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Theorem suplocexprlemex 7721
Description: Lemma for suplocexpr 7724. 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 7714 . . . . 5 (𝜑𝐴P)
61suplocexprlem2b 7713 . . . . 5 (𝐴P → (2nd𝐵) = {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢})
75, 6syl 14 . . . 4 (𝜑 → (2nd𝐵) = {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢})
87opeq2d 3786 . . 3 (𝜑 → ⟨ (1st𝐴), (2nd𝐵)⟩ = ⟨ (1st𝐴), {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢}⟩)
91, 8eqtr4id 2229 . 2 (𝜑𝐵 = ⟨ (1st𝐴), (2nd𝐵)⟩)
10 suplocexprlemell 7712 . . . . . . . . 9 (𝑠 (1st𝐴) ↔ ∃𝑡𝐴 𝑠 ∈ (1st𝑡))
1110biimpi 120 . . . . . . . 8 (𝑠 (1st𝐴) → ∃𝑡𝐴 𝑠 ∈ (1st𝑡))
1211adantl 277 . . . . . . 7 ((𝜑𝑠 (1st𝐴)) → ∃𝑡𝐴 𝑠 ∈ (1st𝑡))
135ad2antrr 488 . . . . . . . . . 10 (((𝜑𝑠 (1st𝐴)) ∧ (𝑡𝐴𝑠 ∈ (1st𝑡))) → 𝐴P)
14 simprl 529 . . . . . . . . . 10 (((𝜑𝑠 (1st𝐴)) ∧ (𝑡𝐴𝑠 ∈ (1st𝑡))) → 𝑡𝐴)
1513, 14sseldd 3157 . . . . . . . . 9 (((𝜑𝑠 (1st𝐴)) ∧ (𝑡𝐴𝑠 ∈ (1st𝑡))) → 𝑡P)
16 prop 7474 . . . . . . . . 9 (𝑡P → ⟨(1st𝑡), (2nd𝑡)⟩ ∈ P)
1715, 16syl 14 . . . . . . . 8 (((𝜑𝑠 (1st𝐴)) ∧ (𝑡𝐴𝑠 ∈ (1st𝑡))) → ⟨(1st𝑡), (2nd𝑡)⟩ ∈ P)
18 simprr 531 . . . . . . . 8 (((𝜑𝑠 (1st𝐴)) ∧ (𝑡𝐴𝑠 ∈ (1st𝑡))) → 𝑠 ∈ (1st𝑡))
19 elprnql 7480 . . . . . . . 8 ((⟨(1st𝑡), (2nd𝑡)⟩ ∈ P𝑠 ∈ (1st𝑡)) → 𝑠Q)
2017, 18, 19syl2anc 411 . . . . . . 7 (((𝜑𝑠 (1st𝐴)) ∧ (𝑡𝐴𝑠 ∈ (1st𝑡))) → 𝑠Q)
2112, 20rexlimddv 2599 . . . . . 6 ((𝜑𝑠 (1st𝐴)) → 𝑠Q)
2221ex 115 . . . . 5 (𝜑 → (𝑠 (1st𝐴) → 𝑠Q))
2322ssrdv 3162 . . . 4 (𝜑 (1st𝐴) ⊆ Q)
24 ssrab2 3241 . . . . 5 {𝑢Q ∣ ∃𝑤 (2nd𝐴)𝑤 <Q 𝑢} ⊆ Q
257, 24eqsstrdi 3208 . . . 4 (𝜑 → (2nd𝐵) ⊆ Q)
262, 3, 4suplocexprlemml 7715 . . . . 5 (𝜑 → ∃𝑞Q 𝑞 (1st𝐴))
272, 3, 4, 1suplocexprlemmu 7717 . . . . 5 (𝜑 → ∃𝑟Q 𝑟 ∈ (2nd𝐵))
2826, 27jca 306 . . . 4 (𝜑 → (∃𝑞Q 𝑞 (1st𝐴) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐵)))
2923, 25, 28jca31 309 . . 3 (𝜑 → (( (1st𝐴) ⊆ Q ∧ (2nd𝐵) ⊆ Q) ∧ (∃𝑞Q 𝑞 (1st𝐴) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐵))))
302, 3, 4suplocexprlemrl 7716 . . . . 5 (𝜑 → ∀𝑞Q (𝑞 (1st𝐴) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 (1st𝐴))))
312, 3, 4, 1suplocexprlemru 7718 . . . . 5 (𝜑 → ∀𝑟Q (𝑟 ∈ (2nd𝐵) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐵))))
3230, 31jca 306 . . . 4 (𝜑 → (∀𝑞Q (𝑞 (1st𝐴) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 (1st𝐴))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐵) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐵)))))
332, 3, 4, 1suplocexprlemdisj 7719 . . . 4 (𝜑 → ∀𝑞Q ¬ (𝑞 (1st𝐴) ∧ 𝑞 ∈ (2nd𝐵)))
342, 3, 4, 1suplocexprlemloc 7720 . . . 4 (𝜑 → ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 (1st𝐴) ∨ 𝑟 ∈ (2nd𝐵))))
3532, 33, 343jca 1177 . . 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 7473 . . 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 417 . 2 (𝜑 → ⟨ (1st𝐴), (2nd𝐵)⟩ ∈ P)
389, 37eqeltrd 2254 1 (𝜑𝐵P)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708  w3a 978   = wceq 1353  wex 1492  wcel 2148  wral 2455  wrex 2456  {crab 2459  wss 3130  cop 3596   cuni 3810   cint 3845   class class class wbr 4004  cima 4630  cfv 5217  1st c1st 6139  2nd c2nd 6140  Qcnq 7279   <Q cltq 7284  Pcnp 7290  <P cltp 7294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-eprel 4290  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-1o 6417  df-2o 6418  df-oadd 6421  df-omul 6422  df-er 6535  df-ec 6537  df-qs 6541  df-ni 7303  df-pli 7304  df-mi 7305  df-lti 7306  df-plpq 7343  df-mpq 7344  df-enq 7346  df-nqqs 7347  df-plqqs 7348  df-mqqs 7349  df-1nqqs 7350  df-rq 7351  df-ltnqqs 7352  df-enq0 7423  df-nq0 7424  df-0nq0 7425  df-plq0 7426  df-mq0 7427  df-inp 7465  df-iltp 7469
This theorem is referenced by:  suplocexprlemub  7722  suplocexpr  7724
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