MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iblss Structured version   Visualization version   GIF version
Theorem iblss 25654
Description: A subset of an integrable function is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.)
Hypotheses
Ref Expression
iblss.1 (šœ‘ ā†’ š“ āŠ† šµ)
iblss.2 (šœ‘ ā†’ š“ āˆˆ dom vol)
iblss.3 ((šœ‘ āˆ§ š‘„ āˆˆ šµ) ā†’ š¶ āˆˆ š‘‰)
iblss.4 (šœ‘ ā†’ (š‘„ āˆˆ šµ ā†¦ š¶) āˆˆ šæ1)
Assertion
Ref Expression
iblss (šœ‘ ā†’ (š‘„ āˆˆ š“ ā†¦ š¶) āˆˆ šæ1)
Distinct variable groups:   š‘„,š“   š‘„,šµ   šœ‘,š‘„   š‘„,š‘‰
Allowed substitution hint:   š¶(š‘„)
Proof of Theorem iblss
Dummy variable š‘˜ is distinct from all other variables.
StepHypRef Expression
1 iblss.1 . . . 4 (šœ‘ ā†’ š“ āŠ† šµ)
21resmptd 6040 . . 3 (šœ‘ ā†’ ((š‘„ āˆˆ šµ ā†¦ š¶) ā†¾ š“) = (š‘„ āˆˆ š“ ā†¦ š¶))
3 iblss.4 . . . . 5 (šœ‘ ā†’ (š‘„ āˆˆ šµ ā†¦ š¶) āˆˆ šæ1)
4 iblmbf 25617 . . . . 5 ((š‘„ āˆˆ šµ ā†¦ š¶) āˆˆ šæ1 ā†’ (š‘„ āˆˆ šµ ā†¦ š¶) āˆˆ MblFn)
53, 4syl 17 . . . 4 (šœ‘ ā†’ (š‘„ āˆˆ šµ ā†¦ š¶) āˆˆ MblFn)
6 iblss.2 . . . 4 (šœ‘ ā†’ š“ āˆˆ dom vol)
7 mbfres 25493 . . . 4 (((š‘„ āˆˆ šµ ā†¦ š¶) āˆˆ MblFn āˆ§ š“ āˆˆ dom vol) ā†’ ((š‘„ āˆˆ šµ ā†¦ š¶) ā†¾ š“) āˆˆ MblFn)
85, 6, 7syl2anc 583 . . 3 (šœ‘ ā†’ ((š‘„ āˆˆ šµ ā†¦ š¶) ā†¾ š“) āˆˆ MblFn)
92, 8eqeltrrd 2833 . 2 (šœ‘ ā†’ (š‘„ āˆˆ š“ ā†¦ š¶) āˆˆ MblFn)
10 ifan 4581 . . . . . 6 if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0) = if(š‘„ āˆˆ š“, if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0), 0)
111sselda 3982 . . . . . . . . 9 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ š‘„ āˆˆ šµ)
1211ad4ant14 749 . . . . . . . 8 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ ā„) āˆ§ š‘„ āˆˆ š“) ā†’ š‘„ āˆˆ šµ)
13 iblss.3 . . . . . . . . . . . . . . 15 ((šœ‘ āˆ§ š‘„ āˆˆ šµ) ā†’ š¶ āˆˆ š‘‰)
145, 13mbfmptcl 25485 . . . . . . . . . . . . . 14 ((šœ‘ āˆ§ š‘„ āˆˆ šµ) ā†’ š¶ āˆˆ ā„‚)
1514ad4ant14 749 . . . . . . . . . . . . 13 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ ā„) āˆ§ š‘„ āˆˆ šµ) ā†’ š¶ āˆˆ ā„‚)
16 ax-icn 11175 . . . . . . . . . . . . . 14 i āˆˆ ā„‚
17 ine0 11656 . . . . . . . . . . . . . 14 i ā‰  0
18 elfzelz 13508 . . . . . . . . . . . . . . 15 (š‘˜ āˆˆ (0...3) ā†’ š‘˜ āˆˆ ā„¤)
1918ad3antlr 728 . . . . . . . . . . . . . 14 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ ā„) āˆ§ š‘„ āˆˆ šµ) ā†’ š‘˜ āˆˆ ā„¤)
20 expclz 14057 . . . . . . . . . . . . . 14 ((i āˆˆ ā„‚ āˆ§ i ā‰  0 āˆ§ š‘˜ āˆˆ ā„¤) ā†’ (iā†‘š‘˜) āˆˆ ā„‚)
2116, 17, 19, 20mp3an12i 1464 . . . . . . . . . . . . 13 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ ā„) āˆ§ š‘„ āˆˆ šµ) ā†’ (iā†‘š‘˜) āˆˆ ā„‚)
22 expne0i 14067 . . . . . . . . . . . . . 14 ((i āˆˆ ā„‚ āˆ§ i ā‰  0 āˆ§ š‘˜ āˆˆ ā„¤) ā†’ (iā†‘š‘˜) ā‰  0)
2316, 17, 19, 22mp3an12i 1464 . . . . . . . . . . . . 13 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ ā„) āˆ§ š‘„ āˆˆ šµ) ā†’ (iā†‘š‘˜) ā‰  0)
2415, 21, 23divcld 11997 . . . . . . . . . . . 12 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ ā„) āˆ§ š‘„ āˆˆ šµ) ā†’ (š¶ / (iā†‘š‘˜)) āˆˆ ā„‚)
2524recld 15148 . . . . . . . . . . 11 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ ā„) āˆ§ š‘„ āˆˆ šµ) ā†’ (ā„œā€˜(š¶ / (iā†‘š‘˜))) āˆˆ ā„)
26 0re 11223 . . . . . . . . . . 11 0 āˆˆ ā„
27 ifcl 4573 . . . . . . . . . . 11 (((ā„œā€˜(š¶ / (iā†‘š‘˜))) āˆˆ ā„ āˆ§ 0 āˆˆ ā„) ā†’ if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0) āˆˆ ā„)
2825, 26, 27sylancl 585 . . . . . . . . . 10 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ ā„) āˆ§ š‘„ āˆˆ šµ) ā†’ if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0) āˆˆ ā„)
2928rexrd 11271 . . . . . . . . 9 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ ā„) āˆ§ š‘„ āˆˆ šµ) ā†’ if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0) āˆˆ ā„*)
30 max1 13171 . . . . . . . . . 10 ((0 āˆˆ ā„ āˆ§ (ā„œā€˜(š¶ / (iā†‘š‘˜))) āˆˆ ā„) ā†’ 0 ā‰¤ if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0))
3126, 25, 30sylancr 586 . . . . . . . . 9 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ ā„) āˆ§ š‘„ āˆˆ šµ) ā†’ 0 ā‰¤ if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0))
32 elxrge0 13441 . . . . . . . . 9 (if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0) āˆˆ (0[,]+āˆž) ā†” (if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0) āˆˆ ā„* āˆ§ 0 ā‰¤ if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0)))
3329, 31, 32sylanbrc 582 . . . . . . . 8 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ ā„) āˆ§ š‘„ āˆˆ šµ) ā†’ if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0) āˆˆ (0[,]+āˆž))
3412, 33syldan 590 . . . . . . 7 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ ā„) āˆ§ š‘„ āˆˆ š“) ā†’ if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0) āˆˆ (0[,]+āˆž))
35 0e0iccpnf 13443 . . . . . . . 8 0 āˆˆ (0[,]+āˆž)
3635a1i 11 . . . . . . 7 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ ā„) āˆ§ Ā¬ š‘„ āˆˆ š“) ā†’ 0 āˆˆ (0[,]+āˆž))
3734, 36ifclda 4563 . . . . . 6 (((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ ā„) ā†’ if(š‘„ āˆˆ š“, if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0), 0) āˆˆ (0[,]+āˆž))
3810, 37eqeltrid 2836 . . . . 5 (((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ ā„) ā†’ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0) āˆˆ (0[,]+āˆž))
3938fmpttd 7116 . . . 4 ((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) ā†’ (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0)):ā„āŸ¶(0[,]+āˆž))
40 eqidd 2732 . . . . . 6 (šœ‘ ā†’ (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ šµ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0)) = (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ šµ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0)))
41 eqidd 2732 . . . . . 6 ((šœ‘ āˆ§ š‘„ āˆˆ šµ) ā†’ (ā„œā€˜(š¶ / (iā†‘š‘˜))) = (ā„œā€˜(š¶ / (iā†‘š‘˜))))
4240, 41, 3, 13iblitg 25618 . . . . 5 ((šœ‘ āˆ§ š‘˜ āˆˆ ā„¤) ā†’ (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ šµ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0))) āˆˆ ā„)
4318, 42sylan2 592 . . . 4 ((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) ā†’ (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ šµ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0))) āˆˆ ā„)
44 ifan 4581 . . . . . . 7 if((š‘„ āˆˆ šµ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0) = if(š‘„ āˆˆ šµ, if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0), 0)
4535a1i 11 . . . . . . . 8 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ ā„) āˆ§ Ā¬ š‘„ āˆˆ šµ) ā†’ 0 āˆˆ (0[,]+āˆž))
4633, 45ifclda 4563 . . . . . . 7 (((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ ā„) ā†’ if(š‘„ āˆˆ šµ, if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0), 0) āˆˆ (0[,]+āˆž))
4744, 46eqeltrid 2836 . . . . . 6 (((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ ā„) ā†’ if((š‘„ āˆˆ šµ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0) āˆˆ (0[,]+āˆž))
4847fmpttd 7116 . . . . 5 ((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) ā†’ (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ šµ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0)):ā„āŸ¶(0[,]+āˆž))
4928leidd 11787 . . . . . . . . . . 11 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ ā„) āˆ§ š‘„ āˆˆ šµ) ā†’ if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0) ā‰¤ if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0))
50 breq1 5151 . . . . . . . . . . . 12 (if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0) = if(š‘„ āˆˆ š“, if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0), 0) ā†’ (if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0) ā‰¤ if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0) ā†” if(š‘„ āˆˆ š“, if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0), 0) ā‰¤ if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0)))
51 breq1 5151 . . . . . . . . . . . 12 (0 = if(š‘„ āˆˆ š“, if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0), 0) ā†’ (0 ā‰¤ if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0) ā†” if(š‘„ āˆˆ š“, if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0), 0) ā‰¤ if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0)))
5250, 51ifboth 4567 . . . . . . . . . . 11 ((if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0) ā‰¤ if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0) āˆ§ 0 ā‰¤ if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0)) ā†’ if(š‘„ āˆˆ š“, if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0), 0) ā‰¤ if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0))
5349, 31, 52syl2anc 583 . . . . . . . . . 10 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ ā„) āˆ§ š‘„ āˆˆ šµ) ā†’ if(š‘„ āˆˆ š“, if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0), 0) ā‰¤ if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0))
54 iftrue 4534 . . . . . . . . . . 11 (š‘„ āˆˆ šµ ā†’ if(š‘„ āˆˆ šµ, if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0), 0) = if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0))
5554adantl 481 . . . . . . . . . 10 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ ā„) āˆ§ š‘„ āˆˆ šµ) ā†’ if(š‘„ āˆˆ šµ, if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0), 0) = if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0))
5653, 55breqtrrd 5176 . . . . . . . . 9 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ ā„) āˆ§ š‘„ āˆˆ šµ) ā†’ if(š‘„ āˆˆ š“, if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0), 0) ā‰¤ if(š‘„ āˆˆ šµ, if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0), 0))
57 0le0 12320 . . . . . . . . . . 11 0 ā‰¤ 0
5857a1i 11 . . . . . . . . . 10 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ ā„) āˆ§ Ā¬ š‘„ āˆˆ šµ) ā†’ 0 ā‰¤ 0)
5912stoic1a 1773 . . . . . . . . . . 11 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ ā„) āˆ§ Ā¬ š‘„ āˆˆ šµ) ā†’ Ā¬ š‘„ āˆˆ š“)
6059iffalsed 4539 . . . . . . . . . 10 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ ā„) āˆ§ Ā¬ š‘„ āˆˆ šµ) ā†’ if(š‘„ āˆˆ š“, if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0), 0) = 0)
61 iffalse 4537 . . . . . . . . . . 11 (Ā¬ š‘„ āˆˆ šµ ā†’ if(š‘„ āˆˆ šµ, if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0), 0) = 0)
6261adantl 481 . . . . . . . . . 10 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ ā„) āˆ§ Ā¬ š‘„ āˆˆ šµ) ā†’ if(š‘„ āˆˆ šµ, if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0), 0) = 0)
6358, 60, 623brtr4d 5180 . . . . . . . . 9 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ ā„) āˆ§ Ā¬ š‘„ āˆˆ šµ) ā†’ if(š‘„ āˆˆ š“, if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0), 0) ā‰¤ if(š‘„ āˆˆ šµ, if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0), 0))
6456, 63pm2.61dan 810 . . . . . . . 8 (((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ ā„) ā†’ if(š‘„ āˆˆ š“, if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0), 0) ā‰¤ if(š‘„ āˆˆ šµ, if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0), 0))
6564, 10, 443brtr4g 5182 . . . . . . 7 (((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ ā„) ā†’ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0) ā‰¤ if((š‘„ āˆˆ šµ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0))
6665ralrimiva 3145 . . . . . 6 ((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) ā†’ āˆ€š‘„ āˆˆ ā„ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0) ā‰¤ if((š‘„ āˆˆ šµ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0))
67 reex 11207 . . . . . . . 8 ā„ āˆˆ V
6867a1i 11 . . . . . . 7 ((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) ā†’ ā„ āˆˆ V)
69 eqidd 2732 . . . . . . 7 ((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) ā†’ (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0)) = (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0)))
70 eqidd 2732 . . . . . . 7 ((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) ā†’ (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ šµ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0)) = (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ šµ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0)))
7168, 38, 47, 69, 70ofrfval2 7695 . . . . . 6 ((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) ā†’ ((š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0)) āˆ˜r ā‰¤ (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ šµ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0)) ā†” āˆ€š‘„ āˆˆ ā„ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0) ā‰¤ if((š‘„ āˆˆ šµ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0)))
7266, 71mpbird 257 . . . . 5 ((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) ā†’ (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0)) āˆ˜r ā‰¤ (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ šµ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0)))
73 itg2le 25589 . . . . 5 (((š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0)):ā„āŸ¶(0[,]+āˆž) āˆ§ (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ šµ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0)):ā„āŸ¶(0[,]+āˆž) āˆ§ (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0)) āˆ˜r ā‰¤ (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ šµ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0))) ā†’ (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0))) ā‰¤ (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ šµ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0))))
7439, 48, 72, 73syl3anc 1370 . . . 4 ((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) ā†’ (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0))) ā‰¤ (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ šµ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0))))
75 itg2lecl 25588 . . . 4 (((š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0)):ā„āŸ¶(0[,]+āˆž) āˆ§ (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ šµ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0))) āˆˆ ā„ āˆ§ (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0))) ā‰¤ (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ šµ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0)))) ā†’ (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0))) āˆˆ ā„)
7639, 43, 74, 75syl3anc 1370 . . 3 ((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) ā†’ (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0))) āˆˆ ā„)
7776ralrimiva 3145 . 2 (šœ‘ ā†’ āˆ€š‘˜ āˆˆ (0...3)(āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0))) āˆˆ ā„)
78 eqidd 2732 . . 3 (šœ‘ ā†’ (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0)) = (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0)))
79 eqidd 2732 . . 3 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (ā„œā€˜(š¶ / (iā†‘š‘˜))) = (ā„œā€˜(š¶ / (iā†‘š‘˜))))
8011, 14syldan 590 . . 3 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ š¶ āˆˆ ā„‚)
8178, 79, 80isibl2 25616 . 2 (šœ‘ ā†’ ((š‘„ āˆˆ š“ ā†¦ š¶) āˆˆ šæ1 ā†” ((š‘„ āˆˆ š“ ā†¦ š¶) āˆˆ MblFn āˆ§ āˆ€š‘˜ āˆˆ (0...3)(āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0))) āˆˆ ā„)))
829, 77, 81mpbir2and 710 1 (šœ‘ ā†’ (š‘„ āˆˆ š“ ā†¦ š¶) āˆˆ šæ1)
Colors of variables: wff setvar class
Syntax hints:  Ā¬ wn 3   ā†’ wi 4   āˆ§ wa 395   = wceq 1540   āˆˆ wcel 2105   ā‰  wne 2939  āˆ€wral 3060  Vcvv 3473   āŠ† wss 3948  ifcif 4528   class class class wbr 5148   ā†¦ cmpt 5231  dom cdm 5676   ā†¾ cres 5678  āŸ¶wf 6539  ā€˜cfv 6543  (class class class)co 7412   āˆ˜r cofr 7673  ā„‚cc 11114  ā„cr 11115  0cc0 11116  ici 11118  +āˆžcpnf 11252  ā„*cxr 11254   ā‰¤ cle 11256   / cdiv 11878  3c3 12275  ā„¤cz 12565  [,]cicc 13334  ...cfz 13491  ā†‘cexp 14034  ā„œcre 15051  volcvol 25312  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-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-inf2 9642  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-int 4951  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-se 5632  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-isom 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7674  df-ofr 7675  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-2o 8473  df-er 8709  df-map 8828  df-pm 8829  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-sup 9443  df-inf 9444  df-oi 9511  df-dju 9902  df-card 9940  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-q 12940  df-rp 12982  df-xadd 13100  df-ioo 13335  df-ico 13337  df-icc 13338  df-fz 13492  df-fzo 13635  df-fl 13764  df-mod 13842  df-seq 13974  df-exp 14035  df-hash 14298  df-cj 15053  df-re 15054  df-im 15055  df-sqrt 15189  df-abs 15190  df-clim 15439  df-sum 15640  df-xmet 21226  df-met 21227  df-ovol 25313  df-vol 25314  df-mbf 25468  df-itg1 25469  df-itg2 25470  df-ibl 25471
This theorem is referenced by:  itgss3  25664  itgless  25666  bddmulibl  25688  itgcn  25694  ditgcl  25707  ditgswap  25708  ditgsplitlem  25709  ftc1lem1  25890  ftc1lem2  25891  ftc1a  25892  ftc1lem4  25894  ftc2  25899  ftc2ditglem  25900  itgsubstlem  25903  itgpowd  25905  fdvposlt  34075  fdvposle  34077  circlemeth  34116  ftc1cnnclem  37023  ftc1anc  37033  ftc2nc  37034  areacirc  37045  lcmineqlem10  41370  lcmineqlem12  41372  lhe4.4ex1a  43551  itgsin0pilem1  45125  iblioosinexp  45128  itgsinexplem1  45129  itgsinexp  45130  itgcoscmulx  45144  itgsincmulx  45149  iblcncfioo  45153  dirkeritg  45277  fourierdlem87  45368  fourierdlem95  45376  fourierdlem103  45384  fourierdlem104  45385  fourierdlem107  45388  fourierdlem111  45392  fourierdlem112  45393  sqwvfoura  45403  sqwvfourb  45404  etransclem18  45427
  Copyright terms: Public domain W3C validator