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Theorem leisorel 14457
Description: Version of isorel 7333 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 14456 . . . 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 7333 . . . 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 3944   class class class wbr 5149  cfv 6549   Isom wiso 6550  *cxr 11279   < clt 11280  cle 11281
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 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-le 11286
This theorem is referenced by:  seqcoll  14461  isercolllem2  15648  isercoll  15650  summolem2a  15697  prodmolem2a  15914  xrhmeo  24915  fourierdlem52  45684
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