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Theorem ltexprlemlol 7600
Description: The lower cut of our constructed difference is lower. Lemma for ltexpri 7611. (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 528 . . . . . 6 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) → 𝑞Q)
2 simprrr 540 . . . . . . 7 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) → (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))
32simpld 112 . . . . . 6 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) → 𝑦 ∈ (2nd𝐴))
4 simprl 529 . . . . . . . 8 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) → 𝑞 <Q 𝑟)
5 simpll 527 . . . . . . . . 9 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) → 𝐴<P 𝐵)
6 ltrelpr 7503 . . . . . . . . . . . 12 <P ⊆ (P × P)
76brel 4678 . . . . . . . . . . 11 (𝐴<P 𝐵 → (𝐴P𝐵P))
87simpld 112 . . . . . . . . . 10 (𝐴<P 𝐵𝐴P)
9 prop 7473 . . . . . . . . . . 11 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
10 elprnqu 7480 . . . . . . . . . . 11 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (2nd𝐴)) → 𝑦Q)
119, 10sylan 283 . . . . . . . . . 10 ((𝐴P𝑦 ∈ (2nd𝐴)) → 𝑦Q)
128, 11sylan 283 . . . . . . . . 9 ((𝐴<P 𝐵𝑦 ∈ (2nd𝐴)) → 𝑦Q)
135, 3, 12syl2anc 411 . . . . . . . 8 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) → 𝑦Q)
14 ltanqi 7400 . . . . . . . 8 ((𝑞 <Q 𝑟𝑦Q) → (𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟))
154, 13, 14syl2anc 411 . . . . . . 7 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) → (𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟))
167simprd 114 . . . . . . . . 9 (𝐴<P 𝐵𝐵P)
175, 16syl 14 . . . . . . . 8 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) → 𝐵P)
182simprd 114 . . . . . . . 8 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) → (𝑦 +Q 𝑟) ∈ (1st𝐵))
19 prop 7473 . . . . . . . . 9 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
20 prcdnql 7482 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)) → ((𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟) → (𝑦 +Q 𝑞) ∈ (1st𝐵)))
2119, 20sylan 283 . . . . . . . 8 ((𝐵P ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)) → ((𝑦 +Q 𝑞) <Q (𝑦 +Q 𝑟) → (𝑦 +Q 𝑞) ∈ (1st𝐵)))
2217, 18, 21syl2anc 411 . . . . . . 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 310 . . . . 5 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))) → (𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
2524eximi 1600 . . . 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 7596 . . . . . . . . 9 (𝑟 ∈ (1st𝐶) ↔ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))
28 19.42v 1906 . . . . . . . . 9 (∃𝑦(𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))) ↔ (𝑟Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))
2927, 28bitr4i 187 . . . . . . . 8 (𝑟 ∈ (1st𝐶) ↔ ∃𝑦(𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))
3029anbi2i 457 . . . . . . 7 ((𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)) ↔ (𝑞 <Q 𝑟 ∧ ∃𝑦(𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))))
31 19.42v 1906 . . . . . . 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 187 . . . . . 6 ((𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)) ↔ ∃𝑦(𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵)))))
3332anbi2i 457 . . . . 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 1906 . . . . 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 187 . . . 4 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶))) ↔ ∃𝑦((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟 ∧ (𝑟Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑟) ∈ (1st𝐵))))))
3626ltexprlemell 7596 . . . . 5 (𝑞 ∈ (1st𝐶) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
37 19.42v 1906 . . . . 5 (∃𝑦(𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))) ↔ (𝑞Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
3836, 37bitr4i 187 . . . 4 (𝑞 ∈ (1st𝐶) ↔ ∃𝑦(𝑞Q ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑞) ∈ (1st𝐵))))
3925, 35, 383imtr4i 201 . . 3 (((𝐴<P 𝐵𝑞Q) ∧ (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶))) → 𝑞 ∈ (1st𝐶))
4039ex 115 . 2 ((𝐴<P 𝐵𝑞Q) → ((𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)) → 𝑞 ∈ (1st𝐶)))
4140rexlimdvw 2598 1 ((𝐴<P 𝐵𝑞Q) → (∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)) → 𝑞 ∈ (1st𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wex 1492  wcel 2148  wrex 2456  {crab 2459  cop 3595   class class class wbr 4003  cfv 5216  (class class class)co 5874  1st c1st 6138  2nd c2nd 6139  Qcnq 7278   +Q cplq 7280   <Q cltq 7283  Pcnp 7289  <P cltp 7293
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 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
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 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-lti 7305  df-plpq 7342  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-ltnqqs 7351  df-inp 7464  df-iltp 7468
This theorem is referenced by:  ltexprlemrnd  7603
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