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Theorem suplocexprlemrl 7639
Description: Lemma for suplocexpr 7647. The lower cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-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)))
Assertion
Ref Expression
suplocexprlemrl (𝜑 → ∀𝑞Q (𝑞 (1st𝐴) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 (1st𝐴))))
Distinct variable groups:   𝐴,𝑟   𝑥,𝐴,𝑦   𝜑,𝑞,𝑟   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑧)   𝐴(𝑧,𝑞)

Proof of Theorem suplocexprlemrl
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 suplocexprlemell 7635 . . . . . . 7 (𝑞 (1st𝐴) ↔ ∃𝑠𝐴 𝑞 ∈ (1st𝑠))
21biimpi 119 . . . . . 6 (𝑞 (1st𝐴) → ∃𝑠𝐴 𝑞 ∈ (1st𝑠))
32adantl 275 . . . . 5 (((𝜑𝑞Q) ∧ 𝑞 (1st𝐴)) → ∃𝑠𝐴 𝑞 ∈ (1st𝑠))
4 suplocexpr.m . . . . . . . . . . 11 (𝜑 → ∃𝑥 𝑥𝐴)
5 suplocexpr.ub . . . . . . . . . . 11 (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)
6 suplocexpr.loc . . . . . . . . . . 11 (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))
74, 5, 6suplocexprlemss 7637 . . . . . . . . . 10 (𝜑𝐴P)
87ad3antrrr 484 . . . . . . . . 9 ((((𝜑𝑞Q) ∧ 𝑞 (1st𝐴)) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) → 𝐴P)
9 simprl 521 . . . . . . . . 9 ((((𝜑𝑞Q) ∧ 𝑞 (1st𝐴)) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) → 𝑠𝐴)
108, 9sseldd 3129 . . . . . . . 8 ((((𝜑𝑞Q) ∧ 𝑞 (1st𝐴)) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) → 𝑠P)
11 prop 7397 . . . . . . . 8 (𝑠P → ⟨(1st𝑠), (2nd𝑠)⟩ ∈ P)
1210, 11syl 14 . . . . . . 7 ((((𝜑𝑞Q) ∧ 𝑞 (1st𝐴)) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) → ⟨(1st𝑠), (2nd𝑠)⟩ ∈ P)
13 simprr 522 . . . . . . 7 ((((𝜑𝑞Q) ∧ 𝑞 (1st𝐴)) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) → 𝑞 ∈ (1st𝑠))
14 prnmaxl 7410 . . . . . . 7 ((⟨(1st𝑠), (2nd𝑠)⟩ ∈ P𝑞 ∈ (1st𝑠)) → ∃𝑟 ∈ (1st𝑠)𝑞 <Q 𝑟)
1512, 13, 14syl2anc 409 . . . . . 6 ((((𝜑𝑞Q) ∧ 𝑞 (1st𝐴)) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) → ∃𝑟 ∈ (1st𝑠)𝑞 <Q 𝑟)
16 ltrelnq 7287 . . . . . . . . 9 <Q ⊆ (Q × Q)
1716brel 4640 . . . . . . . 8 (𝑞 <Q 𝑟 → (𝑞Q𝑟Q))
1817simprd 113 . . . . . . 7 (𝑞 <Q 𝑟𝑟Q)
1918ad2antll 483 . . . . . 6 (((((𝜑𝑞Q) ∧ 𝑞 (1st𝐴)) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) ∧ (𝑟 ∈ (1st𝑠) ∧ 𝑞 <Q 𝑟)) → 𝑟Q)
20 simprr 522 . . . . . . 7 (((((𝜑𝑞Q) ∧ 𝑞 (1st𝐴)) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) ∧ (𝑟 ∈ (1st𝑠) ∧ 𝑞 <Q 𝑟)) → 𝑞 <Q 𝑟)
21 simplrl 525 . . . . . . . . 9 (((((𝜑𝑞Q) ∧ 𝑞 (1st𝐴)) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) ∧ (𝑟 ∈ (1st𝑠) ∧ 𝑞 <Q 𝑟)) → 𝑠𝐴)
22 simprl 521 . . . . . . . . 9 (((((𝜑𝑞Q) ∧ 𝑞 (1st𝐴)) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) ∧ (𝑟 ∈ (1st𝑠) ∧ 𝑞 <Q 𝑟)) → 𝑟 ∈ (1st𝑠))
23 rspe 2506 . . . . . . . . 9 ((𝑠𝐴𝑟 ∈ (1st𝑠)) → ∃𝑠𝐴 𝑟 ∈ (1st𝑠))
2421, 22, 23syl2anc 409 . . . . . . . 8 (((((𝜑𝑞Q) ∧ 𝑞 (1st𝐴)) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) ∧ (𝑟 ∈ (1st𝑠) ∧ 𝑞 <Q 𝑟)) → ∃𝑠𝐴 𝑟 ∈ (1st𝑠))
25 suplocexprlemell 7635 . . . . . . . 8 (𝑟 (1st𝐴) ↔ ∃𝑠𝐴 𝑟 ∈ (1st𝑠))
2624, 25sylibr 133 . . . . . . 7 (((((𝜑𝑞Q) ∧ 𝑞 (1st𝐴)) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) ∧ (𝑟 ∈ (1st𝑠) ∧ 𝑞 <Q 𝑟)) → 𝑟 (1st𝐴))
2720, 26jca 304 . . . . . 6 (((((𝜑𝑞Q) ∧ 𝑞 (1st𝐴)) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) ∧ (𝑟 ∈ (1st𝑠) ∧ 𝑞 <Q 𝑟)) → (𝑞 <Q 𝑟𝑟 (1st𝐴)))
2815, 19, 27reximssdv 2561 . . . . 5 ((((𝜑𝑞Q) ∧ 𝑞 (1st𝐴)) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 (1st𝐴)))
293, 28rexlimddv 2579 . . . 4 (((𝜑𝑞Q) ∧ 𝑞 (1st𝐴)) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 (1st𝐴)))
3029ex 114 . . 3 ((𝜑𝑞Q) → (𝑞 (1st𝐴) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 (1st𝐴))))
31 simprr 522 . . . . . . 7 (((𝜑𝑞Q) ∧ (𝑞 <Q 𝑟𝑟 (1st𝐴))) → 𝑟 (1st𝐴))
3231, 25sylib 121 . . . . . 6 (((𝜑𝑞Q) ∧ (𝑞 <Q 𝑟𝑟 (1st𝐴))) → ∃𝑠𝐴 𝑟 ∈ (1st𝑠))
33 simprl 521 . . . . . . . . 9 ((((𝜑𝑞Q) ∧ (𝑞 <Q 𝑟𝑟 (1st𝐴))) ∧ (𝑠𝐴𝑟 ∈ (1st𝑠))) → 𝑠𝐴)
34 simplrl 525 . . . . . . . . . 10 ((((𝜑𝑞Q) ∧ (𝑞 <Q 𝑟𝑟 (1st𝐴))) ∧ (𝑠𝐴𝑟 ∈ (1st𝑠))) → 𝑞 <Q 𝑟)
357ad3antrrr 484 . . . . . . . . . . . . 13 ((((𝜑𝑞Q) ∧ (𝑞 <Q 𝑟𝑟 (1st𝐴))) ∧ (𝑠𝐴𝑟 ∈ (1st𝑠))) → 𝐴P)
3635, 33sseldd 3129 . . . . . . . . . . . 12 ((((𝜑𝑞Q) ∧ (𝑞 <Q 𝑟𝑟 (1st𝐴))) ∧ (𝑠𝐴𝑟 ∈ (1st𝑠))) → 𝑠P)
3736, 11syl 14 . . . . . . . . . . 11 ((((𝜑𝑞Q) ∧ (𝑞 <Q 𝑟𝑟 (1st𝐴))) ∧ (𝑠𝐴𝑟 ∈ (1st𝑠))) → ⟨(1st𝑠), (2nd𝑠)⟩ ∈ P)
38 simprr 522 . . . . . . . . . . 11 ((((𝜑𝑞Q) ∧ (𝑞 <Q 𝑟𝑟 (1st𝐴))) ∧ (𝑠𝐴𝑟 ∈ (1st𝑠))) → 𝑟 ∈ (1st𝑠))
39 prcdnql 7406 . . . . . . . . . . 11 ((⟨(1st𝑠), (2nd𝑠)⟩ ∈ P𝑟 ∈ (1st𝑠)) → (𝑞 <Q 𝑟𝑞 ∈ (1st𝑠)))
4037, 38, 39syl2anc 409 . . . . . . . . . 10 ((((𝜑𝑞Q) ∧ (𝑞 <Q 𝑟𝑟 (1st𝐴))) ∧ (𝑠𝐴𝑟 ∈ (1st𝑠))) → (𝑞 <Q 𝑟𝑞 ∈ (1st𝑠)))
4134, 40mpd 13 . . . . . . . . 9 ((((𝜑𝑞Q) ∧ (𝑞 <Q 𝑟𝑟 (1st𝐴))) ∧ (𝑠𝐴𝑟 ∈ (1st𝑠))) → 𝑞 ∈ (1st𝑠))
42 19.8a 1570 . . . . . . . . 9 ((𝑠𝐴𝑞 ∈ (1st𝑠)) → ∃𝑠(𝑠𝐴𝑞 ∈ (1st𝑠)))
4333, 41, 42syl2anc 409 . . . . . . . 8 ((((𝜑𝑞Q) ∧ (𝑞 <Q 𝑟𝑟 (1st𝐴))) ∧ (𝑠𝐴𝑟 ∈ (1st𝑠))) → ∃𝑠(𝑠𝐴𝑞 ∈ (1st𝑠)))
44 df-rex 2441 . . . . . . . 8 (∃𝑠𝐴 𝑞 ∈ (1st𝑠) ↔ ∃𝑠(𝑠𝐴𝑞 ∈ (1st𝑠)))
4543, 44sylibr 133 . . . . . . 7 ((((𝜑𝑞Q) ∧ (𝑞 <Q 𝑟𝑟 (1st𝐴))) ∧ (𝑠𝐴𝑟 ∈ (1st𝑠))) → ∃𝑠𝐴 𝑞 ∈ (1st𝑠))
4645, 1sylibr 133 . . . . . 6 ((((𝜑𝑞Q) ∧ (𝑞 <Q 𝑟𝑟 (1st𝐴))) ∧ (𝑠𝐴𝑟 ∈ (1st𝑠))) → 𝑞 (1st𝐴))
4732, 46rexlimddv 2579 . . . . 5 (((𝜑𝑞Q) ∧ (𝑞 <Q 𝑟𝑟 (1st𝐴))) → 𝑞 (1st𝐴))
4847ex 114 . . . 4 ((𝜑𝑞Q) → ((𝑞 <Q 𝑟𝑟 (1st𝐴)) → 𝑞 (1st𝐴)))
4948rexlimdvw 2578 . . 3 ((𝜑𝑞Q) → (∃𝑟Q (𝑞 <Q 𝑟𝑟 (1st𝐴)) → 𝑞 (1st𝐴)))
5030, 49impbid 128 . 2 ((𝜑𝑞Q) → (𝑞 (1st𝐴) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 (1st𝐴))))
5150ralrimiva 2530 1 (𝜑 → ∀𝑞Q (𝑞 (1st𝐴) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 (1st𝐴))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wo 698  wex 1472  wcel 2128  wral 2435  wrex 2436  wss 3102  cop 3564   cuni 3774   class class class wbr 3967  cima 4591  cfv 5172  1st c1st 6088  2nd c2nd 6089  Qcnq 7202   <Q cltq 7207  Pcnp 7213  <P cltp 7217
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4081  ax-sep 4084  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-iinf 4549
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-id 4255  df-iom 4552  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180  df-1st 6090  df-2nd 6091  df-qs 6488  df-ni 7226  df-nqqs 7270  df-ltnqqs 7275  df-inp 7388  df-iltp 7392
This theorem is referenced by:  suplocexprlemex  7644
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