Theorem cytpval 37787

Description: Substitutions for the Nth cyclotomic polynomial. (Contributed by Stefan O'Rear, 5-Sep-2015.)

Hypotheses

Ref	Expression
cytpval.t	|- T = ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) )
cytpval.o	|- O = ( od ` T )
cytpval.p	|- P = (Poly1 ` ℂfld )
cytpval.x	|- X = (var1 ` ℂfld )
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.

Step	Hyp	Ref	Expression
1		cytpval.p	. . . . . . 7 |- P = (Poly1 ` ℂfld )
2	1	eqcomi 2631	. . . . . 6 |- (Poly1 ` ℂfld ) = P
3	2	fveq2i 6194	. . . . 5 |- (mulGrp ` (Poly1 ` ℂfld ) ) = (mulGrp ` P )
4		cytpval.q	. . . . 5 |- Q = (mulGrp ` P )
5	3, 4	eqtr4i 2647	. . . 4 |- (mulGrp ` (Poly1 ` ℂfld ) ) = Q
6	5	a1i 11	. . 3 |- ( n = N -> (mulGrp ` (Poly1 ` ℂfld ) ) = Q )
7		cytpval.o	. . . . . . . 8 |- O = ( od ` T )
8		cytpval.t	. . . . . . . . 9 |- T = ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) )
9	8	fveq2i 6194	. . . . . . . 8 |- ( od ` T ) = ( od ` ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) )
10	7, 9	eqtri 2644	. . . . . . 7 |- O = ( od ` ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) )
11	10	cnveqi 5297	. . . . . 6 |- `' O = `' ( od ` ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) )
12	11	imaeq1i 5463	. . . . 5 |- ( `' O " { n } ) = ( `' ( od ` ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) ) " { n } )
13		sneq 4187	. . . . . 6 |- ( n = N -> { n } = { N } )
14	13	imaeq2d 5466	. . . . 5 |- ( n = N -> ( `' O " { n } ) = ( `' O " { N } ) )
15	12, 14	syl5eqr 2670	. . . 4 |- ( n = N -> ( `' ( od ` ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) ) " { n } ) = ( `' O " { N } ) )
16		cytpval.x	. . . . . . 7 |- X = (var1 ` ℂfld )
17		cytpval.a	. . . . . . . . 9 |- A = (algSc ` P )
18	1	fveq2i 6194	. . . . . . . . 9 |- (algSc ` P ) = (algSc ` (Poly1 ` ℂfld ) )
19	17, 18	eqtri 2644	. . . . . . . 8 |- A = (algSc ` (Poly1 ` ℂfld ) )
20	19	fveq1i 6192	. . . . . . 7 |- ( A ` r ) = ( (algSc ` (Poly1 ` ℂfld ) ) ` r )
21		cytpval.m	. . . . . . . 8 |- .- = ( -g ` P )
22	1	fveq2i 6194	. . . . . . . 8 |- ( -g ` P ) = ( -g ` (Poly1 ` ℂfld ) )
23	21, 22	eqtri 2644	. . . . . . 7 |- .- = ( -g ` (Poly1 ` ℂfld ) )
24	16, 20, 23	oveq123i 6664	. . . . . 6 |- ( X .- ( A ` r ) ) = ( (var1 ` ℂfld ) ( -g ` (Poly1 ` ℂfld ) ) ( (algSc ` (Poly1 ` ℂfld ) ) ` r ) )
25	24	eqcomi 2631	. . . . 5 |- ( (var1 ` ℂfld ) ( -g ` (Poly1 ` ℂfld ) ) ( (algSc ` (Poly1 ` ℂfld ) ) ` r ) ) = ( X .- ( A ` r ) )
26	25	a1i 11	. . . 4 |- ( n = N -> ( (var1 ` ℂfld ) ( -g ` (Poly1 ` ℂfld ) ) ( (algSc ` (Poly1 ` ℂfld ) ) ` r ) ) = ( X .- ( A ` r ) ) )
27	15, 26	mpteq12dv 4733	. . 3 |- ( n = N -> ( r e. ( `' ( od ` ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) ) " { n } ) |-> ( (var1 ` ℂfld ) ( -g ` (Poly1 ` ℂfld ) ) ( (algSc ` (Poly1 ` ℂfld ) ) ` r ) ) ) = ( r e. ( `' O " { N } ) |-> ( X .- ( A ` r ) ) ) )
28	6, 27	oveq12d 6668	. 2 |- ( n = N -> ( (mulGrp ` (Poly1 ` ℂfld ) ) gsumg ( r e. ( `' ( od ` ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) ) " { n } ) |-> ( (var1 ` ℂfld ) ( -g ` (Poly1 ` ℂfld ) ) ( (algSc ` (Poly1 ` ℂfld ) ) ` r ) ) ) ) = ( Q gsumg ( r e. ( `' O " { N } ) |-> ( X .- ( A ` r ) ) ) ) )
29		df-cytp 37781	. 2 |- CytP = ( n e. NN |-> ( (mulGrp ` (Poly1 ` ℂfld ) ) gsumg ( r e. ( `' ( od ` ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) ) " { n } ) |-> ( (var1 ` ℂfld ) ( -g ` (Poly1 ` ℂfld ) ) ( (algSc ` (Poly1 ` ℂfld ) ) ` r ) ) ) ) )
30		ovex 6678	. 2 |- ( Q gsumg ( r e. ( `' O " { N } ) |-> ( X .- ( A ` r ) ) ) ) e. _V
31	28, 29, 30	fvmpt 6282	1 |- ( N e. NN -> (CytP ` N ) = ( Q gsumg ( r e. ( `' O " { N } ) |-> ( X .- ( A ` r ) ) ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   \ cdif 3571   {csn 4177   |-> cmpt 4729   `'ccnv 5113   "cima 5117   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   NNcn 11020   ↾_s cress 15858   gsum_g cgsu 16101   -gcsg 17424   odcod 17944  mulGrpcmgp 18489  algSccascl 19311  var₁cv1 19546  Poly₁cpl1 19547  ℂ_fldccnfld 19746  CytPccytp 37780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-cytp 37781

This theorem is referenced by: (None)