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Theorem ltexprlemupu 7360
Description: The upper cut of our constructed difference is upper. Lemma for ltexpri 7369. (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 502 . . . . . 6 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) → 𝑟Q)
2 simprrr 512 . . . . . . 7 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) → (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))
32simpld 111 . . . . . 6 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) → 𝑦 ∈ (1st𝐴))
4 simprl 503 . . . . . . . 8 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) → 𝑞 <Q 𝑟)
5 simpll 501 . . . . . . . . 9 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) → 𝐴<P 𝐵)
6 simprrl 511 . . . . . . . . . 10 ((𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))) → 𝑦 ∈ (1st𝐴))
76adantl 273 . . . . . . . . 9 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) → 𝑦 ∈ (1st𝐴))
8 ltrelpr 7261 . . . . . . . . . . . . 13 <P ⊆ (P × P)
98brel 4551 . . . . . . . . . . . 12 (𝐴<P 𝐵 → (𝐴P𝐵P))
109simpld 111 . . . . . . . . . . 11 (𝐴<P 𝐵𝐴P)
11 prop 7231 . . . . . . . . . . 11 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
1210, 11syl 14 . . . . . . . . . 10 (𝐴<P 𝐵 → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
13 elprnql 7237 . . . . . . . . . 10 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (1st𝐴)) → 𝑦Q)
1412, 13sylan 279 . . . . . . . . 9 ((𝐴<P 𝐵𝑦 ∈ (1st𝐴)) → 𝑦Q)
155, 7, 14syl2anc 406 . . . . . . . 8 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) → 𝑦Q)
16 ltanqi 7158 . . . . . . . 8 ((𝑞 <Q 𝑟𝑦Q) → (𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟))
174, 15, 16syl2anc 406 . . . . . . 7 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) → (𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟))
189simprd 113 . . . . . . . . 9 (𝐴<P 𝐵𝐵P)
195, 18syl 14 . . . . . . . 8 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) → 𝐵P)
202simprd 113 . . . . . . . 8 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) → (𝑦 +Q 𝑞) ∈ (2nd𝐵))
21 prop 7231 . . . . . . . . 9 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
22 prcunqu 7241 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)) → ((𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟) → (𝑦 +Q 𝑟) ∈ (2nd𝐵)))
2321, 22sylan 279 . . . . . . . 8 ((𝐵P ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)) → ((𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟) → (𝑦 +Q 𝑟) ∈ (2nd𝐵)))
2419, 20, 23syl2anc 406 . . . . . . 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 306 . . . . 5 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) → (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))))
2726eximi 1562 . . . 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 7355 . . . . . . . . 9 (𝑞 ∈ (2nd𝐶) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))
30 19.42v 1860 . . . . . . . . 9 (∃𝑦(𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))
3129, 30bitr4i 186 . . . . . . . 8 (𝑞 ∈ (2nd𝐶) ↔ ∃𝑦(𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))
3231anbi2i 450 . . . . . . 7 ((𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)) ↔ (𝑞 <Q 𝑟 ∧ ∃𝑦(𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
33 19.42v 1860 . . . . . . 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 186 . . . . . 6 ((𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)) ↔ ∃𝑦(𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
3534anbi2i 450 . . . . 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 1860 . . . . 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 186 . . . 4 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶))) ↔ ∃𝑦((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))))
3828ltexprlemelu 7355 . . . . 5 (𝑟 ∈ (2nd𝐶) ↔ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))))
39 19.42v 1860 . . . . 5 (∃𝑦(𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))) ↔ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))))
4038, 39bitr4i 186 . . . 4 (𝑟 ∈ (2nd𝐶) ↔ ∃𝑦(𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))))
4127, 37, 403imtr4i 200 . . 3 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶))) → 𝑟 ∈ (2nd𝐶))
4241ex 114 . 2 ((𝐴<P 𝐵𝑟Q) → ((𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)) → 𝑟 ∈ (2nd𝐶)))
4342rexlimdvw 2527 1 ((𝐴<P 𝐵𝑟Q) → (∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)) → 𝑟 ∈ (2nd𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1314  wex 1451  wcel 1463  wrex 2391  {crab 2394  cop 3496   class class class wbr 3895  cfv 5081  (class class class)co 5728  1st c1st 5990  2nd c2nd 5991  Qcnq 7036   +Q cplq 7038   <Q cltq 7041  Pcnp 7047  <P cltp 7051
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-eprel 4171  df-id 4175  df-iord 4248  df-on 4250  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-recs 6156  df-irdg 6221  df-oadd 6271  df-omul 6272  df-er 6383  df-ec 6385  df-qs 6389  df-ni 7060  df-pli 7061  df-mi 7062  df-lti 7063  df-plpq 7100  df-enq 7103  df-nqqs 7104  df-plqqs 7105  df-ltnqqs 7109  df-inp 7222  df-iltp 7226
This theorem is referenced by:  ltexprlemrnd  7361
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