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Theorem iblitg 25286
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 714 . . . . . . 7 (((šœ‘ āˆ§ š¾ āˆˆ ā„¤) āˆ§ š‘„ āˆˆ š“) ā†’ š‘‡ = (ā„œā€˜(šµ / (iā†‘š¾))))
5 iexpcyc 14171 . . . . . . . . . 10 (š¾ āˆˆ ā„¤ ā†’ (iā†‘(š¾ mod 4)) = (iā†‘š¾))
65oveq2d 7425 . . . . . . . . 9 (š¾ āˆˆ ā„¤ ā†’ (šµ / (iā†‘(š¾ mod 4))) = (šµ / (iā†‘š¾)))
76fveq2d 6896 . . . . . . . 8 (š¾ āˆˆ ā„¤ ā†’ (ā„œā€˜(šµ / (iā†‘(š¾ mod 4)))) = (ā„œā€˜(šµ / (iā†‘š¾))))
87ad2antlr 726 . . . . . . 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 2776 . . . . . 6 (((šœ‘ āˆ§ š¾ āˆˆ ā„¤) āˆ§ š‘„ āˆˆ š“) ā†’ š‘‡ = (ā„œā€˜(šµ / (iā†‘(š¾ mod 4)))))
109ibllem 25282 . . . . 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 5251 . . . 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 2773 . . 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 6896 . 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 7417 . . . . . . . . . . 11 (š‘˜ = (š¾ mod 4) ā†’ (iā†‘š‘˜) = (iā†‘(š¾ mod 4)))
1514oveq2d 7425 . . . . . . . . . 10 (š‘˜ = (š¾ mod 4) ā†’ (šµ / (iā†‘š‘˜)) = (šµ / (iā†‘(š¾ mod 4))))
1615fveq2d 6896 . . . . . . . . 9 (š‘˜ = (š¾ mod 4) ā†’ (ā„œā€˜(šµ / (iā†‘š‘˜))) = (ā„œā€˜(šµ / (iā†‘(š¾ mod 4)))))
1716breq2d 5161 . . . . . . . 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 4553 . . . . . 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 5251 . . . . 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 6896 . . . 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 2819 . . 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 2734 . . . . . . 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 2734 . . . . . . 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 25284 . . . . . 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 231 . . . . 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 12295 . . . . . 6 4 āˆˆ ā„•
32 zmodfz 13858 . . . . . 6 ((š¾ āˆˆ ā„¤ āˆ§ 4 āˆˆ ā„•) ā†’ (š¾ mod 4) āˆˆ (0...(4 āˆ’ 1)))
3331, 32mpan2 690 . . . . 5 (š¾ āˆˆ ā„¤ ā†’ (š¾ mod 4) āˆˆ (0...(4 āˆ’ 1)))
34 4m1e3 12341 . . . . . 6 (4 āˆ’ 1) = 3
3534oveq2i 7420 . . . . 5 (0...(4 āˆ’ 1)) = (0...3)
3633, 35eleqtrdi 2844 . . . 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 3614 . 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 2834 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 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  1c1 11111  ici 11112   ā‰¤ cle 11249   āˆ’ cmin 11444   / cdiv 11871  ā„•cn 12212  3c3 12268  4c4 12269  ā„¤cz 12558  ...cfz 13484   mod cmo 13834  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-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  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-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  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-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  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-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-sup 9437  df-inf 9438  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-n0 12473  df-z 12559  df-uz 12823  df-rp 12975  df-fz 13485  df-fl 13757  df-mod 13835  df-seq 13967  df-exp 14028  df-ibl 25139
This theorem is referenced by:  itgcl  25301  itgcnlem  25307  iblss  25322  iblss2  25323  itgsplit  25353
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