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Mirrors > Home > MPE Home > Th. List > Mathboxes > fourierdlem29 | Structured version Visualization version GIF version |
Description: Explicit function value for πΎ applied to π΄. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
fourierdlem29.1 | β’ πΎ = (π β (-Ο[,]Ο) β¦ if(π = 0, 1, (π / (2 Β· (sinβ(π / 2)))))) |
Ref | Expression |
---|---|
fourierdlem29 | β’ (π΄ β (-Ο[,]Ο) β (πΎβπ΄) = if(π΄ = 0, 1, (π΄ / (2 Β· (sinβ(π΄ / 2)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2729 | . . 3 β’ (π = π΄ β (π = 0 β π΄ = 0)) | |
2 | id 22 | . . . 4 β’ (π = π΄ β π = π΄) | |
3 | fvoveq1 7436 | . . . . 5 β’ (π = π΄ β (sinβ(π / 2)) = (sinβ(π΄ / 2))) | |
4 | 3 | oveq2d 7429 | . . . 4 β’ (π = π΄ β (2 Β· (sinβ(π / 2))) = (2 Β· (sinβ(π΄ / 2)))) |
5 | 2, 4 | oveq12d 7431 | . . 3 β’ (π = π΄ β (π / (2 Β· (sinβ(π / 2)))) = (π΄ / (2 Β· (sinβ(π΄ / 2))))) |
6 | 1, 5 | ifbieq2d 4551 | . 2 β’ (π = π΄ β if(π = 0, 1, (π / (2 Β· (sinβ(π / 2))))) = if(π΄ = 0, 1, (π΄ / (2 Β· (sinβ(π΄ / 2)))))) |
7 | fourierdlem29.1 | . 2 β’ πΎ = (π β (-Ο[,]Ο) β¦ if(π = 0, 1, (π / (2 Β· (sinβ(π / 2)))))) | |
8 | 1ex 11235 | . . 3 β’ 1 β V | |
9 | ovex 7446 | . . 3 β’ (π΄ / (2 Β· (sinβ(π΄ / 2)))) β V | |
10 | 8, 9 | ifex 4575 | . 2 β’ if(π΄ = 0, 1, (π΄ / (2 Β· (sinβ(π΄ / 2))))) β V |
11 | 6, 7, 10 | fvmpt 6998 | 1 β’ (π΄ β (-Ο[,]Ο) β (πΎβπ΄) = if(π΄ = 0, 1, (π΄ / (2 Β· (sinβ(π΄ / 2)))))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 ifcif 4525 β¦ cmpt 5227 βcfv 6543 (class class class)co 7413 0cc0 11133 1c1 11134 Β· cmul 11138 -cneg 11470 / cdiv 11896 2c2 12292 [,]cicc 13354 sincsin 16034 Οcpi 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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