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Theorem suplocexprlemrl 7715
Description: Lemma for suplocexpr 7723. 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 7711 . . . . . . 7 (𝑞 (1st𝐴) ↔ ∃𝑠𝐴 𝑞 ∈ (1st𝑠))
21biimpi 120 . . . . . 6 (𝑞 (1st𝐴) → ∃𝑠𝐴 𝑞 ∈ (1st𝑠))
32adantl 277 . . . . 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 7713 . . . . . . . . . 10 (𝜑𝐴P)
87ad3antrrr 492 . . . . . . . . 9 ((((𝜑𝑞Q) ∧ 𝑞 (1st𝐴)) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) → 𝐴P)
9 simprl 529 . . . . . . . . 9 ((((𝜑𝑞Q) ∧ 𝑞 (1st𝐴)) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) → 𝑠𝐴)
108, 9sseldd 3156 . . . . . . . 8 ((((𝜑𝑞Q) ∧ 𝑞 (1st𝐴)) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) → 𝑠P)
11 prop 7473 . . . . . . . 8 (𝑠P → ⟨(1st𝑠), (2nd𝑠)⟩ ∈ P)
1210, 11syl 14 . . . . . . 7 ((((𝜑𝑞Q) ∧ 𝑞 (1st𝐴)) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) → ⟨(1st𝑠), (2nd𝑠)⟩ ∈ P)
13 simprr 531 . . . . . . 7 ((((𝜑𝑞Q) ∧ 𝑞 (1st𝐴)) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) → 𝑞 ∈ (1st𝑠))
14 prnmaxl 7486 . . . . . . 7 ((⟨(1st𝑠), (2nd𝑠)⟩ ∈ P𝑞 ∈ (1st𝑠)) → ∃𝑟 ∈ (1st𝑠)𝑞 <Q 𝑟)
1512, 13, 14syl2anc 411 . . . . . 6 ((((𝜑𝑞Q) ∧ 𝑞 (1st𝐴)) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) → ∃𝑟 ∈ (1st𝑠)𝑞 <Q 𝑟)
16 ltrelnq 7363 . . . . . . . . 9 <Q ⊆ (Q × Q)
1716brel 4678 . . . . . . . 8 (𝑞 <Q 𝑟 → (𝑞Q𝑟Q))
1817simprd 114 . . . . . . 7 (𝑞 <Q 𝑟𝑟Q)
1918ad2antll 491 . . . . . 6 (((((𝜑𝑞Q) ∧ 𝑞 (1st𝐴)) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) ∧ (𝑟 ∈ (1st𝑠) ∧ 𝑞 <Q 𝑟)) → 𝑟Q)
20 simprr 531 . . . . . . 7 (((((𝜑𝑞Q) ∧ 𝑞 (1st𝐴)) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) ∧ (𝑟 ∈ (1st𝑠) ∧ 𝑞 <Q 𝑟)) → 𝑞 <Q 𝑟)
21 simplrl 535 . . . . . . . . 9 (((((𝜑𝑞Q) ∧ 𝑞 (1st𝐴)) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) ∧ (𝑟 ∈ (1st𝑠) ∧ 𝑞 <Q 𝑟)) → 𝑠𝐴)
22 simprl 529 . . . . . . . . 9 (((((𝜑𝑞Q) ∧ 𝑞 (1st𝐴)) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) ∧ (𝑟 ∈ (1st𝑠) ∧ 𝑞 <Q 𝑟)) → 𝑟 ∈ (1st𝑠))
23 rspe 2526 . . . . . . . . 9 ((𝑠𝐴𝑟 ∈ (1st𝑠)) → ∃𝑠𝐴 𝑟 ∈ (1st𝑠))
2421, 22, 23syl2anc 411 . . . . . . . 8 (((((𝜑𝑞Q) ∧ 𝑞 (1st𝐴)) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) ∧ (𝑟 ∈ (1st𝑠) ∧ 𝑞 <Q 𝑟)) → ∃𝑠𝐴 𝑟 ∈ (1st𝑠))
25 suplocexprlemell 7711 . . . . . . . 8 (𝑟 (1st𝐴) ↔ ∃𝑠𝐴 𝑟 ∈ (1st𝑠))
2624, 25sylibr 134 . . . . . . 7 (((((𝜑𝑞Q) ∧ 𝑞 (1st𝐴)) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) ∧ (𝑟 ∈ (1st𝑠) ∧ 𝑞 <Q 𝑟)) → 𝑟 (1st𝐴))
2720, 26jca 306 . . . . . 6 (((((𝜑𝑞Q) ∧ 𝑞 (1st𝐴)) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) ∧ (𝑟 ∈ (1st𝑠) ∧ 𝑞 <Q 𝑟)) → (𝑞 <Q 𝑟𝑟 (1st𝐴)))
2815, 19, 27reximssdv 2581 . . . . 5 ((((𝜑𝑞Q) ∧ 𝑞 (1st𝐴)) ∧ (𝑠𝐴𝑞 ∈ (1st𝑠))) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 (1st𝐴)))
293, 28rexlimddv 2599 . . . 4 (((𝜑𝑞Q) ∧ 𝑞 (1st𝐴)) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 (1st𝐴)))
3029ex 115 . . 3 ((𝜑𝑞Q) → (𝑞 (1st𝐴) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 (1st𝐴))))
31 simprr 531 . . . . . . 7 (((𝜑𝑞Q) ∧ (𝑞 <Q 𝑟𝑟 (1st𝐴))) → 𝑟 (1st𝐴))
3231, 25sylib 122 . . . . . 6 (((𝜑𝑞Q) ∧ (𝑞 <Q 𝑟𝑟 (1st𝐴))) → ∃𝑠𝐴 𝑟 ∈ (1st𝑠))
33 simprl 529 . . . . . . . . 9 ((((𝜑𝑞Q) ∧ (𝑞 <Q 𝑟𝑟 (1st𝐴))) ∧ (𝑠𝐴𝑟 ∈ (1st𝑠))) → 𝑠𝐴)
34 simplrl 535 . . . . . . . . . 10 ((((𝜑𝑞Q) ∧ (𝑞 <Q 𝑟𝑟 (1st𝐴))) ∧ (𝑠𝐴𝑟 ∈ (1st𝑠))) → 𝑞 <Q 𝑟)
357ad3antrrr 492 . . . . . . . . . . . . 13 ((((𝜑𝑞Q) ∧ (𝑞 <Q 𝑟𝑟 (1st𝐴))) ∧ (𝑠𝐴𝑟 ∈ (1st𝑠))) → 𝐴P)
3635, 33sseldd 3156 . . . . . . . . . . . 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 531 . . . . . . . . . . 11 ((((𝜑𝑞Q) ∧ (𝑞 <Q 𝑟𝑟 (1st𝐴))) ∧ (𝑠𝐴𝑟 ∈ (1st𝑠))) → 𝑟 ∈ (1st𝑠))
39 prcdnql 7482 . . . . . . . . . . 11 ((⟨(1st𝑠), (2nd𝑠)⟩ ∈ P𝑟 ∈ (1st𝑠)) → (𝑞 <Q 𝑟𝑞 ∈ (1st𝑠)))
4037, 38, 39syl2anc 411 . . . . . . . . . 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 1590 . . . . . . . . 9 ((𝑠𝐴𝑞 ∈ (1st𝑠)) → ∃𝑠(𝑠𝐴𝑞 ∈ (1st𝑠)))
4333, 41, 42syl2anc 411 . . . . . . . 8 ((((𝜑𝑞Q) ∧ (𝑞 <Q 𝑟𝑟 (1st𝐴))) ∧ (𝑠𝐴𝑟 ∈ (1st𝑠))) → ∃𝑠(𝑠𝐴𝑞 ∈ (1st𝑠)))
44 df-rex 2461 . . . . . . . 8 (∃𝑠𝐴 𝑞 ∈ (1st𝑠) ↔ ∃𝑠(𝑠𝐴𝑞 ∈ (1st𝑠)))
4543, 44sylibr 134 . . . . . . 7 ((((𝜑𝑞Q) ∧ (𝑞 <Q 𝑟𝑟 (1st𝐴))) ∧ (𝑠𝐴𝑟 ∈ (1st𝑠))) → ∃𝑠𝐴 𝑞 ∈ (1st𝑠))
4645, 1sylibr 134 . . . . . 6 ((((𝜑𝑞Q) ∧ (𝑞 <Q 𝑟𝑟 (1st𝐴))) ∧ (𝑠𝐴𝑟 ∈ (1st𝑠))) → 𝑞 (1st𝐴))
4732, 46rexlimddv 2599 . . . . 5 (((𝜑𝑞Q) ∧ (𝑞 <Q 𝑟𝑟 (1st𝐴))) → 𝑞 (1st𝐴))
4847ex 115 . . . 4 ((𝜑𝑞Q) → ((𝑞 <Q 𝑟𝑟 (1st𝐴)) → 𝑞 (1st𝐴)))
4948rexlimdvw 2598 . . 3 ((𝜑𝑞Q) → (∃𝑟Q (𝑞 <Q 𝑟𝑟 (1st𝐴)) → 𝑞 (1st𝐴)))
5030, 49impbid 129 . 2 ((𝜑𝑞Q) → (𝑞 (1st𝐴) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 (1st𝐴))))
5150ralrimiva 2550 1 (𝜑 → ∀𝑞Q (𝑞 (1st𝐴) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 (1st𝐴))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wo 708  wex 1492  wcel 2148  wral 2455  wrex 2456  wss 3129  cop 3595   cuni 3809   class class class wbr 4003  cima 4629  cfv 5216  1st c1st 6138  2nd c2nd 6139  Qcnq 7278   <Q cltq 7283  Pcnp 7289  <P cltp 7293
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 4118  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  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-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-1st 6140  df-2nd 6141  df-qs 6540  df-ni 7302  df-nqqs 7346  df-ltnqqs 7351  df-inp 7464  df-iltp 7468
This theorem is referenced by:  suplocexprlemex  7720
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