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Theorem leisorel 14461
Description: Version of isorel 7340 for strictly increasing functions on the reals. (Contributed by Mario Carneiro, 6-Apr-2015.) (Revised by Mario Carneiro, 9-Sep-2015.)
Assertion
Ref Expression
leisorel ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → (𝐶𝐷 ↔ (𝐹𝐶) ≤ (𝐹𝐷)))

Proof of Theorem leisorel
StepHypRef Expression
1 leiso 14460 . . . 4 ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → (𝐹 Isom < , < (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ≤ (𝐴, 𝐵)))
21biimpcd 248 . . 3 (𝐹 Isom < , < (𝐴, 𝐵) → ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → 𝐹 Isom ≤ , ≤ (𝐴, 𝐵)))
3 isorel 7340 . . . 4 ((𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝐶𝐷 ↔ (𝐹𝐶) ≤ (𝐹𝐷)))
43ex 411 . . 3 (𝐹 Isom ≤ , ≤ (𝐴, 𝐵) → ((𝐶𝐴𝐷𝐴) → (𝐶𝐷 ↔ (𝐹𝐶) ≤ (𝐹𝐷))))
52, 4syl6 35 . 2 (𝐹 Isom < , < (𝐴, 𝐵) → ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → ((𝐶𝐴𝐷𝐴) → (𝐶𝐷 ↔ (𝐹𝐶) ≤ (𝐹𝐷)))))
653imp 1108 1 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → (𝐶𝐷 ↔ (𝐹𝐶) ≤ (𝐹𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084  wcel 2098  wss 3949   class class class wbr 5152  cfv 6553   Isom wiso 6554  *cxr 11285   < clt 11286  cle 11287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-isom 6562  df-le 11292
This theorem is referenced by:  seqcoll  14465  isercolllem2  15652  isercoll  15654  summolem2a  15701  prodmolem2a  15918  xrhmeo  24891  fourierdlem52  45575
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