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Theorem iblss 25314
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 6039 . . 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 25277 . . . . 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 25153 . . . 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 585 . . 3 (šœ‘ ā†’ ((š‘„ āˆˆ šµ ā†¦ š¶) ā†¾ š“) āˆˆ MblFn)
92, 8eqeltrrd 2835 . 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 751 . . . . . . . 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 25145 . . . . . . . . . . . . . 14 ((šœ‘ āˆ§ š‘„ āˆˆ šµ) ā†’ š¶ āˆˆ ā„‚)
1514ad4ant14 751 . . . . . . . . . . . . 13 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ ā„) āˆ§ š‘„ āˆˆ šµ) ā†’ š¶ āˆˆ ā„‚)
16 ax-icn 11166 . . . . . . . . . . . . . 14 i āˆˆ ā„‚
17 ine0 11646 . . . . . . . . . . . . . 14 i ā‰  0
18 elfzelz 13498 . . . . . . . . . . . . . . 15 (š‘˜ āˆˆ (0...3) ā†’ š‘˜ āˆˆ ā„¤)
1918ad3antlr 730 . . . . . . . . . . . . . 14 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ ā„) āˆ§ š‘„ āˆˆ šµ) ā†’ š‘˜ āˆˆ ā„¤)
20 expclz 14047 . . . . . . . . . . . . . 14 ((i āˆˆ ā„‚ āˆ§ i ā‰  0 āˆ§ š‘˜ āˆˆ ā„¤) ā†’ (iā†‘š‘˜) āˆˆ ā„‚)
2116, 17, 19, 20mp3an12i 1466 . . . . . . . . . . . . 13 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ ā„) āˆ§ š‘„ āˆˆ šµ) ā†’ (iā†‘š‘˜) āˆˆ ā„‚)
22 expne0i 14057 . . . . . . . . . . . . . 14 ((i āˆˆ ā„‚ āˆ§ i ā‰  0 āˆ§ š‘˜ āˆˆ ā„¤) ā†’ (iā†‘š‘˜) ā‰  0)
2316, 17, 19, 22mp3an12i 1466 . . . . . . . . . . . . 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 11987 . . . . . . . . . . . 12 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ ā„) āˆ§ š‘„ āˆˆ šµ) ā†’ (š¶ / (iā†‘š‘˜)) āˆˆ ā„‚)
2524recld 15138 . . . . . . . . . . 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 11213 . . . . . . . . . . 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 587 . . . . . . . . . 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 11261 . . . . . . . . 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 13161 . . . . . . . . . 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 588 . . . . . . . . 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 13431 . . . . . . . . 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 584 . . . . . . . 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 592 . . . . . . 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 13433 . . . . . . . 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 2838 . . . . 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 7112 . . . 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 2734 . . . . . 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 2734 . . . . . 6 ((šœ‘ āˆ§ š‘„ āˆˆ šµ) ā†’ (ā„œā€˜(š¶ / (iā†‘š‘˜))) = (ā„œā€˜(š¶ / (iā†‘š‘˜))))
4240, 41, 3, 13iblitg 25278 . . . . 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 594 . . . 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 2838 . . . . . 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 7112 . . . . 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 11777 . . . . . . . . . . 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 585 . . . . . . . . . 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 483 . . . . . . . . . 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 12310 . . . . . . . . . . 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 1775 . . . . . . . . . . 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 483 . . . . . . . . . 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 812 . . . . . . . 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 3147 . . . . . 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 11198 . . . . . . . 8 ā„ āˆˆ V
6867a1i 11 . . . . . . 7 ((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) ā†’ ā„ āˆˆ V)
69 eqidd 2734 . . . . . . 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 2734 . . . . . . 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 7688 . . . . . 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 25249 . . . . 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 1372 . . . 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 25248 . . . 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 1372 . . 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 3147 . 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 2734 . . 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 2734 . . 3 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (ā„œā€˜(š¶ / (iā†‘š‘˜))) = (ā„œā€˜(š¶ / (iā†‘š‘˜))))
8011, 14syldan 592 . . 3 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ š¶ āˆˆ ā„‚)
8178, 79, 80isibl2 25276 . 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 712 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 397   = wceq 1542   āˆˆ wcel 2107   ā‰  wne 2941  āˆ€wral 3062  Vcvv 3475   āŠ† wss 3948  ifcif 4528   class class class wbr 5148   ā†¦ cmpt 5231  dom cdm 5676   ā†¾ cres 5678  āŸ¶wf 6537  ā€˜cfv 6541  (class class class)co 7406   āˆ˜r cofr 7666  ā„‚cc 11105  ā„cr 11106  0cc0 11107  ici 11109  +āˆžcpnf 11242  ā„*cxr 11244   ā‰¤ cle 11246   / cdiv 11868  3c3 12265  ā„¤cz 12555  [,]cicc 13324  ...cfz 13481  ā†‘cexp 14024  ā„œcre 15041  volcvol 24972  MblFncmbf 25123  āˆ«2citg2 25125  šæ1cibl 25126
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-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-inf2 9633  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184  ax-pre-sup 11185
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 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 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-isom 6550  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-of 7667  df-ofr 7668  df-om 7853  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-2o 8464  df-er 8700  df-map 8819  df-pm 8820  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-sup 9434  df-inf 9435  df-oi 9502  df-dju 9893  df-card 9931  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-div 11869  df-nn 12210  df-2 12272  df-3 12273  df-4 12274  df-n0 12470  df-z 12556  df-uz 12820  df-q 12930  df-rp 12972  df-xadd 13090  df-ioo 13325  df-ico 13327  df-icc 13328  df-fz 13482  df-fzo 13625  df-fl 13754  df-mod 13832  df-seq 13964  df-exp 14025  df-hash 14288  df-cj 15043  df-re 15044  df-im 15045  df-sqrt 15179  df-abs 15180  df-clim 15429  df-sum 15630  df-xmet 20930  df-met 20931  df-ovol 24973  df-vol 24974  df-mbf 25128  df-itg1 25129  df-itg2 25130  df-ibl 25131
This theorem is referenced by:  itgss3  25324  itgless  25326  bddmulibl  25348  itgcn  25354  ditgcl  25367  ditgswap  25368  ditgsplitlem  25369  ftc1lem1  25544  ftc1lem2  25545  ftc1a  25546  ftc1lem4  25548  ftc2  25553  ftc2ditglem  25554  itgsubstlem  25557  itgpowd  25559  fdvposlt  33600  fdvposle  33602  circlemeth  33641  ftc1cnnclem  36548  ftc1anc  36558  ftc2nc  36559  areacirc  36570  lcmineqlem10  40892  lcmineqlem12  40894  lhe4.4ex1a  43074  itgsin0pilem1  44653  iblioosinexp  44656  itgsinexplem1  44657  itgsinexp  44658  itgcoscmulx  44672  itgsincmulx  44677  iblcncfioo  44681  dirkeritg  44805  fourierdlem87  44896  fourierdlem95  44904  fourierdlem103  44912  fourierdlem104  44913  fourierdlem107  44916  fourierdlem111  44920  fourierdlem112  44921  sqwvfoura  44931  sqwvfourb  44932  etransclem18  44955
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