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Theorem iblss2 25173
Description: Change the domain of an integrability predicate. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Hypotheses
Ref Expression
iblss2.1 (šœ‘ ā†’ š“ āŠ† šµ)
iblss2.2 (šœ‘ ā†’ šµ āˆˆ dom vol)
iblss2.3 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ š¶ āˆˆ š‘‰)
iblss2.4 ((šœ‘ āˆ§ š‘„ āˆˆ (šµ āˆ– š“)) ā†’ š¶ = 0)
iblss2.5 (šœ‘ ā†’ (š‘„ āˆˆ š“ ā†¦ š¶) āˆˆ šæ1)
Assertion
Ref Expression
iblss2 (šœ‘ ā†’ (š‘„ āˆˆ šµ 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 iblss2
Dummy variable š‘˜ is distinct from all other variables.
StepHypRef Expression
1 iblss2.1 . . 3 (šœ‘ ā†’ š“ āŠ† šµ)
2 iblss2.2 . . 3 (šœ‘ ā†’ šµ āˆˆ dom vol)
3 iblss2.3 . . 3 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ š¶ āˆˆ š‘‰)
4 iblss2.4 . . 3 ((šœ‘ āˆ§ š‘„ āˆˆ (šµ āˆ– š“)) ā†’ š¶ = 0)
5 iblss2.5 . . . 4 (šœ‘ ā†’ (š‘„ āˆˆ š“ ā†¦ š¶) āˆˆ šæ1)
6 iblmbf 25135 . . . 4 ((š‘„ āˆˆ š“ ā†¦ š¶) āˆˆ šæ1 ā†’ (š‘„ āˆˆ š“ ā†¦ š¶) āˆˆ MblFn)
75, 6syl 17 . . 3 (šœ‘ ā†’ (š‘„ āˆˆ š“ ā†¦ š¶) āˆˆ MblFn)
81, 2, 3, 4, 7mbfss 25013 . 2 (šœ‘ ā†’ (š‘„ āˆˆ šµ ā†¦ š¶) āˆˆ MblFn)
91adantr 482 . . . . . . . . . . 11 ((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) ā†’ š“ āŠ† šµ)
109sselda 3945 . . . . . . . . . 10 (((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ š“) ā†’ š‘„ āˆˆ šµ)
1110iftrued 4495 . . . . . . . . 9 (((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ š“) 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))
12 iftrue 4493 . . . . . . . . . 10 (š‘„ āˆˆ š“ ā†’ if(š‘„ āˆˆ š“, 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))
1312adantl 483 . . . . . . . . 9 (((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ š“) ā†’ if(š‘„ āˆˆ š“, 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))
1411, 13eqtr4d 2780 . . . . . . . 8 (((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ š‘„ āˆˆ š“) ā†’ if(š‘„ āˆˆ šµ, if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0), 0) = if(š‘„ āˆˆ š“, if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0), 0))
15 ifid 4527 . . . . . . . . 9 if(š‘„ āˆˆ šµ, 0, 0) = 0
16 simplll 774 . . . . . . . . . . . . . . . . 17 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ Ā¬ š‘„ āˆˆ š“) āˆ§ š‘„ āˆˆ šµ) ā†’ šœ‘)
17 simpr 486 . . . . . . . . . . . . . . . . . 18 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ Ā¬ š‘„ āˆˆ š“) āˆ§ š‘„ āˆˆ šµ) ā†’ š‘„ āˆˆ šµ)
18 simplr 768 . . . . . . . . . . . . . . . . . 18 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ Ā¬ š‘„ āˆˆ š“) āˆ§ š‘„ āˆˆ šµ) ā†’ Ā¬ š‘„ āˆˆ š“)
1917, 18eldifd 3922 . . . . . . . . . . . . . . . . 17 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ Ā¬ š‘„ āˆˆ š“) āˆ§ š‘„ āˆˆ šµ) ā†’ š‘„ āˆˆ (šµ āˆ– š“))
2016, 19, 4syl2anc 585 . . . . . . . . . . . . . . . 16 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ Ā¬ š‘„ āˆˆ š“) āˆ§ š‘„ āˆˆ šµ) ā†’ š¶ = 0)
2120oveq1d 7373 . . . . . . . . . . . . . . 15 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ Ā¬ š‘„ āˆˆ š“) āˆ§ š‘„ āˆˆ šµ) ā†’ (š¶ / (iā†‘š‘˜)) = (0 / (iā†‘š‘˜)))
22 simpllr 775 . . . . . . . . . . . . . . . 16 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ Ā¬ š‘„ āˆˆ š“) āˆ§ š‘„ āˆˆ šµ) ā†’ š‘˜ āˆˆ (0...3))
23 elfzelz 13442 . . . . . . . . . . . . . . . 16 (š‘˜ āˆˆ (0...3) ā†’ š‘˜ āˆˆ ā„¤)
24 ax-icn 11111 . . . . . . . . . . . . . . . . 17 i āˆˆ ā„‚
25 ine0 11591 . . . . . . . . . . . . . . . . 17 i ā‰  0
26 expclz 13991 . . . . . . . . . . . . . . . . . 18 ((i āˆˆ ā„‚ āˆ§ i ā‰  0 āˆ§ š‘˜ āˆˆ ā„¤) ā†’ (iā†‘š‘˜) āˆˆ ā„‚)
27 expne0i 14001 . . . . . . . . . . . . . . . . . 18 ((i āˆˆ ā„‚ āˆ§ i ā‰  0 āˆ§ š‘˜ āˆˆ ā„¤) ā†’ (iā†‘š‘˜) ā‰  0)
2826, 27div0d 11931 . . . . . . . . . . . . . . . . 17 ((i āˆˆ ā„‚ āˆ§ i ā‰  0 āˆ§ š‘˜ āˆˆ ā„¤) ā†’ (0 / (iā†‘š‘˜)) = 0)
2924, 25, 28mp3an12 1452 . . . . . . . . . . . . . . . 16 (š‘˜ āˆˆ ā„¤ ā†’ (0 / (iā†‘š‘˜)) = 0)
3022, 23, 293syl 18 . . . . . . . . . . . . . . 15 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ Ā¬ š‘„ āˆˆ š“) āˆ§ š‘„ āˆˆ šµ) ā†’ (0 / (iā†‘š‘˜)) = 0)
3121, 30eqtrd 2777 . . . . . . . . . . . . . 14 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ Ā¬ š‘„ āˆˆ š“) āˆ§ š‘„ āˆˆ šµ) ā†’ (š¶ / (iā†‘š‘˜)) = 0)
3231fveq2d 6847 . . . . . . . . . . . . 13 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ Ā¬ š‘„ āˆˆ š“) āˆ§ š‘„ āˆˆ šµ) ā†’ (ā„œā€˜(š¶ / (iā†‘š‘˜))) = (ā„œā€˜0))
33 re0 15038 . . . . . . . . . . . . 13 (ā„œā€˜0) = 0
3432, 33eqtrdi 2793 . . . . . . . . . . . 12 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ Ā¬ š‘„ āˆˆ š“) āˆ§ š‘„ āˆˆ šµ) ā†’ (ā„œā€˜(š¶ / (iā†‘š‘˜))) = 0)
3534ifeq1d 4506 . . . . . . . . . . 11 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ Ā¬ š‘„ āˆˆ š“) āˆ§ š‘„ āˆˆ šµ) ā†’ if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0) = if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0, 0))
36 ifid 4527 . . . . . . . . . . 11 if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0, 0) = 0
3735, 36eqtrdi 2793 . . . . . . . . . 10 ((((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ Ā¬ š‘„ āˆˆ š“) āˆ§ š‘„ āˆˆ šµ) ā†’ if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0) = 0)
3837ifeq1da 4518 . . . . . . . . 9 (((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ Ā¬ š‘„ āˆˆ š“) ā†’ if(š‘„ āˆˆ šµ, if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0), 0) = if(š‘„ āˆˆ šµ, 0, 0))
39 iffalse 4496 . . . . . . . . . 10 (Ā¬ š‘„ āˆˆ š“ ā†’ if(š‘„ āˆˆ š“, if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0), 0) = 0)
4039adantl 483 . . . . . . . . 9 (((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ Ā¬ š‘„ āˆˆ š“) ā†’ if(š‘„ āˆˆ š“, if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0), 0) = 0)
4115, 38, 403eqtr4a 2803 . . . . . . . 8 (((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) āˆ§ Ā¬ š‘„ āˆˆ š“) ā†’ if(š‘„ āˆˆ šµ, if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0), 0) = if(š‘„ āˆˆ š“, if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0), 0))
4214, 41pm2.61dan 812 . . . . . . 7 ((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) ā†’ if(š‘„ āˆˆ šµ, if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0), 0) = if(š‘„ āˆˆ š“, if(0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0), 0))
43 ifan 4540 . . . . . . 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)
44 ifan 4540 . . . . . . 7 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)
4542, 43, 443eqtr4g 2802 . . . . . 6 ((šœ‘ āˆ§ š‘˜ āˆˆ (0...3)) ā†’ if((š‘„ āˆˆ šµ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0) = if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0))
4645mpteq2dv 5208 . . . . 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)) = (š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0)))
4746fveq2d 6847 . . . 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))) = (āˆ«2ā€˜(š‘„ āˆˆ ā„ ā†¦ if((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(š¶ / (iā†‘š‘˜)))), (ā„œā€˜(š¶ / (iā†‘š‘˜))), 0))))
48 eqidd 2738 . . . . . 6 (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)))
49 eqidd 2738 . . . . . 6 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ (ā„œā€˜(š¶ / (iā†‘š‘˜))) = (ā„œā€˜(š¶ / (iā†‘š‘˜))))
5048, 49, 5, 3iblitg 25136 . . . . 5 ((šœ‘ āˆ§ š‘˜ āˆˆ ā„¤) 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„)
5123, 50sylan2 594 . . . 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))) āˆˆ ā„)
5247, 51eqeltrd 2838 . . 3 ((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„)
5352ralrimiva 3144 . 2 (šœ‘ ā†’ āˆ€š‘˜ āˆˆ (0...3)(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„)
54 eqidd 2738 . . 3 (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)))
55 eqidd 2738 . . 3 ((šœ‘ āˆ§ š‘„ āˆˆ šµ) ā†’ (ā„œā€˜(š¶ / (iā†‘š‘˜))) = (ā„œā€˜(š¶ / (iā†‘š‘˜))))
56 elun 4109 . . . . . 6 (š‘„ āˆˆ (š“ āˆŖ (šµ āˆ– š“)) ā†” (š‘„ āˆˆ š“ āˆØ š‘„ āˆˆ (šµ āˆ– š“)))
57 undif2 4437 . . . . . . . 8 (š“ āˆŖ (šµ āˆ– š“)) = (š“ āˆŖ šµ)
58 ssequn1 4141 . . . . . . . . 9 (š“ āŠ† šµ ā†” (š“ āˆŖ šµ) = šµ)
591, 58sylib 217 . . . . . . . 8 (šœ‘ ā†’ (š“ āˆŖ šµ) = šµ)
6057, 59eqtrid 2789 . . . . . . 7 (šœ‘ ā†’ (š“ āˆŖ (šµ āˆ– š“)) = šµ)
6160eleq2d 2824 . . . . . 6 (šœ‘ ā†’ (š‘„ āˆˆ (š“ āˆŖ (šµ āˆ– š“)) ā†” š‘„ āˆˆ šµ))
6256, 61bitr3id 285 . . . . 5 (šœ‘ ā†’ ((š‘„ āˆˆ š“ āˆØ š‘„ āˆˆ (šµ āˆ– š“)) ā†” š‘„ āˆˆ šµ))
6362biimpar 479 . . . 4 ((šœ‘ āˆ§ š‘„ āˆˆ šµ) ā†’ (š‘„ āˆˆ š“ āˆØ š‘„ āˆˆ (šµ āˆ– š“)))
647, 3mbfmptcl 25003 . . . . 5 ((šœ‘ āˆ§ š‘„ āˆˆ š“) ā†’ š¶ āˆˆ ā„‚)
65 0cn 11148 . . . . . 6 0 āˆˆ ā„‚
664, 65eqeltrdi 2846 . . . . 5 ((šœ‘ āˆ§ š‘„ āˆˆ (šµ āˆ– š“)) ā†’ š¶ āˆˆ ā„‚)
6764, 66jaodan 957 . . . 4 ((šœ‘ āˆ§ (š‘„ āˆˆ š“ āˆØ š‘„ āˆˆ (šµ āˆ– š“))) ā†’ š¶ āˆˆ ā„‚)
6863, 67syldan 592 . . 3 ((šœ‘ āˆ§ š‘„ āˆˆ šµ) ā†’ š¶ āˆˆ ā„‚)
6954, 55, 68isibl2 25134 . 2 (šœ‘ ā†’ ((š‘„ āˆˆ šµ ā†¦ š¶) āˆˆ šæ1 ā†” ((š‘„ āˆˆ šµ ā†¦ š¶) āˆˆ MblFn āˆ§ āˆ€š‘˜ āˆˆ (0...3)(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„)))
708, 53, 69mpbir2and 712 1 (šœ‘ ā†’ (š‘„ āˆˆ šµ ā†¦ š¶) āˆˆ šæ1)
Colors of variables: wff setvar class
Syntax hints:  Ā¬ wn 3   ā†’ wi 4   āˆ§ wa 397   āˆØ wo 846   āˆ§ w3a 1088   = wceq 1542   āˆˆ wcel 2107   ā‰  wne 2944  āˆ€wral 3065   āˆ– cdif 3908   āˆŖ cun 3909   āŠ† wss 3911  ifcif 4487   class class class wbr 5106   ā†¦ cmpt 5189  dom cdm 5634  ā€˜cfv 6497  (class class class)co 7358  ā„‚cc 11050  ā„cr 11051  0cc0 11052  ici 11054   ā‰¤ cle 11191   / cdiv 11813  3c3 12210  ā„¤cz 12500  ...cfz 13425  ā†‘cexp 13968  ā„œcre 14983  volcvol 24830  MblFncmbf 24981  āˆ«2citg2 24983  šæ1cibl 24984
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 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-inf2 9578  ax-cnex 11108  ax-resscn 11109  ax-1cn 11110  ax-icn 11111  ax-addcl 11112  ax-addrcl 11113  ax-mulcl 11114  ax-mulrcl 11115  ax-mulcom 11116  ax-addass 11117  ax-mulass 11118  ax-distr 11119  ax-i2m1 11120  ax-1ne0 11121  ax-1rid 11122  ax-rnegex 11123  ax-rrecex 11124  ax-cnre 11125  ax-pre-lttri 11126  ax-pre-lttrn 11127  ax-pre-ltadd 11128  ax-pre-mulgt0 11129  ax-pre-sup 11130
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7618  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-2o 8414  df-er 8649  df-map 8768  df-pm 8769  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-sup 9379  df-inf 9380  df-oi 9447  df-dju 9838  df-card 9876  df-pnf 11192  df-mnf 11193  df-xr 11194  df-ltxr 11195  df-le 11196  df-sub 11388  df-neg 11389  df-div 11814  df-nn 12155  df-2 12217  df-3 12218  df-4 12219  df-n0 12415  df-z 12501  df-uz 12765  df-q 12875  df-rp 12917  df-xadd 13035  df-ioo 13269  df-ico 13271  df-icc 13272  df-fz 13426  df-fzo 13569  df-fl 13698  df-mod 13776  df-seq 13908  df-exp 13969  df-hash 14232  df-cj 14985  df-re 14986  df-im 14987  df-sqrt 15121  df-abs 15122  df-clim 15371  df-sum 15572  df-xmet 20792  df-met 20793  df-ovol 24831  df-vol 24832  df-mbf 24986  df-ibl 24989
This theorem is referenced by:  itgss3  25182  itgless  25184  ftc1anclem5  36158  ftc1anclem6  36159  areacirc  36174  arearect  41552  areaquad  41553
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