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Theorem ltexprlemrnd 7420
Description: Our constructed difference is rounded. Lemma for ltexpri 7428. (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 7416 . . . . 5 ((𝐴<P 𝐵𝑞Q𝑞 ∈ (1st𝐶)) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)))
323expia 1183 . . . 4 ((𝐴<P 𝐵𝑞Q) → (𝑞 ∈ (1st𝐶) → ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶))))
41ltexprlemlol 7417 . . . 4 ((𝐴<P 𝐵𝑞Q) → (∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶)) → 𝑞 ∈ (1st𝐶)))
53, 4impbid 128 . . 3 ((𝐴<P 𝐵𝑞Q) → (𝑞 ∈ (1st𝐶) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶))))
65ralrimiva 2505 . 2 (𝐴<P 𝐵 → ∀𝑞Q (𝑞 ∈ (1st𝐶) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶))))
71ltexprlemopu 7418 . . . . 5 ((𝐴<P 𝐵𝑟Q𝑟 ∈ (2nd𝐶)) → ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)))
873expia 1183 . . . 4 ((𝐴<P 𝐵𝑟Q) → (𝑟 ∈ (2nd𝐶) → ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶))))
91ltexprlemupu 7419 . . . 4 ((𝐴<P 𝐵𝑟Q) → (∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)) → 𝑟 ∈ (2nd𝐶)))
108, 9impbid 128 . . 3 ((𝐴<P 𝐵𝑟Q) → (𝑟 ∈ (2nd𝐶) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶))))
1110ralrimiva 2505 . 2 (𝐴<P 𝐵 → ∀𝑟Q (𝑟 ∈ (2nd𝐶) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶))))
126, 11jca 304 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 103  wb 104   = wceq 1331  wex 1468  wcel 1480  wral 2416  wrex 2417  {crab 2420  cop 3530   class class class wbr 3929  cfv 5123  (class class class)co 5774  1st c1st 6036  2nd c2nd 6037  Qcnq 7095   +Q cplq 7097   <Q cltq 7100  <P cltp 7110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7119  df-pli 7120  df-mi 7121  df-lti 7122  df-plpq 7159  df-mpq 7160  df-enq 7162  df-nqqs 7163  df-plqqs 7164  df-mqqs 7165  df-1nqqs 7166  df-ltnqqs 7168  df-inp 7281  df-iltp 7285
This theorem is referenced by:  ltexprlempr  7423
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