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Theorem isibl2 25284
Description: The predicate "š¹ is integrable" when š¹ is a mapping operation. (Contributed by Mario Carneiro, 31-Jul-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‘˜))))
isibl2.3 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ šµ āˆˆ š‘‰)
Assertion
Ref Expression
isibl2 (šœ‘ ā†’ ((š‘„ āˆˆ š“ ā†¦ šµ) āˆˆ šæ1 ā†” ((š‘„ āˆˆ š“ ā†¦ šµ) āˆˆ MblFn āˆ§ āˆ€š‘˜ āˆˆ (0...3)(āˆ«2ā€˜šŗ) āˆˆ ā„)))
Distinct variable groups:   š‘„,š‘˜,š“   šµ,š‘˜   šœ‘,š‘˜,š‘„   š‘„,š‘‰
Allowed substitution hints:   šµ(š‘„)   š‘‡(š‘„,š‘˜)   šŗ(š‘„,š‘˜)   š‘‰(š‘˜)
Proof of Theorem isibl2
Dummy variable š‘¦ is distinct from all other variables.
StepHypRef Expression
1 isibl.1 . . 3 (šœ‘ ā†’ šŗ = (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ š‘‡), š‘‡, 0)))
2 nfv 1918 . . . . . . 7 ā„²š‘„ š‘¦ āˆˆ š“
3 nfcv 2904 . . . . . . . 8 ā„²š‘„0
4 nfcv 2904 . . . . . . . 8 ā„²š‘„ ā‰¤
5 nfcv 2904 . . . . . . . . 9 ā„²š‘„ā„œ
6 nffvmpt1 6903 . . . . . . . . . 10 ā„²š‘„((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘¦)
7 nfcv 2904 . . . . . . . . . 10 ā„²š‘„ /
8 nfcv 2904 . . . . . . . . . 10 ā„²š‘„(iā†‘š‘˜)
96, 7, 8nfov 7439 . . . . . . . . 9 ā„²š‘„(((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘¦) / (iā†‘š‘˜))
105, 9nffv 6902 . . . . . . . 8 ā„²š‘„(ā„œā€˜(((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘¦) / (iā†‘š‘˜)))
113, 4, 10nfbr 5196 . . . . . . 7 ā„²š‘„0 ā‰¤ (ā„œā€˜(((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘¦) / (iā†‘š‘˜)))
122, 11nfan 1903 . . . . . 6 ā„²š‘„(š‘¦ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘¦) / (iā†‘š‘˜))))
1312, 10, 3nfif 4559 . . . . 5 ā„²š‘„if((š‘¦ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘¦) / (iā†‘š‘˜)))), (ā„œā€˜(((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘¦) / (iā†‘š‘˜))), 0)
14 nfcv 2904 . . . . 5 ā„²š‘¦if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘„) / (iā†‘š‘˜)))), (ā„œā€˜(((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘„) / (iā†‘š‘˜))), 0)
15 eleq1w 2817 . . . . . . 7 (š‘¦ = š‘„ ā†’ (š‘¦ āˆˆ š“ ā†” š‘„ āˆˆ š“))
16 fveq2 6892 . . . . . . . . 9 (š‘¦ = š‘„ ā†’ ((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘¦) = ((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘„))
1716fvoveq1d 7431 . . . . . . . 8 (š‘¦ = š‘„ ā†’ (ā„œā€˜(((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘¦) / (iā†‘š‘˜))) = (ā„œā€˜(((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘„) / (iā†‘š‘˜))))
1817breq2d 5161 . . . . . . 7 (š‘¦ = š‘„ ā†’ (0 ā‰¤ (ā„œā€˜(((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘¦) / (iā†‘š‘˜))) ā†” 0 ā‰¤ (ā„œā€˜(((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘„) / (iā†‘š‘˜)))))
1915, 18anbi12d 632 . . . . . 6 (š‘¦ = š‘„ ā†’ ((š‘¦ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘¦) / (iā†‘š‘˜)))) ā†” (š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘„) / (iā†‘š‘˜))))))
2019, 17ifbieq1d 4553 . . . . 5 (š‘¦ = š‘„ ā†’ if((š‘¦ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘¦) / (iā†‘š‘˜)))), (ā„œā€˜(((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘¦) / (iā†‘š‘˜))), 0) = if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘„) / (iā†‘š‘˜)))), (ā„œā€˜(((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘„) / (iā†‘š‘˜))), 0))
2113, 14, 20cbvmpt 5260 . . . 4 (š‘¦ āˆˆ ā„ ā†¦ if((š‘¦ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘¦) / (iā†‘š‘˜)))), (ā„œā€˜(((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘¦) / (iā†‘š‘˜))), 0)) = (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘„) / (iā†‘š‘˜)))), (ā„œā€˜(((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘„) / (iā†‘š‘˜))), 0))
22 simpr 486 . . . . . . . . 9 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ š‘„ āˆˆ š“)
23 isibl2.3 . . . . . . . . 9 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ šµ āˆˆ š‘‰)
24 eqid 2733 . . . . . . . . . 10 (š‘„ āˆˆ š“ ā†¦ šµ) = (š‘„ āˆˆ š“ ā†¦ šµ)
2524fvmpt2 7010 . . . . . . . . 9 ((š‘„ āˆˆ š“ āˆ§ šµ āˆˆ š‘‰) ā†’ ((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘„) = šµ)
2622, 23, 25syl2anc 585 . . . . . . . 8 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ ((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘„) = šµ)
2726fvoveq1d 7431 . . . . . . 7 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (ā„œā€˜(((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘„) / (iā†‘š‘˜))) = (ā„œā€˜(šµ / (iā†‘š‘˜))))
28 isibl.2 . . . . . . 7 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ š‘‡ = (ā„œā€˜(šµ / (iā†‘š‘˜))))
2927, 28eqtr4d 2776 . . . . . 6 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (ā„œā€˜(((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘„) / (iā†‘š‘˜))) = š‘‡)
3029ibllem 25282 . . . . 5 (šœ‘ ā†’ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘„) / (iā†‘š‘˜)))), (ā„œā€˜(((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘„) / (iā†‘š‘˜))), 0) = if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ š‘‡), š‘‡, 0))
3130mpteq2dv 5251 . . . 4 (šœ‘ ā†’ (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘„) / (iā†‘š‘˜)))), (ā„œā€˜(((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘„) / (iā†‘š‘˜))), 0)) = (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ š‘‡), š‘‡, 0)))
3221, 31eqtrid 2785 . . 3 (šœ‘ ā†’ (š‘¦ āˆˆ ā„ ā†¦ if((š‘¦ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘¦) / (iā†‘š‘˜)))), (ā„œā€˜(((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘¦) / (iā†‘š‘˜))), 0)) = (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ š‘‡), š‘‡, 0)))
331, 32eqtr4d 2776 . 2 (šœ‘ ā†’ šŗ = (š‘¦ āˆˆ ā„ ā†¦ if((š‘¦ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘¦) / (iā†‘š‘˜)))), (ā„œā€˜(((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘¦) / (iā†‘š‘˜))), 0)))
34 eqidd 2734 . 2 ((šœ‘ āˆ§ š‘¦ āˆˆ š“) ā†’ (ā„œā€˜(((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘¦) / (iā†‘š‘˜))) = (ā„œā€˜(((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘¦) / (iā†‘š‘˜))))
3524, 23dmmptd 6696 . 2 (šœ‘ ā†’ dom (š‘„ āˆˆ š“ ā†¦ šµ) = š“)
36 eqidd 2734 . 2 ((šœ‘ āˆ§ š‘¦ āˆˆ š“) ā†’ ((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘¦) = ((š‘„ āˆˆ š“ ā†¦ šµ)ā€˜š‘¦))
3733, 34, 35, 36isibl 25283 1 (šœ‘ ā†’ ((š‘„ āˆˆ š“ ā†¦ šµ) āˆˆ šæ1 ā†” ((š‘„ āˆˆ š“ ā†¦ šµ) 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 397   = wceq 1542   āˆˆ wcel 2107  āˆ€wral 3062  ifcif 4529   class class class wbr 5149   ā†¦ cmpt 5232  ā€˜cfv 6544  (class class class)co 7409  ā„cr 11109  0cc0 11110  ici 11112   ā‰¤ cle 11249   / cdiv 11871  3c3 12268  ...cfz 13484  ā†‘cexp 14027  ā„œcre 15044  MblFncmbf 25131  āˆ«2citg2 25133  šæ1cibl 25134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  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-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-ibl 25139
This theorem is referenced by:  iblitg  25286  iblcnlem1  25305  iblss  25322  iblss2  25323  itgeqa  25331  iblconst  25335  iblabsr  25347  iblmulc2  25348  iblmulc2nc  36553  iblsplit  44682
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