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Theorem leisorel 13456
Description: Version of isorel 6740 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 13455 . . . 4 ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → (𝐹 Isom < , < (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ≤ (𝐴, 𝐵)))
21biimpcd 239 . . 3 (𝐹 Isom < , < (𝐴, 𝐵) → ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → 𝐹 Isom ≤ , ≤ (𝐴, 𝐵)))
3 isorel 6740 . . . 4 ((𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝐶𝐷 ↔ (𝐹𝐶) ≤ (𝐹𝐷)))
43ex 449 . . 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 1102 1 ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → (𝐶𝐷 ↔ (𝐹𝐶) ≤ (𝐹𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072  wcel 2139  wss 3715   class class class wbr 4804  cfv 6049   Isom wiso 6050  *cxr 10285   < clt 10286  cle 10287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-le 10292
This theorem is referenced by:  seqcoll  13460  isercolllem2  14615  isercoll  14617  summolem2a  14665  prodmolem2a  14883  xrhmeo  22966  fourierdlem52  40896
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