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Theorem iblitg 24974
Description: If a function is integrable, then the āˆ«2 integrals of the function's decompositions all exist. (Contributed by Mario Carneiro, 7-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Hypotheses
Ref Expression
iblitg.1 (šœ‘ ā†’ šŗ = (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ š‘‡), š‘‡, 0)))
iblitg.2 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ š‘‡ = (ā„œā€˜(šµ / (iā†‘š¾))))
iblitg.3 (šœ‘ ā†’ (š‘„ āˆˆ š“ ā†¦ šµ) āˆˆ šæ1)
iblitg.4 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ šµ āˆˆ š‘‰)
Assertion
Ref Expression
iblitg ((šœ‘ āˆ§ š¾ āˆˆ ā„¤) ā†’ (āˆ«2ā€˜šŗ) āˆˆ ā„)
Distinct variable groups:   š‘„,š“   š‘„,š¾   šœ‘,š‘„   š‘„,š‘‰
Allowed substitution hints:   šµ(š‘„)   š‘‡(š‘„)   šŗ(š‘„)

Proof of Theorem iblitg
Dummy variable š‘˜ is distinct from all other variables.
StepHypRef Expression
1 iblitg.1 . . . . 5 (šœ‘ ā†’ šŗ = (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ š‘‡), š‘‡, 0)))
21adantr 482 . . . 4 ((šœ‘ āˆ§ š¾ āˆˆ ā„¤) ā†’ šŗ = (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ š‘‡), š‘‡, 0)))
3 iblitg.2 . . . . . . . 8 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ š‘‡ = (ā„œā€˜(šµ / (iā†‘š¾))))
43adantlr 713 . . . . . . 7 (((šœ‘ āˆ§ š¾ āˆˆ ā„¤) āˆ§ š‘„ āˆˆ š“) ā†’ š‘‡ = (ā„œā€˜(šµ / (iā†‘š¾))))
5 iexpcyc 13965 . . . . . . . . . 10 (š¾ āˆˆ ā„¤ ā†’ (iā†‘(š¾ mod 4)) = (iā†‘š¾))
65oveq2d 7319 . . . . . . . . 9 (š¾ āˆˆ ā„¤ ā†’ (šµ / (iā†‘(š¾ mod 4))) = (šµ / (iā†‘š¾)))
76fveq2d 6804 . . . . . . . 8 (š¾ āˆˆ ā„¤ ā†’ (ā„œā€˜(šµ / (iā†‘(š¾ mod 4)))) = (ā„œā€˜(šµ / (iā†‘š¾))))
87ad2antlr 725 . . . . . . 7 (((šœ‘ āˆ§ š¾ āˆˆ ā„¤) āˆ§ š‘„ āˆˆ š“) ā†’ (ā„œā€˜(šµ / (iā†‘(š¾ mod 4)))) = (ā„œā€˜(šµ / (iā†‘š¾))))
94, 8eqtr4d 2779 . . . . . 6 (((šœ‘ āˆ§ š¾ āˆˆ ā„¤) āˆ§ š‘„ āˆˆ š“) ā†’ š‘‡ = (ā„œā€˜(šµ / (iā†‘(š¾ mod 4)))))
109ibllem 24970 . . . . 5 ((šœ‘ āˆ§ š¾ āˆˆ ā„¤) ā†’ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ š‘‡), š‘‡, 0) = if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘(š¾ mod 4))))), (ā„œā€˜(šµ / (iā†‘(š¾ mod 4)))), 0))
1110mpteq2dv 5183 . . . 4 ((šœ‘ āˆ§ š¾ āˆˆ ā„¤) ā†’ (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ š‘‡), š‘‡, 0)) = (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘(š¾ mod 4))))), (ā„œā€˜(šµ / (iā†‘(š¾ mod 4)))), 0)))
122, 11eqtrd 2776 . . 3 ((šœ‘ āˆ§ š¾ āˆˆ ā„¤) ā†’ šŗ = (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘(š¾ mod 4))))), (ā„œā€˜(šµ / (iā†‘(š¾ mod 4)))), 0)))
1312fveq2d 6804 . 2 ((šœ‘ āˆ§ š¾ āˆˆ ā„¤) ā†’ (āˆ«2ā€˜šŗ) = (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘(š¾ mod 4))))), (ā„œā€˜(šµ / (iā†‘(š¾ mod 4)))), 0))))
14 oveq2 7311 . . . . . . . . . . 11 (š‘˜ = (š¾ mod 4) ā†’ (iā†‘š‘˜) = (iā†‘(š¾ mod 4)))
1514oveq2d 7319 . . . . . . . . . 10 (š‘˜ = (š¾ mod 4) ā†’ (šµ / (iā†‘š‘˜)) = (šµ / (iā†‘(š¾ mod 4))))
1615fveq2d 6804 . . . . . . . . 9 (š‘˜ = (š¾ mod 4) ā†’ (ā„œā€˜(šµ / (iā†‘š‘˜))) = (ā„œā€˜(šµ / (iā†‘(š¾ mod 4)))))
1716breq2d 5093 . . . . . . . 8 (š‘˜ = (š¾ mod 4) ā†’ (0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜))) ā†” 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘(š¾ mod 4))))))
1817anbi2d 630 . . . . . . 7 (š‘˜ = (š¾ mod 4) ā†’ ((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))) ā†” (š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘(š¾ mod 4)))))))
1918, 16ifbieq1d 4489 . . . . . 6 (š‘˜ = (š¾ mod 4) ā†’ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0) = if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘(š¾ mod 4))))), (ā„œā€˜(šµ / (iā†‘(š¾ mod 4)))), 0))
2019mpteq2dv 5183 . . . . 5 (š‘˜ = (š¾ mod 4) ā†’ (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0)) = (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘(š¾ mod 4))))), (ā„œā€˜(šµ / (iā†‘(š¾ mod 4)))), 0)))
2120fveq2d 6804 . . . 4 (š‘˜ = (š¾ mod 4) ā†’ (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0))) = (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘(š¾ mod 4))))), (ā„œā€˜(šµ / (iā†‘(š¾ mod 4)))), 0))))
2221eleq1d 2821 . . 3 (š‘˜ = (š¾ mod 4) ā†’ ((āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0))) āˆˆ ā„ ā†” (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘(š¾ mod 4))))), (ā„œā€˜(šµ / (iā†‘(š¾ mod 4)))), 0))) āˆˆ ā„))
23 iblitg.3 . . . . . 6 (šœ‘ ā†’ (š‘„ āˆˆ š“ ā†¦ šµ) āˆˆ šæ1)
24 eqidd 2737 . . . . . . 7 (šœ‘ ā†’ (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0)) = (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0)))
25 eqidd 2737 . . . . . . 7 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (ā„œā€˜(šµ / (iā†‘š‘˜))) = (ā„œā€˜(šµ / (iā†‘š‘˜))))
26 iblitg.4 . . . . . . 7 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ šµ āˆˆ š‘‰)
2724, 25, 26isibl2 24972 . . . . . 6 (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‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0))) āˆˆ ā„)))
2823, 27mpbid 232 . . . . 5 (šœ‘ ā†’ ((š‘„ āˆˆ š“ ā†¦ šµ) āˆˆ MblFn āˆ§ āˆ€š‘˜ āˆˆ (0...3)(āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0))) āˆˆ ā„))
2928simprd 497 . . . 4 (šœ‘ ā†’ āˆ€š‘˜ āˆˆ (0...3)(āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0))) āˆˆ ā„)
3029adantr 482 . . 3 ((šœ‘ āˆ§ š¾ āˆˆ ā„¤) ā†’ āˆ€š‘˜ āˆˆ (0...3)(āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))), (ā„œā€˜(šµ / (iā†‘š‘˜))), 0))) āˆˆ ā„)
31 4nn 12098 . . . . . 6 4 āˆˆ ā„•
32 zmodfz 13655 . . . . . 6 ((š¾ āˆˆ ā„¤ āˆ§ 4 āˆˆ ā„•) ā†’ (š¾ mod 4) āˆˆ (0...(4 āˆ’ 1)))
3331, 32mpan2 689 . . . . 5 (š¾ āˆˆ ā„¤ ā†’ (š¾ mod 4) āˆˆ (0...(4 āˆ’ 1)))
34 4m1e3 12144 . . . . . 6 (4 āˆ’ 1) = 3
3534oveq2i 7314 . . . . 5 (0...(4 āˆ’ 1)) = (0...3)
3633, 35eleqtrdi 2847 . . . 4 (š¾ āˆˆ ā„¤ ā†’ (š¾ mod 4) āˆˆ (0...3))
3736adantl 483 . . 3 ((šœ‘ āˆ§ š¾ āˆˆ ā„¤) ā†’ (š¾ mod 4) āˆˆ (0...3))
3822, 30, 37rspcdva 3567 . 2 ((šœ‘ āˆ§ š¾ āˆˆ ā„¤) ā†’ (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘(š¾ mod 4))))), (ā„œā€˜(šµ / (iā†‘(š¾ mod 4)))), 0))) āˆˆ ā„)
3913, 38eqeltrd 2837 1 ((šœ‘ āˆ§ š¾ āˆˆ ā„¤) ā†’ (āˆ«2ā€˜šŗ) āˆˆ ā„)
Colors of variables: wff setvar class
Syntax hints:   ā†’ wi 4   āˆ§ wa 397   = wceq 1539   āˆˆ wcel 2104  āˆ€wral 3062  ifcif 4465   class class class wbr 5081   ā†¦ cmpt 5164  ā€˜cfv 6454  (class class class)co 7303  ā„cr 10912  0cc0 10913  1c1 10914  ici 10915   ā‰¤ cle 11052   āˆ’ cmin 11247   / cdiv 11674  ā„•cn 12015  3c3 12071  4c4 12072  ā„¤cz 12361  ...cfz 13281   mod cmo 13631  ā†‘cexp 13824  ā„œcre 14849  MblFncmbf 24819  āˆ«2citg2 24821  šæ1cibl 24822
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-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7616  ax-cnex 10969  ax-resscn 10970  ax-1cn 10971  ax-icn 10972  ax-addcl 10973  ax-addrcl 10974  ax-mulcl 10975  ax-mulrcl 10976  ax-mulcom 10977  ax-addass 10978  ax-mulass 10979  ax-distr 10980  ax-i2m1 10981  ax-1ne0 10982  ax-1rid 10983  ax-rnegex 10984  ax-rrecex 10985  ax-cnre 10986  ax-pre-lttri 10987  ax-pre-lttrn 10988  ax-pre-ltadd 10989  ax-pre-mulgt0 10990  ax-pre-sup 10991
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3285  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-tr 5199  df-id 5496  df-eprel 5502  df-po 5510  df-so 5511  df-fr 5551  df-we 5553  df-xp 5602  df-rel 5603  df-cnv 5604  df-co 5605  df-dm 5606  df-rn 5607  df-res 5608  df-ima 5609  df-pred 6213  df-ord 6280  df-on 6281  df-lim 6282  df-suc 6283  df-iota 6406  df-fun 6456  df-fn 6457  df-f 6458  df-f1 6459  df-fo 6460  df-f1o 6461  df-fv 6462  df-riota 7260  df-ov 7306  df-oprab 7307  df-mpo 7308  df-om 7741  df-2nd 7860  df-frecs 8124  df-wrecs 8155  df-recs 8229  df-rdg 8268  df-er 8525  df-en 8761  df-dom 8762  df-sdom 8763  df-sup 9241  df-inf 9242  df-pnf 11053  df-mnf 11054  df-xr 11055  df-ltxr 11056  df-le 11057  df-sub 11249  df-neg 11250  df-div 11675  df-nn 12016  df-2 12078  df-3 12079  df-4 12080  df-n0 12276  df-z 12362  df-uz 12625  df-rp 12773  df-fz 13282  df-fl 13554  df-mod 13632  df-seq 13764  df-exp 13825  df-ibl 24827
This theorem is referenced by:  itgcl  24989  itgcnlem  24995  iblss  25010  iblss2  25011  itgsplit  25041
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