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Theorem ltexprlemlol 6728
Description: The lower cut of our constructed difference is lower. Lemma for ltexpri 6739. (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
ltexprlemlol ((𝐴<P 𝐵𝑞Q) → (∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)) → 𝑞 ∈ (1st𝐶)))
Distinct variable groups:   𝑥,𝑦,𝑞,𝑟,𝐴   𝑥,𝐵,𝑦,𝑞,𝑟   𝑥,𝐶,𝑦,𝑞,𝑟

Proof of Theorem ltexprlemlol
StepHypRef Expression
1 simplr 490 . . . . . 6 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) → 𝑞Q)
2 simprrr 500 . . . . . . 7 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) → (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))
32simpld 109 . . . . . 6 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) → 𝑦 ∈ (2nd𝐴))
4 simprl 491 . . . . . . . 8 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) → 𝑞 <Q 𝑟)
5 simpll 489 . . . . . . . . 9 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) → 𝐴<P 𝐵)
6 ltrelpr 6631 . . . . . . . . . . . 12 <P ⊆ (P × P)
76brel 4417 . . . . . . . . . . 11 (𝐴<P 𝐵 → (𝐴P𝐵P))
87simpld 109 . . . . . . . . . 10 (𝐴<P 𝐵𝐴P)
9 prop 6601 . . . . . . . . . . 11 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
10 elprnqu 6608 . . . . . . . . . . 11 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (2nd𝐴)) → 𝑦Q)
119, 10sylan 271 . . . . . . . . . 10 ((𝐴P𝑦 ∈ (2nd𝐴)) → 𝑦Q)
128, 11sylan 271 . . . . . . . . 9 ((𝐴<P 𝐵𝑦 ∈ (2nd𝐴)) → 𝑦Q)
135, 3, 12syl2anc 397 . . . . . . . 8 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) → 𝑦Q)
14 ltanqi 6528 . . . . . . . 8 ((𝑞 <Q 𝑟𝑦Q) → (𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟))
154, 13, 14syl2anc 397 . . . . . . 7 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) → (𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟))
167simprd 111 . . . . . . . . 9 (𝐴<P 𝐵𝐵P)
175, 16syl 14 . . . . . . . 8 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) → 𝐵P)
182simprd 111 . . . . . . . 8 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) → (𝑦 +Q 𝑟) ∈ (1st𝐵))
19 prop 6601 . . . . . . . . 9 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
20 prcdnql 6610 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)) → ((𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟) → (𝑦 +Q 𝑞) ∈ (1st𝐵)))
2119, 20sylan 271 . . . . . . . 8 ((𝐵P ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)) → ((𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟) → (𝑦 +Q 𝑞) ∈ (1st𝐵)))
2217, 18, 21syl2anc 397 . . . . . . 7 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) → ((𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟) → (𝑦 +Q 𝑞) ∈ (1st𝐵)))
2315, 22mpd 13 . . . . . 6 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) → (𝑦 +Q 𝑞) ∈ (1st𝐵))
241, 3, 23jca32 297 . . . . 5 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) → (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
2524eximi 1505 . . . 4 (∃𝑦((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) → ∃𝑦(𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
26 ltexprlem.1 . . . . . . . . . 10 𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩
2726ltexprlemell 6724 . . . . . . . . 9 (𝑟 ∈ (1st𝐶) ↔ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))
28 19.42v 1800 . . . . . . . . 9 (∃𝑦(𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))) ↔ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))
2927, 28bitr4i 180 . . . . . . . 8 (𝑟 ∈ (1st𝐶) ↔ ∃𝑦(𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))
3029anbi2i 438 . . . . . . 7 ((𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)) ↔ (𝑞 <Q 𝑟 ∧ ∃𝑦(𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))))
31 19.42v 1800 . . . . . . 7 (∃𝑦(𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))) ↔ (𝑞 <Q 𝑟 ∧ ∃𝑦(𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))))
3230, 31bitr4i 180 . . . . . 6 ((𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)) ↔ ∃𝑦(𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))))
3332anbi2i 438 . . . . 5 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶))) ↔ ((𝐴<P 𝐵𝑞Q) ∧ ∃𝑦(𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))))
34 19.42v 1800 . . . . 5 (∃𝑦((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) ↔ ((𝐴<P 𝐵𝑞Q) ∧ ∃𝑦(𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))))
3533, 34bitr4i 180 . . . 4 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶))) ↔ ∃𝑦((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))))
3626ltexprlemell 6724 . . . . 5 (𝑞 ∈ (1st𝐶) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
37 19.42v 1800 . . . . 5 (∃𝑦(𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
3836, 37bitr4i 180 . . . 4 (𝑞 ∈ (1st𝐶) ↔ ∃𝑦(𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
3925, 35, 383imtr4i 194 . . 3 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶))) → 𝑞 ∈ (1st𝐶))
4039ex 112 . 2 ((𝐴<P 𝐵𝑞Q) → ((𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)) → 𝑞 ∈ (1st𝐶)))
4140rexlimdvw 2451 1 ((𝐴<P 𝐵𝑞Q) → (∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)) → 𝑞 ∈ (1st𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1257  wex 1395  wcel 1407  wrex 2322  {crab 2325  cop 3403   class class class wbr 3789  cfv 4927  (class class class)co 5537  1st c1st 5790  2nd c2nd 5791  Qcnq 6406   +Q cplq 6408   <Q cltq 6411  Pcnp 6417  <P cltp 6421
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 552  ax-in2 553  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-13 1418  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-coll 3897  ax-sep 3900  ax-nul 3908  ax-pow 3952  ax-pr 3969  ax-un 4195  ax-setind 4287  ax-iinf 4336
This theorem depends on definitions:  df-bi 114  df-dc 752  df-3or 895  df-3an 896  df-tru 1260  df-fal 1263  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ne 2219  df-ral 2326  df-rex 2327  df-reu 2328  df-rab 2330  df-v 2574  df-sbc 2785  df-csb 2878  df-dif 2945  df-un 2947  df-in 2949  df-ss 2956  df-nul 3250  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-int 3641  df-iun 3684  df-br 3790  df-opab 3844  df-mpt 3845  df-tr 3880  df-eprel 4051  df-id 4055  df-iord 4128  df-on 4130  df-suc 4133  df-iom 4339  df-xp 4376  df-rel 4377  df-cnv 4378  df-co 4379  df-dm 4380  df-rn 4381  df-res 4382  df-ima 4383  df-iota 4892  df-fun 4929  df-fn 4930  df-f 4931  df-f1 4932  df-fo 4933  df-f1o 4934  df-fv 4935  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-irdg 5985  df-oadd 6033  df-omul 6034  df-er 6134  df-ec 6136  df-qs 6140  df-ni 6430  df-pli 6431  df-mi 6432  df-lti 6433  df-plpq 6470  df-enq 6473  df-nqqs 6474  df-plqqs 6475  df-ltnqqs 6479  df-inp 6592  df-iltp 6596
This theorem is referenced by:  ltexprlemrnd  6731
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