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Theorem ltexprlemupu 6908
Description: The upper cut of our constructed difference is upper. Lemma for ltexpri 6917. (Contributed by Jim Kingdon, 21-Dec-2019.)
Hypothesis
Ref Expression
ltexprlem.1 𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩
Assertion
Ref Expression
ltexprlemupu ((𝐴<P 𝐵𝑟Q) → (∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)) → 𝑟 ∈ (2nd𝐶)))
Distinct variable groups:   𝑥,𝑦,𝑞,𝑟,𝐴   𝑥,𝐵,𝑦,𝑞,𝑟   𝑥,𝐶,𝑦,𝑞,𝑟

Proof of Theorem ltexprlemupu
StepHypRef Expression
1 simplr 497 . . . . . 6 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) → 𝑟Q)
2 simprrr 507 . . . . . . 7 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) → (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))
32simpld 110 . . . . . 6 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) → 𝑦 ∈ (1st𝐴))
4 simprl 498 . . . . . . . 8 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) → 𝑞 <Q 𝑟)
5 simpll 496 . . . . . . . . 9 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) → 𝐴<P 𝐵)
6 simprrl 506 . . . . . . . . . 10 ((𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))) → 𝑦 ∈ (1st𝐴))
76adantl 271 . . . . . . . . 9 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) → 𝑦 ∈ (1st𝐴))
8 ltrelpr 6809 . . . . . . . . . . . . 13 <P ⊆ (P × P)
98brel 4438 . . . . . . . . . . . 12 (𝐴<P 𝐵 → (𝐴P𝐵P))
109simpld 110 . . . . . . . . . . 11 (𝐴<P 𝐵𝐴P)
11 prop 6779 . . . . . . . . . . 11 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
1210, 11syl 14 . . . . . . . . . 10 (𝐴<P 𝐵 → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
13 elprnql 6785 . . . . . . . . . 10 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (1st𝐴)) → 𝑦Q)
1412, 13sylan 277 . . . . . . . . 9 ((𝐴<P 𝐵𝑦 ∈ (1st𝐴)) → 𝑦Q)
155, 7, 14syl2anc 403 . . . . . . . 8 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) → 𝑦Q)
16 ltanqi 6706 . . . . . . . 8 ((𝑞 <Q 𝑟𝑦Q) → (𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟))
174, 15, 16syl2anc 403 . . . . . . 7 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) → (𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟))
189simprd 112 . . . . . . . . 9 (𝐴<P 𝐵𝐵P)
195, 18syl 14 . . . . . . . 8 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) → 𝐵P)
202simprd 112 . . . . . . . 8 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) → (𝑦 +Q 𝑞) ∈ (2nd𝐵))
21 prop 6779 . . . . . . . . 9 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
22 prcunqu 6789 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)) → ((𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟) → (𝑦 +Q 𝑟) ∈ (2nd𝐵)))
2321, 22sylan 277 . . . . . . . 8 ((𝐵P ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)) → ((𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟) → (𝑦 +Q 𝑟) ∈ (2nd𝐵)))
2419, 20, 23syl2anc 403 . . . . . . 7 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) → ((𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟) → (𝑦 +Q 𝑟) ∈ (2nd𝐵)))
2517, 24mpd 13 . . . . . 6 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) → (𝑦 +Q 𝑟) ∈ (2nd𝐵))
261, 3, 25jca32 303 . . . . 5 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) → (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))))
2726eximi 1532 . . . 4 (∃𝑦((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) → ∃𝑦(𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))))
28 ltexprlem.1 . . . . . . . . . 10 𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩
2928ltexprlemelu 6903 . . . . . . . . 9 (𝑞 ∈ (2nd𝐶) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))
30 19.42v 1829 . . . . . . . . 9 (∃𝑦(𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))
3129, 30bitr4i 185 . . . . . . . 8 (𝑞 ∈ (2nd𝐶) ↔ ∃𝑦(𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))
3231anbi2i 445 . . . . . . 7 ((𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)) ↔ (𝑞 <Q 𝑟 ∧ ∃𝑦(𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
33 19.42v 1829 . . . . . . 7 (∃𝑦(𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))) ↔ (𝑞 <Q 𝑟 ∧ ∃𝑦(𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
3432, 33bitr4i 185 . . . . . 6 ((𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)) ↔ ∃𝑦(𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
3534anbi2i 445 . . . . 5 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶))) ↔ ((𝐴<P 𝐵𝑟Q) ∧ ∃𝑦(𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))))
36 19.42v 1829 . . . . 5 (∃𝑦((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) ↔ ((𝐴<P 𝐵𝑟Q) ∧ ∃𝑦(𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))))
3735, 36bitr4i 185 . . . 4 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶))) ↔ ∃𝑦((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))))
3828ltexprlemelu 6903 . . . . 5 (𝑟 ∈ (2nd𝐶) ↔ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))))
39 19.42v 1829 . . . . 5 (∃𝑦(𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))) ↔ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))))
4038, 39bitr4i 185 . . . 4 (𝑟 ∈ (2nd𝐶) ↔ ∃𝑦(𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))))
4127, 37, 403imtr4i 199 . . 3 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶))) → 𝑟 ∈ (2nd𝐶))
4241ex 113 . 2 ((𝐴<P 𝐵𝑟Q) → ((𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)) → 𝑟 ∈ (2nd𝐶)))
4342rexlimdvw 2485 1 ((𝐴<P 𝐵𝑟Q) → (∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)) → 𝑟 ∈ (2nd𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wex 1422  wcel 1434  wrex 2354  {crab 2357  cop 3419   class class class wbr 3805  cfv 4952  (class class class)co 5563  1st c1st 5816  2nd c2nd 5817  Qcnq 6584   +Q cplq 6586   <Q cltq 6589  Pcnp 6595  <P cltp 6599
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-eprel 4072  df-id 4076  df-iord 4149  df-on 4151  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-irdg 6039  df-oadd 6089  df-omul 6090  df-er 6193  df-ec 6195  df-qs 6199  df-ni 6608  df-pli 6609  df-mi 6610  df-lti 6611  df-plpq 6648  df-enq 6651  df-nqqs 6652  df-plqqs 6653  df-ltnqqs 6657  df-inp 6770  df-iltp 6774
This theorem is referenced by:  ltexprlemrnd  6909
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