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Theorem cytpval 31098
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 2480 . . . . . 6  |-  (Poly1 ` fld )  =  P
32fveq2i 5875 . . . . 5  |-  (mulGrp `  (Poly1 ` fld ) )  =  (mulGrp `  P )
4 cytpval.q . . . . 5  |-  Q  =  (mulGrp `  P )
53, 4eqtr4i 2499 . . . 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 5875 . . . . . . . 8  |-  ( od
`  T )  =  ( od `  (
(mulGrp ` fld )s  ( CC  \  { 0 } ) ) )
107, 9eqtri 2496 . . . . . . 7  |-  O  =  ( od `  (
(mulGrp ` fld )s  ( CC  \  { 0 } ) ) )
1110cnveqi 5183 . . . . . 6  |-  `' O  =  `' ( od `  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) ) )
1211imaeq1i 5340 . . . . 5  |-  ( `' O " { n } )  =  ( `' ( od `  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) ) ) " {
n } )
13 sneq 4043 . . . . . 6  |-  ( n  =  N  ->  { n }  =  { N } )
1413imaeq2d 5343 . . . . 5  |-  ( n  =  N  ->  ( `' O " { n } )  =  ( `' O " { N } ) )
1512, 14syl5eqr 2522 . . . 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 5875 . . . . . . . . 9  |-  (algSc `  P )  =  (algSc `  (Poly1 ` fld ) )
1917, 18eqtri 2496 . . . . . . . 8  |-  A  =  (algSc `  (Poly1 ` fld ) )
2019fveq1i 5873 . . . . . . 7  |-  ( A `
 r )  =  ( (algSc `  (Poly1 ` fld )
) `  r )
21 cytpval.m . . . . . . . 8  |-  .-  =  ( -g `  P )
221fveq2i 5875 . . . . . . . 8  |-  ( -g `  P )  =  (
-g `  (Poly1 ` fld ) )
2321, 22eqtri 2496 . . . . . . 7  |-  .-  =  ( -g `  (Poly1 ` fld ) )
2416, 20, 23oveq123i 6309 . . . . . 6  |-  ( X 
.-  ( A `  r ) )  =  ( (var1 ` fld ) ( -g `  (Poly1 ` fld )
) ( (algSc `  (Poly1 ` fld ) ) `  r
) )
2524eqcomi 2480 . . . . 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 4531 . . 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 6313 . 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 31092 . 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 6320 . 2  |-  ( Q 
gsumg  ( r  e.  ( `' O " { N } )  |->  ( X 
.-  ( A `  r ) ) ) )  e.  _V
3128, 29, 30fvmpt 5957 1  |-  ( N  e.  NN  ->  (CytP `  N )  =  ( Q  gsumg  ( r  e.  ( `' O " { N } )  |->  ( X 
.-  ( A `  r ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767    \ cdif 3478   {csn 4033    |-> cmpt 4511   `'ccnv 5004   "cima 5008   ` cfv 5594  (class class class)co 6295   CCcc 9502   0cc0 9504   NNcn 10548   ↾s cress 14508    gsumg cgsu 14713   -gcsg 15927   odcod 16422  mulGrpcmgp 17013  algSccascl 17830  var1cv1 18085  Poly1cpl1 18086  ℂfldccnfld 18290  CytPccytp 31091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-cytp 31092
This theorem is referenced by: (None)
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