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Theorem ltexprlemupu 7594
Description: The upper cut of our constructed difference is upper. Lemma for ltexpri 7603. (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 528 . . . . . 6 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) → 𝑟Q)
2 simprrr 540 . . . . . . 7 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) → (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))
32simpld 112 . . . . . 6 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) → 𝑦 ∈ (1st𝐴))
4 simprl 529 . . . . . . . 8 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) → 𝑞 <Q 𝑟)
5 simpll 527 . . . . . . . . 9 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) → 𝐴<P 𝐵)
6 simprrl 539 . . . . . . . . . 10 ((𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))) → 𝑦 ∈ (1st𝐴))
76adantl 277 . . . . . . . . 9 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) → 𝑦 ∈ (1st𝐴))
8 ltrelpr 7495 . . . . . . . . . . . . 13 <P ⊆ (P × P)
98brel 4675 . . . . . . . . . . . 12 (𝐴<P 𝐵 → (𝐴P𝐵P))
109simpld 112 . . . . . . . . . . 11 (𝐴<P 𝐵𝐴P)
11 prop 7465 . . . . . . . . . . 11 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
1210, 11syl 14 . . . . . . . . . 10 (𝐴<P 𝐵 → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
13 elprnql 7471 . . . . . . . . . 10 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (1st𝐴)) → 𝑦Q)
1412, 13sylan 283 . . . . . . . . 9 ((𝐴<P 𝐵𝑦 ∈ (1st𝐴)) → 𝑦Q)
155, 7, 14syl2anc 411 . . . . . . . 8 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) → 𝑦Q)
16 ltanqi 7392 . . . . . . . 8 ((𝑞 <Q 𝑟𝑦Q) → (𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟))
174, 15, 16syl2anc 411 . . . . . . 7 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) → (𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟))
189simprd 114 . . . . . . . . 9 (𝐴<P 𝐵𝐵P)
195, 18syl 14 . . . . . . . 8 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) → 𝐵P)
202simprd 114 . . . . . . . 8 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) → (𝑦 +Q 𝑞) ∈ (2nd𝐵))
21 prop 7465 . . . . . . . . 9 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
22 prcunqu 7475 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)) → ((𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟) → (𝑦 +Q 𝑟) ∈ (2nd𝐵)))
2321, 22sylan 283 . . . . . . . 8 ((𝐵P ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)) → ((𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟) → (𝑦 +Q 𝑟) ∈ (2nd𝐵)))
2419, 20, 23syl2anc 411 . . . . . . 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 310 . . . . 5 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))) → (𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))))
2726eximi 1600 . . . 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 7589 . . . . . . . . 9 (𝑞 ∈ (2nd𝐶) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))
30 19.42v 1906 . . . . . . . . 9 (∃𝑦(𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))
3129, 30bitr4i 187 . . . . . . . 8 (𝑞 ∈ (2nd𝐶) ↔ ∃𝑦(𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))
3231anbi2i 457 . . . . . . 7 ((𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)) ↔ (𝑞 <Q 𝑟 ∧ ∃𝑦(𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
33 19.42v 1906 . . . . . . 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 187 . . . . . 6 ((𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)) ↔ ∃𝑦(𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵)))))
3534anbi2i 457 . . . . 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 1906 . . . . 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 187 . . . 4 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶))) ↔ ∃𝑦((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑞Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑞) ∈ (2nd𝐵))))))
3828ltexprlemelu 7589 . . . . 5 (𝑟 ∈ (2nd𝐶) ↔ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))))
39 19.42v 1906 . . . . 5 (∃𝑦(𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))) ↔ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))))
4038, 39bitr4i 187 . . . 4 (𝑟 ∈ (2nd𝐶) ↔ ∃𝑦(𝑟Q ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑟) ∈ (2nd𝐵))))
4127, 37, 403imtr4i 201 . . 3 (((𝐴<P 𝐵𝑟Q) ∧ (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶))) → 𝑟 ∈ (2nd𝐶))
4241ex 115 . 2 ((𝐴<P 𝐵𝑟Q) → ((𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)) → 𝑟 ∈ (2nd𝐶)))
4342rexlimdvw 2598 1 ((𝐴<P 𝐵𝑟Q) → (∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)) → 𝑟 ∈ (2nd𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wex 1492  wcel 2148  wrex 2456  {crab 2459  cop 3594   class class class wbr 4000  cfv 5212  (class class class)co 5869  1st c1st 6133  2nd c2nd 6134  Qcnq 7270   +Q cplq 7272   <Q cltq 7275  Pcnp 7281  <P cltp 7285
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 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  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-ne 2348  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-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-ltnqqs 7343  df-inp 7456  df-iltp 7460
This theorem is referenced by:  ltexprlemrnd  7595
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