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Theorem fourierdlem29 45583
Description: Explicit function value for 𝐾 applied to 𝐴. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypothesis
Ref Expression
fourierdlem29.1 𝐾 = (𝑠 ∈ (-Ο€[,]Ο€) ↦ if(𝑠 = 0, 1, (𝑠 / (2 Β· (sinβ€˜(𝑠 / 2))))))
Assertion
Ref Expression
fourierdlem29 (𝐴 ∈ (-Ο€[,]Ο€) β†’ (πΎβ€˜π΄) = if(𝐴 = 0, 1, (𝐴 / (2 Β· (sinβ€˜(𝐴 / 2))))))
Distinct variable group:   𝐴,𝑠
Allowed substitution hint:   𝐾(𝑠)

Proof of Theorem fourierdlem29
StepHypRef Expression
1 eqeq1 2729 . . 3 (𝑠 = 𝐴 β†’ (𝑠 = 0 ↔ 𝐴 = 0))
2 id 22 . . . 4 (𝑠 = 𝐴 β†’ 𝑠 = 𝐴)
3 fvoveq1 7436 . . . . 5 (𝑠 = 𝐴 β†’ (sinβ€˜(𝑠 / 2)) = (sinβ€˜(𝐴 / 2)))
43oveq2d 7429 . . . 4 (𝑠 = 𝐴 β†’ (2 Β· (sinβ€˜(𝑠 / 2))) = (2 Β· (sinβ€˜(𝐴 / 2))))
52, 4oveq12d 7431 . . 3 (𝑠 = 𝐴 β†’ (𝑠 / (2 Β· (sinβ€˜(𝑠 / 2)))) = (𝐴 / (2 Β· (sinβ€˜(𝐴 / 2)))))
61, 5ifbieq2d 4551 . 2 (𝑠 = 𝐴 β†’ if(𝑠 = 0, 1, (𝑠 / (2 Β· (sinβ€˜(𝑠 / 2))))) = if(𝐴 = 0, 1, (𝐴 / (2 Β· (sinβ€˜(𝐴 / 2))))))
7 fourierdlem29.1 . 2 𝐾 = (𝑠 ∈ (-Ο€[,]Ο€) ↦ if(𝑠 = 0, 1, (𝑠 / (2 Β· (sinβ€˜(𝑠 / 2))))))
8 1ex 11235 . . 3 1 ∈ V
9 ovex 7446 . . 3 (𝐴 / (2 Β· (sinβ€˜(𝐴 / 2)))) ∈ V
108, 9ifex 4575 . 2 if(𝐴 = 0, 1, (𝐴 / (2 Β· (sinβ€˜(𝐴 / 2))))) ∈ V
116, 7, 10fvmpt 6998 1 (𝐴 ∈ (-Ο€[,]Ο€) β†’ (πΎβ€˜π΄) = if(𝐴 = 0, 1, (𝐴 / (2 Β· (sinβ€˜(𝐴 / 2))))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  ifcif 4525   ↦ cmpt 5227  β€˜cfv 6543  (class class class)co 7413  0cc0 11133  1c1 11134   Β· cmul 11138  -cneg 11470   / cdiv 11896  2c2 12292  [,]cicc 13354  sincsin 16034  Ο€cpi 16037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pr 5424  ax-1cn 11191
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3944  df-un 3946  df-ss 3958  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7416
This theorem is referenced by: (None)
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