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Theorem isibl 25517
Description: The predicate "š¹ is integrable". The "integrable" predicate corresponds roughly to the range of validity of āˆ«š“šµ dš‘„, which is to say that the expression āˆ«š“šµ dš‘„ doesn't make sense unless (š‘„ āˆˆ š“ ā†¦ šµ) āˆˆ šæ1. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Hypotheses
Ref Expression
isibl.1 (šœ‘ ā†’ šŗ = (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ š‘‡), š‘‡, 0)))
isibl.2 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ š‘‡ = (ā„œā€˜(šµ / (iā†‘š‘˜))))
isibl.3 (šœ‘ ā†’ dom š¹ = š“)
isibl.4 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (š¹ā€˜š‘„) = šµ)
Assertion
Ref Expression
isibl (šœ‘ ā†’ (š¹ āˆˆ šæ1 ā†” (š¹ āˆˆ MblFn āˆ§ āˆ€š‘˜ āˆˆ (0...3)(āˆ«2ā€˜šŗ) āˆˆ ā„)))
Distinct variable groups:   š‘„,š‘˜,š“   šµ,š‘˜   š‘˜,š¹,š‘„   šœ‘,š‘˜,š‘„
Allowed substitution hints:   šµ(š‘„)   š‘‡(š‘„,š‘˜)   šŗ(š‘„,š‘˜)
Proof of Theorem isibl
Dummy variables š‘“ š‘¦ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6905 . . . . . . . . 9 (ā„œā€˜((š‘“ā€˜š‘„) / (iā†‘š‘˜))) āˆˆ V
2 breq2 5153 . . . . . . . . . . 11 (š‘¦ = (ā„œā€˜((š‘“ā€˜š‘„) / (iā†‘š‘˜))) ā†’ (0 ā‰¤ š‘¦ ā†” 0 ā‰¤ (ā„œā€˜((š‘“ā€˜š‘„) / (iā†‘š‘˜)))))
32anbi2d 627 . . . . . . . . . 10 (š‘¦ = (ā„œā€˜((š‘“ā€˜š‘„) / (iā†‘š‘˜))) ā†’ ((š‘„ āˆˆ dom š‘“ āˆ§ 0 ā‰¤ š‘¦) ā†” (š‘„ āˆˆ dom š‘“ āˆ§ 0 ā‰¤ (ā„œā€˜((š‘“ā€˜š‘„) / (iā†‘š‘˜))))))
4 id 22 . . . . . . . . . 10 (š‘¦ = (ā„œā€˜((š‘“ā€˜š‘„) / (iā†‘š‘˜))) ā†’ š‘¦ = (ā„œā€˜((š‘“ā€˜š‘„) / (iā†‘š‘˜))))
53, 4ifbieq1d 4553 . . . . . . . . 9 (š‘¦ = (ā„œā€˜((š‘“ā€˜š‘„) / (iā†‘š‘˜))) ā†’ if((š‘„ āˆˆ dom š‘“ āˆ§ 0 ā‰¤ š‘¦), š‘¦, 0) = if((š‘„ āˆˆ dom š‘“ āˆ§ 0 ā‰¤ (ā„œā€˜((š‘“ā€˜š‘„) / (iā†‘š‘˜)))), (ā„œā€˜((š‘“ā€˜š‘„) / (iā†‘š‘˜))), 0))
61, 5csbie 3930 . . . . . . . 8 ā¦‹(ā„œā€˜((š‘“ā€˜š‘„) / (iā†‘š‘˜))) / š‘¦ā¦Œif((š‘„ āˆˆ dom š‘“ āˆ§ 0 ā‰¤ š‘¦), š‘¦, 0) = if((š‘„ āˆˆ dom š‘“ āˆ§ 0 ā‰¤ (ā„œā€˜((š‘“ā€˜š‘„) / (iā†‘š‘˜)))), (ā„œā€˜((š‘“ā€˜š‘„) / (iā†‘š‘˜))), 0)
7 dmeq 5904 . . . . . . . . . . 11 (š‘“ = š¹ ā†’ dom š‘“ = dom š¹)
87eleq2d 2817 . . . . . . . . . 10 (š‘“ = š¹ ā†’ (š‘„ āˆˆ dom š‘“ ā†” š‘„ āˆˆ dom š¹))
9 fveq1 6891 . . . . . . . . . . . 12 (š‘“ = š¹ ā†’ (š‘“ā€˜š‘„) = (š¹ā€˜š‘„))
109fvoveq1d 7435 . . . . . . . . . . 11 (š‘“ = š¹ ā†’ (ā„œā€˜((š‘“ā€˜š‘„) / (iā†‘š‘˜))) = (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜))))
1110breq2d 5161 . . . . . . . . . 10 (š‘“ = š¹ ā†’ (0 ā‰¤ (ā„œā€˜((š‘“ā€˜š‘„) / (iā†‘š‘˜))) ā†” 0 ā‰¤ (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜)))))
128, 11anbi12d 629 . . . . . . . . 9 (š‘“ = š¹ ā†’ ((š‘„ āˆˆ dom š‘“ āˆ§ 0 ā‰¤ (ā„œā€˜((š‘“ā€˜š‘„) / (iā†‘š‘˜)))) ā†” (š‘„ āˆˆ dom š¹ āˆ§ 0 ā‰¤ (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜))))))
1312, 10ifbieq1d 4553 . . . . . . . 8 (š‘“ = š¹ ā†’ if((š‘„ āˆˆ dom š‘“ āˆ§ 0 ā‰¤ (ā„œā€˜((š‘“ā€˜š‘„) / (iā†‘š‘˜)))), (ā„œā€˜((š‘“ā€˜š‘„) / (iā†‘š‘˜))), 0) = if((š‘„ āˆˆ dom š¹ āˆ§ 0 ā‰¤ (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜)))), (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜))), 0))
146, 13eqtrid 2782 . . . . . . 7 (š‘“ = š¹ ā†’ ā¦‹(ā„œā€˜((š‘“ā€˜š‘„) / (iā†‘š‘˜))) / š‘¦ā¦Œif((š‘„ āˆˆ dom š‘“ āˆ§ 0 ā‰¤ š‘¦), š‘¦, 0) = if((š‘„ āˆˆ dom š¹ āˆ§ 0 ā‰¤ (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜)))), (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜))), 0))
1514mpteq2dv 5251 . . . . . 6 (š‘“ = š¹ ā†’ (š‘„ āˆˆ ā„ ā†¦ ā¦‹(ā„œā€˜((š‘“ā€˜š‘„) / (iā†‘š‘˜))) / š‘¦ā¦Œif((š‘„ āˆˆ dom š‘“ āˆ§ 0 ā‰¤ š‘¦), š‘¦, 0)) = (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ dom š¹ āˆ§ 0 ā‰¤ (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜)))), (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜))), 0)))
1615fveq2d 6896 . . . . 5 (š‘“ = š¹ ā†’ (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ ā¦‹(ā„œā€˜((š‘“ā€˜š‘„) / (iā†‘š‘˜))) / š‘¦ā¦Œif((š‘„ āˆˆ dom š‘“ āˆ§ 0 ā‰¤ š‘¦), š‘¦, 0))) = (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ dom š¹ āˆ§ 0 ā‰¤ (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜)))), (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜))), 0))))
1716eleq1d 2816 . . . 4 (š‘“ = š¹ ā†’ ((āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ ā¦‹(ā„œā€˜((š‘“ā€˜š‘„) / (iā†‘š‘˜))) / š‘¦ā¦Œif((š‘„ āˆˆ dom š‘“ āˆ§ 0 ā‰¤ š‘¦), š‘¦, 0))) āˆˆ ā„ ā†” (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ dom š¹ āˆ§ 0 ā‰¤ (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜)))), (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜))), 0))) āˆˆ ā„))
1817ralbidv 3175 . . 3 (š‘“ = š¹ ā†’ (āˆ€š‘˜ āˆˆ (0...3)(āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ ā¦‹(ā„œā€˜((š‘“ā€˜š‘„) / (iā†‘š‘˜))) / š‘¦ā¦Œif((š‘„ āˆˆ dom š‘“ āˆ§ 0 ā‰¤ š‘¦), š‘¦, 0))) āˆˆ ā„ ā†” āˆ€š‘˜ āˆˆ (0...3)(āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ dom š¹ āˆ§ 0 ā‰¤ (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜)))), (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜))), 0))) āˆˆ ā„))
19 df-ibl 25373 . . 3 šæ1 = {š‘“ āˆˆ MblFn āˆ£ āˆ€š‘˜ āˆˆ (0...3)(āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ ā¦‹(ā„œā€˜((š‘“ā€˜š‘„) / (iā†‘š‘˜))) / š‘¦ā¦Œif((š‘„ āˆˆ dom š‘“ āˆ§ 0 ā‰¤ š‘¦), š‘¦, 0))) āˆˆ ā„}
2018, 19elrab2 3687 . 2 (š¹ āˆˆ šæ1 ā†” (š¹ āˆˆ MblFn āˆ§ āˆ€š‘˜ āˆˆ (0...3)(āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ dom š¹ āˆ§ 0 ā‰¤ (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜)))), (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜))), 0))) āˆˆ ā„))
21 isibl.3 . . . . . . . . . . . 12 (šœ‘ ā†’ dom š¹ = š“)
2221eleq2d 2817 . . . . . . . . . . 11 (šœ‘ ā†’ (š‘„ āˆˆ dom š¹ ā†” š‘„ āˆˆ š“))
2322anbi1d 628 . . . . . . . . . 10 (šœ‘ ā†’ ((š‘„ āˆˆ dom š¹ āˆ§ 0 ā‰¤ (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜)))) ā†” (š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜))))))
2423ifbid 4552 . . . . . . . . 9 (šœ‘ ā†’ if((š‘„ āˆˆ dom š¹ āˆ§ 0 ā‰¤ (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜)))), (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜))), 0) = if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜)))), (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜))), 0))
25 isibl.4 . . . . . . . . . . . 12 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (š¹ā€˜š‘„) = šµ)
2625fvoveq1d 7435 . . . . . . . . . . 11 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜))) = (ā„œā€˜(šµ / (iā†‘š‘˜))))
27 isibl.2 . . . . . . . . . . 11 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ š‘‡ = (ā„œā€˜(šµ / (iā†‘š‘˜))))
2826, 27eqtr4d 2773 . . . . . . . . . 10 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜))) = š‘‡)
2928ibllem 25516 . . . . . . . . 9 (šœ‘ ā†’ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜)))), (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜))), 0) = if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ š‘‡), š‘‡, 0))
3024, 29eqtrd 2770 . . . . . . . 8 (šœ‘ ā†’ if((š‘„ āˆˆ dom š¹ āˆ§ 0 ā‰¤ (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜)))), (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜))), 0) = if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ š‘‡), š‘‡, 0))
3130mpteq2dv 5251 . . . . . . 7 (šœ‘ ā†’ (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ dom š¹ āˆ§ 0 ā‰¤ (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜)))), (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜))), 0)) = (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ š‘‡), š‘‡, 0)))
32 isibl.1 . . . . . . 7 (šœ‘ ā†’ šŗ = (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ š‘‡), š‘‡, 0)))
3331, 32eqtr4d 2773 . . . . . 6 (šœ‘ ā†’ (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ dom š¹ āˆ§ 0 ā‰¤ (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜)))), (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜))), 0)) = šŗ)
3433fveq2d 6896 . . . . 5 (šœ‘ ā†’ (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ dom š¹ āˆ§ 0 ā‰¤ (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜)))), (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜))), 0))) = (āˆ«2ā€˜šŗ))
3534eleq1d 2816 . . . 4 (šœ‘ ā†’ ((āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ dom š¹ āˆ§ 0 ā‰¤ (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜)))), (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜))), 0))) āˆˆ ā„ ā†” (āˆ«2ā€˜šŗ) āˆˆ ā„))
3635ralbidv 3175 . . 3 (šœ‘ ā†’ (āˆ€š‘˜ āˆˆ (0...3)(āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ dom š¹ āˆ§ 0 ā‰¤ (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜)))), (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜))), 0))) āˆˆ ā„ ā†” āˆ€š‘˜ āˆˆ (0...3)(āˆ«2ā€˜šŗ) āˆˆ ā„))
3736anbi2d 627 . 2 (šœ‘ ā†’ ((š¹ āˆˆ MblFn āˆ§ āˆ€š‘˜ āˆˆ (0...3)(āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ dom š¹ āˆ§ 0 ā‰¤ (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜)))), (ā„œā€˜((š¹ā€˜š‘„) / (iā†‘š‘˜))), 0))) āˆˆ ā„) ā†” (š¹ āˆˆ MblFn āˆ§ āˆ€š‘˜ āˆˆ (0...3)(āˆ«2ā€˜šŗ) āˆˆ ā„)))
3820, 37bitrid 282 1 (šœ‘ ā†’ (š¹ āˆˆ šæ1 ā†” (š¹ āˆˆ MblFn āˆ§ āˆ€š‘˜ āˆˆ (0...3)(āˆ«2ā€˜šŗ) āˆˆ ā„)))
Colors of variables: wff setvar class
Syntax hints:   ā†’ wi 4   ā†” wb 205   āˆ§ wa 394   = wceq 1539   āˆˆ wcel 2104  āˆ€wral 3059  ā¦‹csb 3894  ifcif 4529   class class class wbr 5149   ā†¦ cmpt 5232  dom cdm 5677  ā€˜cfv 6544  (class class class)co 7413  ā„cr 11113  0cc0 11114  ici 11116   ā‰¤ cle 11255   / cdiv 11877  3c3 12274  ...cfz 13490  ā†‘cexp 14033  ā„œcre 15050  MblFncmbf 25365  āˆ«2citg2 25367  šæ1cibl 25368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rab 3431  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-dm 5687  df-iota 6496  df-fv 6552  df-ov 7416  df-ibl 25373
This theorem is referenced by:  isibl2  25518  ibl0  25538  iblempty  44981
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