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Theorem cytpval 27507
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 2442 . . . . . 6  |-  (Poly1 ` fld )  =  P
32fveq2i 5733 . . . . 5  |-  (mulGrp `  (Poly1 ` fld ) )  =  (mulGrp `  P )
4 cytpval.q . . . . 5  |-  Q  =  (mulGrp `  P )
53, 4eqtr4i 2461 . . . 4  |-  (mulGrp `  (Poly1 ` fld ) )  =  Q
65a1i 11 . . 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 5733 . . . . . . . 8  |-  ( od
`  T )  =  ( od `  (
(mulGrp ` fld )s  ( CC  \  { 0 } ) ) )
107, 9eqtri 2458 . . . . . . 7  |-  O  =  ( od `  (
(mulGrp ` fld )s  ( CC  \  { 0 } ) ) )
1110cnveqi 5049 . . . . . 6  |-  `' O  =  `' ( od `  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) ) )
1211imaeq1i 5202 . . . . 5  |-  ( `' O " { n } )  =  ( `' ( od `  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) ) ) " {
n } )
13 sneq 3827 . . . . . 6  |-  ( n  =  N  ->  { n }  =  { N } )
1413imaeq2d 5205 . . . . 5  |-  ( n  =  N  ->  ( `' O " { n } )  =  ( `' O " { N } ) )
1512, 14syl5eqr 2484 . . . 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 5733 . . . . . . . . 9  |-  (algSc `  P )  =  (algSc `  (Poly1 ` fld ) )
1917, 18eqtri 2458 . . . . . . . 8  |-  A  =  (algSc `  (Poly1 ` fld ) )
2019fveq1i 5731 . . . . . . 7  |-  ( A `
 r )  =  ( (algSc `  (Poly1 ` fld )
) `  r )
21 cytpval.m . . . . . . . 8  |-  .-  =  ( -g `  P )
221fveq2i 5733 . . . . . . . 8  |-  ( -g `  P )  =  (
-g `  (Poly1 ` fld ) )
2321, 22eqtri 2458 . . . . . . 7  |-  .-  =  ( -g `  (Poly1 ` fld ) )
2416, 20, 23oveq123i 6097 . . . . . 6  |-  ( X 
.-  ( A `  r ) )  =  ( (var1 ` fld ) ( -g `  (Poly1 ` fld )
) ( (algSc `  (Poly1 ` fld ) ) `  r
) )
2524eqcomi 2442 . . . . 5  |-  ( (var1 ` fld ) ( -g `  (Poly1 ` fld )
) ( (algSc `  (Poly1 ` fld ) ) `  r
) )  =  ( X  .-  ( A `
 r ) )
2625a1i 11 . . . 4  |-  ( n  =  N  ->  (
(var1 ` fld ) ( -g `  (Poly1 ` fld )
) ( (algSc `  (Poly1 ` fld ) ) `  r
) )  =  ( X  .-  ( A `
 r ) ) )
2715, 26mpteq12dv 4289 . . 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 6101 . 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 27501 . 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 6108 . 2  |-  ( Q 
gsumg  ( r  e.  ( `' O " { N } )  |->  ( X 
.-  ( A `  r ) ) ) )  e.  _V
3128, 29, 30fvmpt 5808 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 1653    e. wcel 1726    \ cdif 3319   {csn 3816    e. cmpt 4268   `'ccnv 4879   "cima 4883   ` cfv 5456  (class class class)co 6083   CCcc 8990   0cc0 8992   NNcn 10002   ↾s cress 13472    gsumg cgsu 13726   -gcsg 14690   odcod 15165  mulGrpcmgp 15650  algSccascl 16373  var1cv1 16572  Poly1cpl1 16573  ℂfldccnfld 16705  CytPccytp 27500
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-cytp 27501
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