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Theorem itgvallem 25172
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 7369 . . . . . . . . 9 (š‘˜ = š¾ ā†’ (iā†‘š‘˜) = (iā†‘š¾))
2 itgvallem.1 . . . . . . . . 9 (iā†‘š¾) = š‘‡
31, 2eqtrdi 2789 . . . . . . . 8 (š‘˜ = š¾ ā†’ (iā†‘š‘˜) = š‘‡)
43oveq2d 7377 . . . . . . 7 (š‘˜ = š¾ ā†’ (šµ / (iā†‘š‘˜)) = (šµ / š‘‡))
54fveq2d 6850 . . . . . 6 (š‘˜ = š¾ ā†’ (ā„œā€˜(šµ / (iā†‘š‘˜))) = (ā„œā€˜(šµ / š‘‡)))
65breq2d 5121 . . . . 5 (š‘˜ = š¾ ā†’ (0 ā‰¤ (ā„œā€˜(šµ / (iā†‘š‘˜))) ā†” 0 ā‰¤ (ā„œā€˜(šµ / š‘‡))))
76anbi2d 630 . . . 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 4514 . . 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 5211 . 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 6850 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 397   = wceq 1542   āˆˆ wcel 2107  ifcif 4490   class class class wbr 5109   ā†¦ cmpt 5192  ā€˜cfv 6500  (class class class)co 7361  ā„cr 11058  0cc0 11059  ici 11061   ā‰¤ cle 11198   / cdiv 11820  ā†‘cexp 13976  ā„œcre 14991  āˆ«2citg2 25003
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-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-iota 6452  df-fv 6508  df-ov 7364
This theorem is referenced by:  iblcnlem1  25175  itgcnlem  25177
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