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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 2737 | . . 3 β’ (π = π΄ β (π = 0 β π΄ = 0)) | |
2 | id 22 | . . . 4 β’ (π = π΄ β π = π΄) | |
3 | fvoveq1 7381 | . . . . 5 β’ (π = π΄ β (sinβ(π / 2)) = (sinβ(π΄ / 2))) | |
4 | 3 | oveq2d 7374 | . . . 4 β’ (π = π΄ β (2 Β· (sinβ(π / 2))) = (2 Β· (sinβ(π΄ / 2)))) |
5 | 2, 4 | oveq12d 7376 | . . 3 β’ (π = π΄ β (π / (2 Β· (sinβ(π / 2)))) = (π΄ / (2 Β· (sinβ(π΄ / 2))))) |
6 | 1, 5 | ifbieq2d 4513 | . 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 11156 | . . 3 β’ 1 β V | |
9 | ovex 7391 | . . 3 β’ (π΄ / (2 Β· (sinβ(π΄ / 2)))) β V | |
10 | 8, 9 | ifex 4537 | . 2 β’ if(π΄ = 0, 1, (π΄ / (2 Β· (sinβ(π΄ / 2))))) β V |
11 | 6, 7, 10 | fvmpt 6949 | 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 1542 β wcel 2107 ifcif 4487 β¦ cmpt 5189 βcfv 6497 (class class class)co 7358 0cc0 11056 1c1 11057 Β· cmul 11061 -cneg 11391 / cdiv 11817 2c2 12213 [,]cicc 13273 sincsin 15951 Οcpi 15954 |
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 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-1cn 11114 |
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-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fv 6505 df-ov 7361 |
This theorem is referenced by: (None) |
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