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Theorem ltexprlemupu 7163
Description: The upper cut of our constructed difference is upper. Lemma for ltexpri 7172. (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 7064 . . . . . . . . . . . . 13 <P ⊆ (P × P)
98brel 4490 . . . . . . . . . . . 12 (𝐴<P 𝐵 → (𝐴P𝐵P))
109simpld 110 . . . . . . . . . . 11 (𝐴<P 𝐵𝐴P)
11 prop 7034 . . . . . . . . . . 11 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
1210, 11syl 14 . . . . . . . . . 10 (𝐴<P 𝐵 → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
13 elprnql 7040 . . . . . . . . . 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 6961 . . . . . . . 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 7034 . . . . . . . . 9 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
22 prcunqu 7044 . . . . . . . . 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 1536 . . . 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 7158 . . . . . . . . 9 (𝑞 ∈ (2nd𝐶) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))
30 19.42v 1834 . . . . . . . . 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 1834 . . . . . . 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 1834 . . . . 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 7158 . . . . 5 (𝑟 ∈ (2nd𝐶) ↔ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))))
39 19.42v 1834 . . . . 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 2492 1 ((𝐴<P 𝐵𝑟Q) → (∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)) → 𝑟 ∈ (2nd𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wex 1426  wcel 1438  wrex 2360  {crab 2363  cop 3449   class class class wbr 3845  cfv 5015  (class class class)co 5652  1st c1st 5909  2nd c2nd 5910  Qcnq 6839   +Q cplq 6841   <Q cltq 6844  Pcnp 6850  <P cltp 6854
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-eprel 4116  df-id 4120  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-oadd 6185  df-omul 6186  df-er 6292  df-ec 6294  df-qs 6298  df-ni 6863  df-pli 6864  df-mi 6865  df-lti 6866  df-plpq 6903  df-enq 6906  df-nqqs 6907  df-plqqs 6908  df-ltnqqs 6912  df-inp 7025  df-iltp 7029
This theorem is referenced by:  ltexprlemrnd  7164
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