Theorem coe1tm 19643

Description: Coefficient vector of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)

Hypotheses

Ref	Expression
coe1tm.z	|- .0. = ( 0g ` R )
coe1tm.k	|- K = ( Base ` R )
coe1tm.p	|- P = (Poly1 ` R )
coe1tm.x	|- X = (var1 ` R )
coe1tm.m	|- .x. = ( .s ` P )
coe1tm.n	|- N = (mulGrp ` P )
coe1tm.e	|- .^ = (.g ` N )

Assertion

Ref	Expression
coe1tm	|- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> (coe1 ` ( C .x. ( D .^ X ) ) ) = ( x e. NN0 |-> if ( x = D , C , .0. ) ) )

Distinct variable groups:   x, .0.   x, C   x, D   x, K   x, .^   x, N   x, P   x, X   x, R   x, .x.

Proof of Theorem coe1tm

Dummy variables a b y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		coe1tm.k	. . . 4 |- K = ( Base ` R )
2		coe1tm.p	. . . 4 |- P = (Poly1 ` R )
3		coe1tm.x	. . . 4 |- X = (var1 ` R )
4		coe1tm.m	. . . 4 |- .x. = ( .s ` P )
5		coe1tm.n	. . . 4 |- N = (mulGrp ` P )
6		coe1tm.e	. . . 4 |- .^ = (.g ` N )
7		eqid 2622	. . . 4 |- ( Base ` P ) = ( Base ` P )
8	1, 2, 3, 4, 5, 6, 7	ply1tmcl 19642	. . 3 |- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> ( C .x. ( D .^ X ) ) e. ( Base ` P ) )
9		eqid 2622	. . . 4 |- (coe1 ` ( C .x. ( D .^ X ) ) ) = (coe1 ` ( C .x. ( D .^ X ) ) )
10		eqid 2622	. . . 4 |- ( x e. NN0 |-> ( 1o X. { x } ) ) = ( x e. NN0 |-> ( 1o X. { x } ) )
11	9, 7, 2, 10	coe1fval2 19580	. . 3 |- ( ( C .x. ( D .^ X ) ) e. ( Base ` P ) -> (coe1 ` ( C .x. ( D .^ X ) ) ) = ( ( C .x. ( D .^ X ) ) o. ( x e. NN0 |-> ( 1o X. { x } ) ) ) )
12	8, 11	syl 17	. 2 |- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> (coe1 ` ( C .x. ( D .^ X ) ) ) = ( ( C .x. ( D .^ X ) ) o. ( x e. NN0 |-> ( 1o X. { x } ) ) ) )
13		fconst6g 6094	. . . . 5 |- ( x e. NN0 -> ( 1o X. { x } ) : 1o --> NN0 )
14		nn0ex 11298	. . . . . 6 |- NN0 e. _V
15		1on 7567	. . . . . . 7 |- 1o e. On
16	15	elexi 3213	. . . . . 6 |- 1o e. _V
17	14, 16	elmap 7886	. . . . 5 |- ( ( 1o X. { x } ) e. ( NN0 ^m 1o ) <-> ( 1o X. { x } ) : 1o --> NN0 )
18	13, 17	sylibr 224	. . . 4 |- ( x e. NN0 -> ( 1o X. { x } ) e. ( NN0 ^m 1o ) )
19	18	adantl 482	. . 3 |- ( ( ( R e. Ring /\ C e. K /\ D e. NN0 ) /\ x e. NN0 ) -> ( 1o X. { x } ) e. ( NN0 ^m 1o ) )
20		eqidd 2623	. . 3 |- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> ( x e. NN0 |-> ( 1o X. { x } ) ) = ( x e. NN0 |-> ( 1o X. { x } ) ) )
21		eqid 2622	. . . . . . . 8 |- (.g ` (mulGrp ` ( 1o mPoly R ) ) ) = (.g ` (mulGrp ` ( 1o mPoly R ) ) )
22	5, 7	mgpbas 18495	. . . . . . . . 9 |- ( Base ` P ) = ( Base ` N )
23	22	a1i 11	. . . . . . . 8 |- ( R e. Ring -> ( Base ` P ) = ( Base ` N ) )
24		eqid 2622	. . . . . . . . . 10 |- (mulGrp ` ( 1o mPoly R ) ) = (mulGrp ` ( 1o mPoly R ) )
25		eqid 2622	. . . . . . . . . . 11 |- (PwSer1 ` R ) = (PwSer1 ` R )
26	2, 25, 7	ply1bas 19565	. . . . . . . . . 10 |- ( Base ` P ) = ( Base ` ( 1o mPoly R ) )
27	24, 26	mgpbas 18495	. . . . . . . . 9 |- ( Base ` P ) = ( Base ` (mulGrp ` ( 1o mPoly R ) ) )
28	27	a1i 11	. . . . . . . 8 |- ( R e. Ring -> ( Base ` P ) = ( Base ` (mulGrp ` ( 1o mPoly R ) ) ) )
29		ssv 3625	. . . . . . . . 9 |- ( Base ` P ) C_ _V
30	29	a1i 11	. . . . . . . 8 |- ( R e. Ring -> ( Base ` P ) C_ _V )
31		ovexd 6680	. . . . . . . 8 |- ( ( R e. Ring /\ ( x e. _V /\ y e. _V ) ) -> ( x ( +g ` N ) y ) e. _V )
32		eqid 2622	. . . . . . . . . . . 12 |- ( .r ` P ) = ( .r ` P )
33	5, 32	mgpplusg 18493	. . . . . . . . . . 11 |- ( .r ` P ) = ( +g ` N )
34		eqid 2622	. . . . . . . . . . . . 13 |- ( 1o mPoly R ) = ( 1o mPoly R )
35	2, 34, 32	ply1mulr 19597	. . . . . . . . . . . 12 |- ( .r ` P ) = ( .r ` ( 1o mPoly R ) )
36	24, 35	mgpplusg 18493	. . . . . . . . . . 11 |- ( .r ` P ) = ( +g ` (mulGrp ` ( 1o mPoly R ) ) )
37	33, 36	eqtr3i 2646	. . . . . . . . . 10 |- ( +g ` N ) = ( +g ` (mulGrp ` ( 1o mPoly R ) ) )
38	37	a1i 11	. . . . . . . . 9 |- ( R e. Ring -> ( +g ` N ) = ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) )
39	38	oveqdr 6674	. . . . . . . 8 |- ( ( R e. Ring /\ ( x e. _V /\ y e. _V ) ) -> ( x ( +g ` N ) y ) = ( x ( +g ` (mulGrp ` ( 1o mPoly R ) ) ) y ) )
40	6, 21, 23, 28, 30, 31, 39	mulgpropd 17584	. . . . . . 7 |- ( R e. Ring -> .^ = (.g ` (mulGrp ` ( 1o mPoly R ) ) ) )
41	40	3ad2ant1 1082	. . . . . 6 |- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> .^ = (.g ` (mulGrp ` ( 1o mPoly R ) ) ) )
42		eqidd 2623	. . . . . 6 |- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> D = D )
43	3	vr1val 19562	. . . . . . 7 |- X = ( ( 1o mVar R ) ` (/) )
44	43	a1i 11	. . . . . 6 |- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> X = ( ( 1o mVar R ) ` (/) ) )
45	41, 42, 44	oveq123d 6671	. . . . 5 |- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> ( D .^ X ) = ( D (.g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) )
46	45	oveq2d 6666	. . . 4 |- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> ( C .x. ( D .^ X ) ) = ( C .x. ( D (.g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) ) )
47		psr1baslem 19555	. . . . . 6 |- ( NN0 ^m 1o ) = { a e. ( NN0 ^m 1o ) | ( `' a " NN ) e. Fin }
48		coe1tm.z	. . . . . 6 |- .0. = ( 0g ` R )
49		eqid 2622	. . . . . 6 |- ( 1r ` R ) = ( 1r ` R )
50	15	a1i 11	. . . . . 6 |- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> 1o e. On )
51		eqid 2622	. . . . . 6 |- ( 1o mVar R ) = ( 1o mVar R )
52		simp1 1061	. . . . . 6 |- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> R e. Ring )
53		0lt1o 7584	. . . . . . 7 |- (/) e. 1o
54	53	a1i 11	. . . . . 6 |- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> (/) e. 1o )
55		simp3 1063	. . . . . 6 |- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> D e. NN0 )
56	34, 47, 48, 49, 50, 24, 21, 51, 52, 54, 55	mplcoe3 19466	. . . . 5 |- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> ( y e. ( NN0 ^m 1o ) |-> if ( y = ( b e. 1o |-> if ( b = (/) , D , 0 ) ) , ( 1r ` R ) , .0. ) ) = ( D (.g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) )
57	56	oveq2d 6666	. . . 4 |- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> ( C .x. ( y e. ( NN0 ^m 1o ) |-> if ( y = ( b e. 1o |-> if ( b = (/) , D , 0 ) ) , ( 1r ` R ) , .0. ) ) ) = ( C .x. ( D (.g ` (mulGrp ` ( 1o mPoly R ) ) ) ( ( 1o mVar R ) ` (/) ) ) ) )
58	2, 34, 4	ply1vsca 19596	. . . . 5 |- .x. = ( .s ` ( 1o mPoly R ) )
59		elsni 4194	. . . . . . . . . . 11 |- ( b e. { (/) } -> b = (/) )
60		df1o2 7572	. . . . . . . . . . 11 |- 1o = { (/) }
61	59, 60	eleq2s 2719	. . . . . . . . . 10 |- ( b e. 1o -> b = (/) )
62	61	iftrued 4094	. . . . . . . . 9 |- ( b e. 1o -> if ( b = (/) , D , 0 ) = D )
63	62	adantl 482	. . . . . . . 8 |- ( ( ( R e. Ring /\ C e. K /\ D e. NN0 ) /\ b e. 1o ) -> if ( b = (/) , D , 0 ) = D )
64	63	mpteq2dva 4744	. . . . . . 7 |- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> ( b e. 1o |-> if ( b = (/) , D , 0 ) ) = ( b e. 1o |-> D ) )
65		fconstmpt 5163	. . . . . . 7 |- ( 1o X. { D } ) = ( b e. 1o |-> D )
66	64, 65	syl6eqr 2674	. . . . . 6 |- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> ( b e. 1o |-> if ( b = (/) , D , 0 ) ) = ( 1o X. { D } ) )
67		fconst6g 6094	. . . . . . . 8 |- ( D e. NN0 -> ( 1o X. { D } ) : 1o --> NN0 )
68	14, 16	elmap 7886	. . . . . . . 8 |- ( ( 1o X. { D } ) e. ( NN0 ^m 1o ) <-> ( 1o X. { D } ) : 1o --> NN0 )
69	67, 68	sylibr 224	. . . . . . 7 |- ( D e. NN0 -> ( 1o X. { D } ) e. ( NN0 ^m 1o ) )
70	69	3ad2ant3 1084	. . . . . 6 |- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> ( 1o X. { D } ) e. ( NN0 ^m 1o ) )
71	66, 70	eqeltrd 2701	. . . . 5 |- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> ( b e. 1o |-> if ( b = (/) , D , 0 ) ) e. ( NN0 ^m 1o ) )
72		simp2 1062	. . . . 5 |- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> C e. K )
73	34, 58, 47, 49, 48, 1, 50, 52, 71, 72	mplmon2 19493	. . . 4 |- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> ( C .x. ( y e. ( NN0 ^m 1o ) |-> if ( y = ( b e. 1o |-> if ( b = (/) , D , 0 ) ) , ( 1r ` R ) , .0. ) ) ) = ( y e. ( NN0 ^m 1o ) |-> if ( y = ( b e. 1o |-> if ( b = (/) , D , 0 ) ) , C , .0. ) ) )
74	46, 57, 73	3eqtr2d 2662	. . 3 |- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> ( C .x. ( D .^ X ) ) = ( y e. ( NN0 ^m 1o ) |-> if ( y = ( b e. 1o |-> if ( b = (/) , D , 0 ) ) , C , .0. ) ) )
75		eqeq1 2626	. . . 4 |- ( y = ( 1o X. { x } ) -> ( y = ( b e. 1o |-> if ( b = (/) , D , 0 ) ) <-> ( 1o X. { x } ) = ( b e. 1o |-> if ( b = (/) , D , 0 ) ) ) )
76	75	ifbid 4108	. . 3 |- ( y = ( 1o X. { x } ) -> if ( y = ( b e. 1o |-> if ( b = (/) , D , 0 ) ) , C , .0. ) = if ( ( 1o X. { x } ) = ( b e. 1o |-> if ( b = (/) , D , 0 ) ) , C , .0. ) )
77	19, 20, 74, 76	fmptco 6396	. 2 |- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> ( ( C .x. ( D .^ X ) ) o. ( x e. NN0 |-> ( 1o X. { x } ) ) ) = ( x e. NN0 |-> if ( ( 1o X. { x } ) = ( b e. 1o |-> if ( b = (/) , D , 0 ) ) , C , .0. ) ) )
78	66	adantr 481	. . . . . 6 |- ( ( ( R e. Ring /\ C e. K /\ D e. NN0 ) /\ x e. NN0 ) -> ( b e. 1o |-> if ( b = (/) , D , 0 ) ) = ( 1o X. { D } ) )
79	78	eqeq2d 2632	. . . . 5 |- ( ( ( R e. Ring /\ C e. K /\ D e. NN0 ) /\ x e. NN0 ) -> ( ( 1o X. { x } ) = ( b e. 1o |-> if ( b = (/) , D , 0 ) ) <-> ( 1o X. { x } ) = ( 1o X. { D } ) ) )
80		fveq1 6190	. . . . . . 7 |- ( ( 1o X. { x } ) = ( 1o X. { D } ) -> ( ( 1o X. { x } ) ` (/) ) = ( ( 1o X. { D } ) ` (/) ) )
81		vex 3203	. . . . . . . . . 10 |- x e. _V
82	81	fvconst2 6469	. . . . . . . . 9 |- ( (/) e. 1o -> ( ( 1o X. { x } ) ` (/) ) = x )
83	53, 82	mp1i 13	. . . . . . . 8 |- ( ( ( R e. Ring /\ C e. K /\ D e. NN0 ) /\ x e. NN0 ) -> ( ( 1o X. { x } ) ` (/) ) = x )
84		simpl3 1066	. . . . . . . . 9 |- ( ( ( R e. Ring /\ C e. K /\ D e. NN0 ) /\ x e. NN0 ) -> D e. NN0 )
85		fvconst2g 6467	. . . . . . . . 9 |- ( ( D e. NN0 /\ (/) e. 1o ) -> ( ( 1o X. { D } ) ` (/) ) = D )
86	84, 53, 85	sylancl 694	. . . . . . . 8 |- ( ( ( R e. Ring /\ C e. K /\ D e. NN0 ) /\ x e. NN0 ) -> ( ( 1o X. { D } ) ` (/) ) = D )
87	83, 86	eqeq12d 2637	. . . . . . 7 |- ( ( ( R e. Ring /\ C e. K /\ D e. NN0 ) /\ x e. NN0 ) -> ( ( ( 1o X. { x } ) ` (/) ) = ( ( 1o X. { D } ) ` (/) ) <-> x = D ) )
88	80, 87	syl5ib 234	. . . . . 6 |- ( ( ( R e. Ring /\ C e. K /\ D e. NN0 ) /\ x e. NN0 ) -> ( ( 1o X. { x } ) = ( 1o X. { D } ) -> x = D ) )
89		sneq 4187	. . . . . . 7 |- ( x = D -> { x } = { D } )
90	89	xpeq2d 5139	. . . . . 6 |- ( x = D -> ( 1o X. { x } ) = ( 1o X. { D } ) )
91	88, 90	impbid1 215	. . . . 5 |- ( ( ( R e. Ring /\ C e. K /\ D e. NN0 ) /\ x e. NN0 ) -> ( ( 1o X. { x } ) = ( 1o X. { D } ) <-> x = D ) )
92	79, 91	bitrd 268	. . . 4 |- ( ( ( R e. Ring /\ C e. K /\ D e. NN0 ) /\ x e. NN0 ) -> ( ( 1o X. { x } ) = ( b e. 1o |-> if ( b = (/) , D , 0 ) ) <-> x = D ) )
93	92	ifbid 4108	. . 3 |- ( ( ( R e. Ring /\ C e. K /\ D e. NN0 ) /\ x e. NN0 ) -> if ( ( 1o X. { x } ) = ( b e. 1o |-> if ( b = (/) , D , 0 ) ) , C , .0. ) = if ( x = D , C , .0. ) )
94	93	mpteq2dva 4744	. 2 |- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> ( x e. NN0 |-> if ( ( 1o X. { x } ) = ( b e. 1o |-> if ( b = (/) , D , 0 ) ) , C , .0. ) ) = ( x e. NN0 |-> if ( x = D , C , .0. ) ) )
95	12, 77, 94	3eqtrd 2660	1 |- ( ( R e. Ring /\ C e. K /\ D e. NN0 ) -> (coe1 ` ( C .x. ( D .^ X ) ) ) = ( x e. NN0 |-> if ( x = D , C , .0. ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ifcif 4086   {csn 4177   |-> cmpt 4729   X. cxp 5112   o. ccom 5118   Oncon0 5723   -->wf 5884   ` cfv 5888  (class class class)co 6650   1oc1o 7553   ^m cmap 7857   0cc0 9936   NN0cn0 11292   Basecbs 15857   +g cplusg 15941   .rcmulr 15942   .scvsca 15945   0gc0g 16100  ._gcmg 17540  mulGrpcmgp 18489   1rcur 18501   Ringcrg 18547   mVar cmvr 19352   mPoly cmpl 19353  PwSer₁cps1 19545  var₁cv1 19546  Poly₁cpl1 19547  coe₁cco1 19548

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-tset 15960  df-ple 15961  df-0g 16102  df-gsum 16103  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-subrg 18778  df-lmod 18865  df-lss 18933  df-psr 19356  df-mvr 19357  df-mpl 19358  df-opsr 19360  df-psr1 19550  df-vr1 19551  df-ply1 19552  df-coe1 19553

This theorem is referenced by:  coe1tmfv1  19644  coe1tmfv2  19645  coe1scl  19657  gsummoncoe1  19674  decpmatid  20575  monmatcollpw  20584  mp2pm2mplem4  20614