (𝑡𝐴𝑠 ∈ (1st𝑡))) → ⟨(1st𝑡), (2nd𝑡)⟩ ∈ P)
18 simprr 527 . . . . . . . 8 (((𝜑𝑠 (1st𝐴)) ∧ (𝑡𝐴𝑠 ∈ (1st𝑡))) → 𝑠 ∈ (1st𝑡))
19 elprnql 7430 . . . . . . . 8 ((⟨(1st𝑡), (2nd𝑡)⟩ ∈ P𝑠 ∈ (1st𝑡)) → 𝑠Q)
2017, 18, 19syl2anc 409 . . . . . . 7 (((𝜑𝑠 (1st𝐴)) ∧ (𝑡𝐴𝑠 ∈ (1st𝑡))) → 𝑠Q)
2112, 20rexlimddv 2592 . . . . . 6 ((𝜑𝑠 (1st𝐴)) → 𝑠Q)
2221ex 114 . . . . 5 (𝜑 → (𝑠 (1st𝐴) → 𝑠Q))
2322ssrdv 3153 . . . 4 (𝜑 (1st𝐴) ⊆ Q)
24 ssrab2 3232 . . . . 5 {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢} ⊆ Q
257, 24eqsstrdi 3199 . . . 4 (𝜑 → (2nd𝐵) ⊆ Q)
262, 3, 4suplocexprlemml 7665 . . . . 5 (𝜑 → ∃𝑞Q 𝑞 (1st𝐴))
272, 3, 4, 1suplocexprlemmu 7667 . . . . 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))))
302, 3, 4suplocexprlemrl 7666 . . . . 5 (𝜑 → ∀𝑞Q (𝑞 (1st𝐴) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 (1st𝐴))))
312, 3, 4, 1suplocexprlemru 7668 . . . . 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)))))
332, 3, 4, 1suplocexprlemdisj 7669 . . . 4 (𝜑 → ∀𝑞Q ¬ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵)))
342, 3, 4, 1suplocexprlemloc 7670 . . . 4 (𝜑 → ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 (1st𝐴) ∨ 𝑟 ∈ (2nd𝐵))))
3532, 33, 343jca 1172 . . 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 7423 . . 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 415 . 2 (𝜑 → ⟨ (1st𝐴), (2nd𝐵)⟩ ∈ P)
389, 37eqeltrd 2247 1 (𝜑𝐵P)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 703  w3a 973   = wceq 1348  wex 1485  wcel 2141  wral 2448  wrex 2449  {crab 2452  wss 3121  cop 3584   cuni 3794   cint 3829   class class class wbr 3987  cima 4612  cfv 5196  1st c1st 6114  2nd c2nd 6115  Qcnq 7229   <Q cltq 7234  Pcnp 7240  <P cltp 7244
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 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-irdg 6346  df-1o 6392  df-2o 6393  df-oadd 6396  df-omul 6397  df-er 6509  df-ec 6511  df-qs 6515  df-ni 7253  df-pli 7254  df-mi 7255  df-lti 7256  df-plpq 7293  df-mpq 7294  df-enq 7296  df-nqqs 7297  df-plqqs 7298  df-mqqs 7299  df-1nqqs 7300  df-rq 7301  df-ltnqqs 7302  df-enq0 7373  df-nq0 7374  df-0nq0 7375  df-plq0 7376  df-mq0 7377  df-inp 7415  df-iltp 7419
This theorem is referenced by:  suplocexprlemub  7672  suplocexpr  7674
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