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Theorem cytpval 27528
Description: Substitutions for the Nth cyclotomic polynomial. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Hypotheses
Ref Expression
cytpval.t  |-  T  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )
cytpval.o  |-  O  =  ( od `  T
)
cytpval.p  |-  P  =  (Poly1 ` fld )
cytpval.x  |-  X  =  (var1 ` fld )
cytpval.q  |-  Q  =  (mulGrp `  P )
cytpval.m  |-  .-  =  ( -g `  P )
cytpval.a  |-  A  =  (algSc `  P )
Assertion
Ref Expression
cytpval  |-  ( N  e.  NN  ->  (CytP `  N )  =  ( Q  gsumg  ( r  e.  ( `' O " { N } )  |->  ( X 
.-  ( A `  r ) ) ) ) )
Distinct variable group:    N, r
Allowed substitution hints:    A( r)    P( r)    Q( r)    T( r)    .- ( r)    O( r)    X( r)

Proof of Theorem cytpval
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 cytpval.p . . . . . . 7  |-  P  =  (Poly1 ` fld )
21eqcomi 2287 . . . . . 6  |-  (Poly1 ` fld )  =  P
32fveq2i 5528 . . . . 5  |-  (mulGrp `  (Poly1 ` fld ) )  =  (mulGrp `  P )
4 cytpval.q . . . . 5  |-  Q  =  (mulGrp `  P )
53, 4eqtr4i 2306 . . . 4  |-  (mulGrp `  (Poly1 ` fld ) )  =  Q
65a1i 10 . . 3  |-  ( n  =  N  ->  (mulGrp `  (Poly1 ` fld ) )  =  Q )
7 cytpval.o . . . . . . . 8  |-  O  =  ( od `  T
)
8 cytpval.t . . . . . . . . 9  |-  T  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )
98fveq2i 5528 . . . . . . . 8  |-  ( od
`  T )  =  ( od `  (
(mulGrp ` fld )s  ( CC  \  { 0 } ) ) )
107, 9eqtri 2303 . . . . . . 7  |-  O  =  ( od `  (
(mulGrp ` fld )s  ( CC  \  { 0 } ) ) )
1110cnveqi 4856 . . . . . 6  |-  `' O  =  `' ( od `  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) ) )
1211imaeq1i 5009 . . . . 5  |-  ( `' O " { n } )  =  ( `' ( od `  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) ) ) " {
n } )
13 sneq 3651 . . . . . 6  |-  ( n  =  N  ->  { n }  =  { N } )
1413imaeq2d 5012 . . . . 5  |-  ( n  =  N  ->  ( `' O " { n } )  =  ( `' O " { N } ) )
1512, 14syl5eqr 2329 . . . 4  |-  ( n  =  N  ->  ( `' ( od `  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) ) ) " {
n } )  =  ( `' O " { N } ) )
16 cytpval.x . . . . . . 7  |-  X  =  (var1 ` fld )
17 cytpval.a . . . . . . . . 9  |-  A  =  (algSc `  P )
181fveq2i 5528 . . . . . . . . 9  |-  (algSc `  P )  =  (algSc `  (Poly1 ` fld ) )
1917, 18eqtri 2303 . . . . . . . 8  |-  A  =  (algSc `  (Poly1 ` fld ) )
2019fveq1i 5526 . . . . . . 7  |-  ( A `
 r )  =  ( (algSc `  (Poly1 ` fld )
) `  r )
21 cytpval.m . . . . . . . 8  |-  .-  =  ( -g `  P )
221fveq2i 5528 . . . . . . . 8  |-  ( -g `  P )  =  (
-g `  (Poly1 ` fld ) )
2321, 22eqtri 2303 . . . . . . 7  |-  .-  =  ( -g `  (Poly1 ` fld ) )
2416, 20, 23oveq123i 5872 . . . . . 6  |-  ( X 
.-  ( A `  r ) )  =  ( (var1 ` fld ) ( -g `  (Poly1 ` fld )
) ( (algSc `  (Poly1 ` fld ) ) `  r
) )
2524eqcomi 2287 . . . . 5  |-  ( (var1 ` fld ) ( -g `  (Poly1 ` fld )
) ( (algSc `  (Poly1 ` fld ) ) `  r
) )  =  ( X  .-  ( A `
 r ) )
2625a1i 10 . . . 4  |-  ( n  =  N  ->  (
(var1 ` fld ) ( -g `  (Poly1 ` fld )
) ( (algSc `  (Poly1 ` fld ) ) `  r
) )  =  ( X  .-  ( A `
 r ) ) )
2715, 26mpteq12dv 4098 . . 3  |-  ( n  =  N  ->  (
r  e.  ( `' ( od `  (
(mulGrp ` fld )s  ( CC  \  { 0 } ) ) ) " {
n } )  |->  ( (var1 ` fld ) ( -g `  (Poly1 ` fld )
) ( (algSc `  (Poly1 ` fld ) ) `  r
) ) )  =  ( r  e.  ( `' O " { N } )  |->  ( X 
.-  ( A `  r ) ) ) )
286, 27oveq12d 5876 . 2  |-  ( n  =  N  ->  (
(mulGrp `  (Poly1 ` fld ) )  gsumg  ( r  e.  ( `' ( od `  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) ) ) " {
n } )  |->  ( (var1 ` fld ) ( -g `  (Poly1 ` fld )
) ( (algSc `  (Poly1 ` fld ) ) `  r
) ) ) )  =  ( Q  gsumg  ( r  e.  ( `' O " { N } ) 
|->  ( X  .-  ( A `  r )
) ) ) )
29 df-cytp 27522 . 2  |- CytP  =  ( n  e.  NN  |->  ( (mulGrp `  (Poly1 ` fld ) )  gsumg  ( r  e.  ( `' ( od `  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) ) ) " {
n } )  |->  ( (var1 ` fld ) ( -g `  (Poly1 ` fld )
) ( (algSc `  (Poly1 ` fld ) ) `  r
) ) ) ) )
30 ovex 5883 . 2  |-  ( Q 
gsumg  ( r  e.  ( `' O " { N } )  |->  ( X 
.-  ( A `  r ) ) ) )  e.  _V
3128, 29, 30fvmpt 5602 1  |-  ( N  e.  NN  ->  (CytP `  N )  =  ( Q  gsumg  ( r  e.  ( `' O " { N } )  |->  ( X 
.-  ( A `  r ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    \ cdif 3149   {csn 3640    e. cmpt 4077   `'ccnv 4688   "cima 4692   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   NNcn 9746   ↾s cress 13149    gsumg cgsu 13401   -gcsg 14365   odcod 14840  mulGrpcmgp 15325  algSccascl 16052  var1cv1 16251  Poly1cpl1 16252  ℂfldccnfld 16377  CytPccytp 27521
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-cytp 27522
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