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Theorem itgvallem 25301
Description: Substitution lemma. (Contributed by Mario Carneiro, 7-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Hypothesis
Ref Expression
itgvallem.1 (iā†‘š¾) = š‘‡
Assertion
Ref Expression
itgvallem (š‘˜ = š¾ 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 / š‘‡))), (ā„œā€˜(šµ / š‘‡)), 0))))
Distinct variable groups:   š‘„,š‘˜   š‘„,š¾
Allowed substitution hints:   š“(š‘„,š‘˜)   šµ(š‘„,š‘˜)   š‘‡(š‘„,š‘˜)   š¾(š‘˜)
Proof of Theorem itgvallem
StepHypRef Expression
1 oveq2 7416 . . . . . . . . 9 (š‘˜ = š¾ ā†’ (iā†‘š‘˜) = (iā†‘š¾))
2 itgvallem.1 . . . . . . . . 9 (iā†‘š¾) = š‘‡
31, 2eqtrdi 2788 . . . . . . . 8 (š‘˜ = š¾ ā†’ (iā†‘š‘˜) = š‘‡)
43oveq2d 7424 . . . . . . 7 (š‘˜ = š¾ ā†’ (šµ / (iā†‘š‘˜)) = (šµ / š‘‡))
54fveq2d 6895 . . . . . 6 (š‘˜ = š¾ ā†’ (ā„œā€˜(šµ / (iā†‘š‘˜))) = (ā„œā€˜(šµ / š‘‡)))
65breq2d 5160 . . . . 5 (š‘˜ = š¾ ā†’ (0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜))) ā†” 0 ā‰¤ (ā„œā€˜(šµ / š‘‡))))
76anbi2d 629 . . . 4 (š‘˜ = š¾ ā†’ ((š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜)))) ā†” (š‘„ āˆˆ š“ āˆ§ 0 ā‰¤ (ā„œā€˜(šµ / š‘‡)))))
87, 5ifbieq1d 4552 . . 3 (š‘˜ = š¾ 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 / š‘‡))), (ā„œā€˜(šµ / š‘‡)), 0))
98mpteq2dv 5250 . 2 (š‘˜ = š¾ 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 / š‘‡))), (ā„œā€˜(šµ / š‘‡)), 0)))
109fveq2d 6895 1 (š‘˜ = š¾ 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 / š‘‡))), (ā„œā€˜(šµ / š‘‡)), 0))))
Colors of variables: wff setvar class
Syntax hints:   ā†’ wi 4   āˆ§ wa 396   = wceq 1541   āˆˆ wcel 2106  ifcif 4528   class class class wbr 5148   ā†¦ cmpt 5231  ā€˜cfv 6543  (class class class)co 7408  ā„cr 11108  0cc0 11109  ici 11111   ā‰¤ cle 11248   / cdiv 11870  ā†‘cexp 14026  ā„œcre 15043  āˆ«2citg2 25132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-iota 6495  df-fv 6551  df-ov 7411
This theorem is referenced by:  iblcnlem1  25304  itgcnlem  25306
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