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Theorem iblitg 25618
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 480 . . . 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 712 . . . . . . 7 (((šœ‘ āˆ§ š¾ āˆˆ ā„¤) āˆ§ š‘„ āˆˆ š“) ā†’ š‘‡ = (ā„œā€˜(šµ / (iā†‘š¾))))
5 iexpcyc 14178 . . . . . . . . . 10 (š¾ āˆˆ ā„¤ ā†’ (iā†‘(š¾ mod 4)) = (iā†‘š¾))
65oveq2d 7428 . . . . . . . . 9 (š¾ āˆˆ ā„¤ ā†’ (šµ / (iā†‘(š¾ mod 4))) = (šµ / (iā†‘š¾)))
76fveq2d 6895 . . . . . . . 8 (š¾ āˆˆ ā„¤ ā†’ (ā„œā€˜(šµ / (iā†‘(š¾ mod 4)))) = (ā„œā€˜(šµ / (iā†‘š¾))))
87ad2antlr 724 . . . . . . 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 2774 . . . . . 6 (((šœ‘ āˆ§ š¾ āˆˆ ā„¤) āˆ§ š‘„ āˆˆ š“) ā†’ š‘‡ = (ā„œā€˜(šµ / (iā†‘(š¾ mod 4)))))
109ibllem 25614 . . . . 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 5250 . . . 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 2771 . . 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 6895 . 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 7420 . . . . . . . . . . 11 (š‘˜ = (š¾ mod 4) ā†’ (iā†‘š‘˜) = (iā†‘(š¾ mod 4)))
1514oveq2d 7428 . . . . . . . . . 10 (š‘˜ = (š¾ mod 4) ā†’ (šµ / (iā†‘š‘˜)) = (šµ / (iā†‘(š¾ mod 4))))
1615fveq2d 6895 . . . . . . . . 9 (š‘˜ = (š¾ mod 4) ā†’ (ā„œā€˜(šµ / (iā†‘š‘˜))) = (ā„œā€˜(šµ / (iā†‘(š¾ mod 4)))))
1716breq2d 5160 . . . . . . . 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 628 . . . . . . 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 4552 . . . . . 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 5250 . . . . 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 6895 . . . 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 2817 . . 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 2732 . . . . . . 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 2732 . . . . . . 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 25616 . . . . . 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 495 . . . 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 480 . . 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 12302 . . . . . 6 4 āˆˆ ā„•
32 zmodfz 13865 . . . . . 6 ((š¾ āˆˆ ā„¤ āˆ§ 4 āˆˆ ā„•) ā†’ (š¾ mod 4) āˆˆ (0...(4 āˆ’ 1)))
3331, 32mpan2 688 . . . . 5 (š¾ āˆˆ ā„¤ ā†’ (š¾ mod 4) āˆˆ (0...(4 āˆ’ 1)))
34 4m1e3 12348 . . . . . 6 (4 āˆ’ 1) = 3
3534oveq2i 7423 . . . . 5 (0...(4 āˆ’ 1)) = (0...3)
3633, 35eleqtrdi 2842 . . . 4 (š¾ āˆˆ ā„¤ ā†’ (š¾ mod 4) āˆˆ (0...3))
3736adantl 481 . . 3 ((šœ‘ āˆ§ š¾ āˆˆ ā„¤) ā†’ (š¾ mod 4) āˆˆ (0...3))
3822, 30, 37rspcdva 3613 . 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 2832 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 395   = wceq 1540   āˆˆ wcel 2105  āˆ€wral 3060  ifcif 4528   class class class wbr 5148   ā†¦ cmpt 5231  ā€˜cfv 6543  (class class class)co 7412  ā„cr 11115  0cc0 11116  1c1 11117  ici 11118   ā‰¤ cle 11256   āˆ’ cmin 11451   / cdiv 11878  ā„•cn 12219  3c3 12275  4c4 12276  ā„¤cz 12565  ...cfz 13491   mod cmo 13841  ā†‘cexp 14034  ā„œcre 15051  MblFncmbf 25463  āˆ«2citg2 25465  šæ1cibl 25466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193  ax-pre-sup 11194
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-er 8709  df-en 8946  df-dom 8947  df-sdom 8948  df-sup 9443  df-inf 9444  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-div 11879  df-nn 12220  df-2 12282  df-3 12283  df-4 12284  df-n0 12480  df-z 12566  df-uz 12830  df-rp 12982  df-fz 13492  df-fl 13764  df-mod 13842  df-seq 13974  df-exp 14035  df-ibl 25471
This theorem is referenced by:  itgcl  25633  itgcnlem  25639  iblss  25654  iblss2  25655  itgsplit  25685
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