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Theorem cytpval 27631
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 2300 . . . . . 6  |-  (Poly1 ` fld )  =  P
32fveq2i 5544 . . . . 5  |-  (mulGrp `  (Poly1 ` fld ) )  =  (mulGrp `  P )
4 cytpval.q . . . . 5  |-  Q  =  (mulGrp `  P )
53, 4eqtr4i 2319 . . . 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 5544 . . . . . . . 8  |-  ( od
`  T )  =  ( od `  (
(mulGrp ` fld )s  ( CC  \  { 0 } ) ) )
107, 9eqtri 2316 . . . . . . 7  |-  O  =  ( od `  (
(mulGrp ` fld )s  ( CC  \  { 0 } ) ) )
1110cnveqi 4872 . . . . . 6  |-  `' O  =  `' ( od `  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) ) )
1211imaeq1i 5025 . . . . 5  |-  ( `' O " { n } )  =  ( `' ( od `  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) ) ) " {
n } )
13 sneq 3664 . . . . . 6  |-  ( n  =  N  ->  { n }  =  { N } )
1413imaeq2d 5028 . . . . 5  |-  ( n  =  N  ->  ( `' O " { n } )  =  ( `' O " { N } ) )
1512, 14syl5eqr 2342 . . . 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 5544 . . . . . . . . 9  |-  (algSc `  P )  =  (algSc `  (Poly1 ` fld ) )
1917, 18eqtri 2316 . . . . . . . 8  |-  A  =  (algSc `  (Poly1 ` fld ) )
2019fveq1i 5542 . . . . . . 7  |-  ( A `
 r )  =  ( (algSc `  (Poly1 ` fld )
) `  r )
21 cytpval.m . . . . . . . 8  |-  .-  =  ( -g `  P )
221fveq2i 5544 . . . . . . . 8  |-  ( -g `  P )  =  (
-g `  (Poly1 ` fld ) )
2321, 22eqtri 2316 . . . . . . 7  |-  .-  =  ( -g `  (Poly1 ` fld ) )
2416, 20, 23oveq123i 5888 . . . . . 6  |-  ( X 
.-  ( A `  r ) )  =  ( (var1 ` fld ) ( -g `  (Poly1 ` fld )
) ( (algSc `  (Poly1 ` fld ) ) `  r
) )
2524eqcomi 2300 . . . . 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 4114 . . 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 5892 . 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 27625 . 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 5899 . 2  |-  ( Q 
gsumg  ( r  e.  ( `' O " { N } )  |->  ( X 
.-  ( A `  r ) ) ) )  e.  _V
3128, 29, 30fvmpt 5618 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 1632    e. wcel 1696    \ cdif 3162   {csn 3653    e. cmpt 4093   `'ccnv 4704   "cima 4708   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753   NNcn 9762   ↾s cress 13165    gsumg cgsu 13417   -gcsg 14381   odcod 14856  mulGrpcmgp 15341  algSccascl 16068  var1cv1 16267  Poly1cpl1 16268  ℂfldccnfld 16393  CytPccytp 27624
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-cytp 27625
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