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Theorem cytpval 29748
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 2467 . . . . . 6  |-  (Poly1 ` fld )  =  P
32fveq2i 5805 . . . . 5  |-  (mulGrp `  (Poly1 ` fld ) )  =  (mulGrp `  P )
4 cytpval.q . . . . 5  |-  Q  =  (mulGrp `  P )
53, 4eqtr4i 2486 . . . 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 5805 . . . . . . . 8  |-  ( od
`  T )  =  ( od `  (
(mulGrp ` fld )s  ( CC  \  { 0 } ) ) )
107, 9eqtri 2483 . . . . . . 7  |-  O  =  ( od `  (
(mulGrp ` fld )s  ( CC  \  { 0 } ) ) )
1110cnveqi 5125 . . . . . 6  |-  `' O  =  `' ( od `  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) ) )
1211imaeq1i 5277 . . . . 5  |-  ( `' O " { n } )  =  ( `' ( od `  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) ) ) " {
n } )
13 sneq 3998 . . . . . 6  |-  ( n  =  N  ->  { n }  =  { N } )
1413imaeq2d 5280 . . . . 5  |-  ( n  =  N  ->  ( `' O " { n } )  =  ( `' O " { N } ) )
1512, 14syl5eqr 2509 . . . 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 5805 . . . . . . . . 9  |-  (algSc `  P )  =  (algSc `  (Poly1 ` fld ) )
1917, 18eqtri 2483 . . . . . . . 8  |-  A  =  (algSc `  (Poly1 ` fld ) )
2019fveq1i 5803 . . . . . . 7  |-  ( A `
 r )  =  ( (algSc `  (Poly1 ` fld )
) `  r )
21 cytpval.m . . . . . . . 8  |-  .-  =  ( -g `  P )
221fveq2i 5805 . . . . . . . 8  |-  ( -g `  P )  =  (
-g `  (Poly1 ` fld ) )
2321, 22eqtri 2483 . . . . . . 7  |-  .-  =  ( -g `  (Poly1 ` fld ) )
2416, 20, 23oveq123i 6217 . . . . . 6  |-  ( X 
.-  ( A `  r ) )  =  ( (var1 ` fld ) ( -g `  (Poly1 ` fld )
) ( (algSc `  (Poly1 ` fld ) ) `  r
) )
2524eqcomi 2467 . . . . 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 4481 . . 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 6221 . 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 29742 . 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 6228 . 2  |-  ( Q 
gsumg  ( r  e.  ( `' O " { N } )  |->  ( X 
.-  ( A `  r ) ) ) )  e.  _V
3128, 29, 30fvmpt 5886 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 1370    e. wcel 1758    \ cdif 3436   {csn 3988    |-> cmpt 4461   `'ccnv 4950   "cima 4954   ` cfv 5529  (class class class)co 6203   CCcc 9395   0cc0 9397   NNcn 10437   ↾s cress 14297    gsumg cgsu 14502   -gcsg 15536   odcod 16153  mulGrpcmgp 16723  algSccascl 17516  var1cv1 17766  Poly1cpl1 17767  ℂfldccnfld 17953  CytPccytp 29741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fv 5537  df-ov 6206  df-cytp 29742
This theorem is referenced by: (None)
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