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Theorem leisorel 14031
Description: Version of isorel 7140 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 14030 . . . 4 ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → (𝐹 Isom < , < (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ≤ (𝐴, 𝐵)))
21biimpcd 252 . . 3 (𝐹 Isom < , < (𝐴, 𝐵) → ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → 𝐹 Isom ≤ , ≤ (𝐴, 𝐵)))
3 isorel 7140 . . . 4 ((𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝐶𝐷 ↔ (𝐹𝐶) ≤ (𝐹𝐷)))
43ex 416 . . 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 1113 1 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → (𝐶𝐷 ↔ (𝐹𝐶) ≤ (𝐹𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089  wcel 2110  wss 3871   class class class wbr 5058  cfv 6385   Isom wiso 6386  *cxr 10871   < clt 10872  cle 10873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5197  ax-nul 5204  ax-pr 5327
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3415  df-dif 3874  df-un 3876  df-in 3878  df-ss 3888  df-nul 4243  df-if 4445  df-sn 4547  df-pr 4549  df-op 4553  df-uni 4825  df-br 5059  df-opab 5121  df-id 5460  df-xp 5562  df-rel 5563  df-cnv 5564  df-co 5565  df-dm 5566  df-rn 5567  df-res 5568  df-ima 5569  df-iota 6343  df-fun 6387  df-fn 6388  df-f 6389  df-f1 6390  df-fo 6391  df-f1o 6392  df-fv 6393  df-isom 6394  df-le 10878
This theorem is referenced by:  seqcoll  14035  isercolllem2  15234  isercoll  15236  summolem2a  15284  prodmolem2a  15501  xrhmeo  23848  fourierdlem52  43382
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