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Theorem ltexprlemrnd 6761
Description: Our constructed difference is rounded. Lemma for ltexpri 6769. (Contributed by Jim Kingdon, 17-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
ltexprlemrnd (𝐴<P 𝐵 → (∀𝑞Q (𝑞 ∈ (1st𝐶) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐶) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)))))
Distinct variable groups:   𝑥,𝑦,𝑞,𝑟,𝐴   𝑥,𝐵,𝑦,𝑞,𝑟   𝑥,𝐶,𝑦,𝑞,𝑟

Proof of Theorem ltexprlemrnd
StepHypRef Expression
1 ltexprlem.1 . . . . . 6 𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩
21ltexprlemopl 6757 . . . . 5 ((𝐴<P 𝐵𝑞Q𝑞 ∈ (1st𝐶)) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)))
323expia 1117 . . . 4 ((𝐴<P 𝐵𝑞Q) → (𝑞 ∈ (1st𝐶) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶))))
41ltexprlemlol 6758 . . . 4 ((𝐴<P 𝐵𝑞Q) → (∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)) → 𝑞 ∈ (1st𝐶)))
53, 4impbid 124 . . 3 ((𝐴<P 𝐵𝑞Q) → (𝑞 ∈ (1st𝐶) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶))))
65ralrimiva 2409 . 2 (𝐴<P 𝐵 → ∀𝑞Q (𝑞 ∈ (1st𝐶) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶))))
71ltexprlemopu 6759 . . . . 5 ((𝐴<P 𝐵𝑟Q𝑟 ∈ (2nd𝐶)) → ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)))
873expia 1117 . . . 4 ((𝐴<P 𝐵𝑟Q) → (𝑟 ∈ (2nd𝐶) → ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶))))
91ltexprlemupu 6760 . . . 4 ((𝐴<P 𝐵𝑟Q) → (∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)) → 𝑟 ∈ (2nd𝐶)))
108, 9impbid 124 . . 3 ((𝐴<P 𝐵𝑟Q) → (𝑟 ∈ (2nd𝐶) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶))))
1110ralrimiva 2409 . 2 (𝐴<P 𝐵 → ∀𝑟Q (𝑟 ∈ (2nd𝐶) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶))))
126, 11jca 294 1 (𝐴<P 𝐵 → (∀𝑞Q (𝑞 ∈ (1st𝐶) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐶) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wex 1397  wcel 1409  wral 2323  wrex 2324  {crab 2327  cop 3406   class class class wbr 3792  cfv 4930  (class class class)co 5540  1st c1st 5793  2nd c2nd 5794  Qcnq 6436   +Q cplq 6438   <Q cltq 6441  <P cltp 6451
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 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-ltnqqs 6509  df-inp 6622  df-iltp 6626
This theorem is referenced by:  ltexprlempr  6764
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