Theorem ply1mulgsum 42178

Description: The product of two polynomials expressed as group sum of scaled monomials. (Contributed by AV, 20-Oct-2019.)

Hypotheses

Ref	Expression
ply1mulgsum.p	|- P = (Poly1 ` R )
ply1mulgsum.b	|- B = ( Base ` P )
ply1mulgsum.a	|- A = (coe1 ` K )
ply1mulgsum.c	|- C = (coe1 ` L )
ply1mulgsum.x	|- X = (var1 ` R )
ply1mulgsum.pm	|- .X. = ( .r ` P )
ply1mulgsum.sm	|- .x. = ( .s ` P )
ply1mulgsum.rm	|- .* = ( .r ` R )
ply1mulgsum.m	|- M = (mulGrp ` P )
ply1mulgsum.e	|- .^ = (.g ` M )

Assertion

Ref	Expression
ply1mulgsum	|- ( ( R e. Ring /\ K e. B /\ L e. B ) -> ( K .X. L ) = ( P gsumg ( k e. NN0 |-> ( ( R gsumg ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) ) .x. ( k .^ X ) ) ) ) )

Distinct variable groups:   A, l   B, l   C, l   K, l   L, l   R, l   A, k   B, k   C, k   k, K   k, L   R, k   .* , k, l   k, X   .^ , k   .x. , k   P, k   .* , l

Allowed substitution hints:   P( l)   .x. ( l)   .X. ( k, l)   .^ ( l)   M( k, l)   X( l)

Proof of Theorem ply1mulgsum

Dummy variables n i m are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ply1mulgsum.p	. . . . . . 7 |- P = (Poly1 ` R )
2		ply1mulgsum.pm	. . . . . . 7 |- .X. = ( .r ` P )
3		ply1mulgsum.rm	. . . . . . 7 |- .* = ( .r ` R )
4		ply1mulgsum.b	. . . . . . 7 |- B = ( Base ` P )
5	1, 2, 3, 4	coe1mul 19640	. . . . . 6 |- ( ( R e. Ring /\ K e. B /\ L e. B ) -> (coe1 ` ( K .X. L ) ) = ( m e. NN0 |-> ( R gsumg ( i e. ( 0 ... m ) |-> ( ( (coe1 ` K ) ` i ) .* ( (coe1 ` L ) ` ( m - i ) ) ) ) ) ) )
6	5	adantr 481	. . . . 5 |- ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ n e. NN0 ) -> (coe1 ` ( K .X. L ) ) = ( m e. NN0 |-> ( R gsumg ( i e. ( 0 ... m ) |-> ( ( (coe1 ` K ) ` i ) .* ( (coe1 ` L ) ` ( m - i ) ) ) ) ) ) )
7	6	fveq1d 6193	. . . 4 |- ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ n e. NN0 ) -> ( (coe1 ` ( K .X. L ) ) ` n ) = ( ( m e. NN0 |-> ( R gsumg ( i e. ( 0 ... m ) |-> ( ( (coe1 ` K ) ` i ) .* ( (coe1 ` L ) ` ( m - i ) ) ) ) ) ) ` n ) )
8		eqidd 2623	. . . . 5 |- ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ n e. NN0 ) -> ( m e. NN0 |-> ( R gsumg ( i e. ( 0 ... m ) |-> ( ( (coe1 ` K ) ` i ) .* ( (coe1 ` L ) ` ( m - i ) ) ) ) ) ) = ( m e. NN0 |-> ( R gsumg ( i e. ( 0 ... m ) |-> ( ( (coe1 ` K ) ` i ) .* ( (coe1 ` L ) ` ( m - i ) ) ) ) ) ) )
9		oveq2 6658	. . . . . . . 8 |- ( m = n -> ( 0 ... m ) = ( 0 ... n ) )
10		oveq1 6657	. . . . . . . . . 10 |- ( m = n -> ( m - i ) = ( n - i ) )
11	10	fveq2d 6195	. . . . . . . . 9 |- ( m = n -> ( (coe1 ` L ) ` ( m - i ) ) = ( (coe1 ` L ) ` ( n - i ) ) )
12	11	oveq2d 6666	. . . . . . . 8 |- ( m = n -> ( ( (coe1 ` K ) ` i ) .* ( (coe1 ` L ) ` ( m - i ) ) ) = ( ( (coe1 ` K ) ` i ) .* ( (coe1 ` L ) ` ( n - i ) ) ) )
13	9, 12	mpteq12dv 4733	. . . . . . 7 |- ( m = n -> ( i e. ( 0 ... m ) |-> ( ( (coe1 ` K ) ` i ) .* ( (coe1 ` L ) ` ( m - i ) ) ) ) = ( i e. ( 0 ... n ) |-> ( ( (coe1 ` K ) ` i ) .* ( (coe1 ` L ) ` ( n - i ) ) ) ) )
14	13	oveq2d 6666	. . . . . 6 |- ( m = n -> ( R gsumg ( i e. ( 0 ... m ) |-> ( ( (coe1 ` K ) ` i ) .* ( (coe1 ` L ) ` ( m - i ) ) ) ) ) = ( R gsumg ( i e. ( 0 ... n ) |-> ( ( (coe1 ` K ) ` i ) .* ( (coe1 ` L ) ` ( n - i ) ) ) ) ) )
15	14	adantl 482	. . . . 5 |- ( ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ n e. NN0 ) /\ m = n ) -> ( R gsumg ( i e. ( 0 ... m ) |-> ( ( (coe1 ` K ) ` i ) .* ( (coe1 ` L ) ` ( m - i ) ) ) ) ) = ( R gsumg ( i e. ( 0 ... n ) |-> ( ( (coe1 ` K ) ` i ) .* ( (coe1 ` L ) ` ( n - i ) ) ) ) ) )
16		simpr 477	. . . . 5 |- ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ n e. NN0 ) -> n e. NN0 )
17		ovexd 6680	. . . . 5 |- ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ n e. NN0 ) -> ( R gsumg ( i e. ( 0 ... n ) |-> ( ( (coe1 ` K ) ` i ) .* ( (coe1 ` L ) ` ( n - i ) ) ) ) ) e. _V )
18	8, 15, 16, 17	fvmptd 6288	. . . 4 |- ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ n e. NN0 ) -> ( ( m e. NN0 |-> ( R gsumg ( i e. ( 0 ... m ) |-> ( ( (coe1 ` K ) ` i ) .* ( (coe1 ` L ) ` ( m - i ) ) ) ) ) ) ` n ) = ( R gsumg ( i e. ( 0 ... n ) |-> ( ( (coe1 ` K ) ` i ) .* ( (coe1 ` L ) ` ( n - i ) ) ) ) ) )
19		ply1mulgsum.x	. . . . . 6 |- X = (var1 ` R )
20		ply1mulgsum.e	. . . . . . 7 |- .^ = (.g ` M )
21		ply1mulgsum.m	. . . . . . . 8 |- M = (mulGrp ` P )
22	21	fveq2i 6194	. . . . . . 7 |- (.g ` M ) = (.g ` (mulGrp ` P ) )
23	20, 22	eqtri 2644	. . . . . 6 |- .^ = (.g ` (mulGrp ` P ) )
24		simp1 1061	. . . . . . 7 |- ( ( R e. Ring /\ K e. B /\ L e. B ) -> R e. Ring )
25	24	adantr 481	. . . . . 6 |- ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ n e. NN0 ) -> R e. Ring )
26		eqid 2622	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
27		ply1mulgsum.sm	. . . . . 6 |- .x. = ( .s ` P )
28		eqid 2622	. . . . . 6 |- ( 0g ` R ) = ( 0g ` R )
29		ringcmn 18581	. . . . . . . . . 10 |- ( R e. Ring -> R e. CMnd )
30	29	3ad2ant1 1082	. . . . . . . . 9 |- ( ( R e. Ring /\ K e. B /\ L e. B ) -> R e. CMnd )
31	30	ad2antrr 762	. . . . . . . 8 |- ( ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ n e. NN0 ) /\ k e. NN0 ) -> R e. CMnd )
32		fzfid 12772	. . . . . . . 8 |- ( ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ n e. NN0 ) /\ k e. NN0 ) -> ( 0 ... k ) e. Fin )
33		simpll1 1100	. . . . . . . . . . 11 |- ( ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ n e. NN0 ) /\ k e. NN0 ) -> R e. Ring )
34	33	adantr 481	. . . . . . . . . 10 |- ( ( ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ n e. NN0 ) /\ k e. NN0 ) /\ l e. ( 0 ... k ) ) -> R e. Ring )
35		simp2 1062	. . . . . . . . . . . 12 |- ( ( R e. Ring /\ K e. B /\ L e. B ) -> K e. B )
36	35	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ n e. NN0 ) /\ k e. NN0 ) -> K e. B )
37		elfznn0 12433	. . . . . . . . . . 11 |- ( l e. ( 0 ... k ) -> l e. NN0 )
38		ply1mulgsum.a	. . . . . . . . . . . 12 |- A = (coe1 ` K )
39	38, 4, 1, 26	coe1fvalcl 19582	. . . . . . . . . . 11 |- ( ( K e. B /\ l e. NN0 ) -> ( A ` l ) e. ( Base ` R ) )
40	36, 37, 39	syl2an 494	. . . . . . . . . 10 |- ( ( ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ n e. NN0 ) /\ k e. NN0 ) /\ l e. ( 0 ... k ) ) -> ( A ` l ) e. ( Base ` R ) )
41		simp3 1063	. . . . . . . . . . . 12 |- ( ( R e. Ring /\ K e. B /\ L e. B ) -> L e. B )
42	41	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ n e. NN0 ) /\ k e. NN0 ) -> L e. B )
43		fznn0sub 12373	. . . . . . . . . . 11 |- ( l e. ( 0 ... k ) -> ( k - l ) e. NN0 )
44		ply1mulgsum.c	. . . . . . . . . . . 12 |- C = (coe1 ` L )
45	44, 4, 1, 26	coe1fvalcl 19582	. . . . . . . . . . 11 |- ( ( L e. B /\ ( k - l ) e. NN0 ) -> ( C ` ( k - l ) ) e. ( Base ` R ) )
46	42, 43, 45	syl2an 494	. . . . . . . . . 10 |- ( ( ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ n e. NN0 ) /\ k e. NN0 ) /\ l e. ( 0 ... k ) ) -> ( C ` ( k - l ) ) e. ( Base ` R ) )
47	26, 3	ringcl 18561	. . . . . . . . . 10 |- ( ( R e. Ring /\ ( A ` l ) e. ( Base ` R ) /\ ( C ` ( k - l ) ) e. ( Base ` R ) ) -> ( ( A ` l ) .* ( C ` ( k - l ) ) ) e. ( Base ` R ) )
48	34, 40, 46, 47	syl3anc 1326	. . . . . . . . 9 |- ( ( ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ n e. NN0 ) /\ k e. NN0 ) /\ l e. ( 0 ... k ) ) -> ( ( A ` l ) .* ( C ` ( k - l ) ) ) e. ( Base ` R ) )
49	48	ralrimiva 2966	. . . . . . . 8 |- ( ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ n e. NN0 ) /\ k e. NN0 ) -> A. l e. ( 0 ... k ) ( ( A ` l ) .* ( C ` ( k - l ) ) ) e. ( Base ` R ) )
50	26, 31, 32, 49	gsummptcl 18366	. . . . . . 7 |- ( ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ n e. NN0 ) /\ k e. NN0 ) -> ( R gsumg ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) ) e. ( Base ` R ) )
51	50	ralrimiva 2966	. . . . . 6 |- ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ n e. NN0 ) -> A. k e. NN0 ( R gsumg ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) ) e. ( Base ` R ) )
52	1, 4, 38, 44, 19, 2, 27, 3, 21, 20	ply1mulgsumlem3 42176	. . . . . . 7 |- ( ( R e. Ring /\ K e. B /\ L e. B ) -> ( k e. NN0 |-> ( R gsumg ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) ) ) finSupp ( 0g ` R ) )
53	52	adantr 481	. . . . . 6 |- ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ n e. NN0 ) -> ( k e. NN0 |-> ( R gsumg ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) ) ) finSupp ( 0g ` R ) )
54	1, 4, 19, 23, 25, 26, 27, 28, 51, 53, 16	gsummoncoe1 19674	. . . . 5 |- ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ n e. NN0 ) -> ( (coe1 ` ( P gsumg ( k e. NN0 |-> ( ( R gsumg ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) ) .x. ( k .^ X ) ) ) ) ) ` n ) = [_ n / k ]_ ( R gsumg ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) ) )
55		vex 3203	. . . . . 6 |- n e. _V
56		csbov2g 6691	. . . . . . 7 |- ( n e. _V -> [_ n / k ]_ ( R gsumg ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) ) = ( R gsumg [_ n / k ]_ ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) ) )
57		id 22	. . . . . . . . 9 |- ( n e. _V -> n e. _V )
58		oveq2 6658	. . . . . . . . . . 11 |- ( k = n -> ( 0 ... k ) = ( 0 ... n ) )
59		oveq1 6657	. . . . . . . . . . . . 13 |- ( k = n -> ( k - l ) = ( n - l ) )
60	59	fveq2d 6195	. . . . . . . . . . . 12 |- ( k = n -> ( C ` ( k - l ) ) = ( C ` ( n - l ) ) )
61	60	oveq2d 6666	. . . . . . . . . . 11 |- ( k = n -> ( ( A ` l ) .* ( C ` ( k - l ) ) ) = ( ( A ` l ) .* ( C ` ( n - l ) ) ) )
62	58, 61	mpteq12dv 4733	. . . . . . . . . 10 |- ( k = n -> ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) = ( l e. ( 0 ... n ) |-> ( ( A ` l ) .* ( C ` ( n - l ) ) ) ) )
63	62	adantl 482	. . . . . . . . 9 |- ( ( n e. _V /\ k = n ) -> ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) = ( l e. ( 0 ... n ) |-> ( ( A ` l ) .* ( C ` ( n - l ) ) ) ) )
64	57, 63	csbied 3560	. . . . . . . 8 |- ( n e. _V -> [_ n / k ]_ ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) = ( l e. ( 0 ... n ) |-> ( ( A ` l ) .* ( C ` ( n - l ) ) ) ) )
65	64	oveq2d 6666	. . . . . . 7 |- ( n e. _V -> ( R gsumg [_ n / k ]_ ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) ) = ( R gsumg ( l e. ( 0 ... n ) |-> ( ( A ` l ) .* ( C ` ( n - l ) ) ) ) ) )
66	56, 65	eqtrd 2656	. . . . . 6 |- ( n e. _V -> [_ n / k ]_ ( R gsumg ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) ) = ( R gsumg ( l e. ( 0 ... n ) |-> ( ( A ` l ) .* ( C ` ( n - l ) ) ) ) ) )
67	55, 66	mp1i 13	. . . . 5 |- ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ n e. NN0 ) -> [_ n / k ]_ ( R gsumg ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) ) = ( R gsumg ( l e. ( 0 ... n ) |-> ( ( A ` l ) .* ( C ` ( n - l ) ) ) ) ) )
68		fveq2 6191	. . . . . . . . . 10 |- ( l = i -> ( A ` l ) = ( A ` i ) )
69	38	fveq1i 6192	. . . . . . . . . 10 |- ( A ` i ) = ( (coe1 ` K ) ` i )
70	68, 69	syl6eq 2672	. . . . . . . . 9 |- ( l = i -> ( A ` l ) = ( (coe1 ` K ) ` i ) )
71		oveq2 6658	. . . . . . . . . . 11 |- ( l = i -> ( n - l ) = ( n - i ) )
72	71	fveq2d 6195	. . . . . . . . . 10 |- ( l = i -> ( C ` ( n - l ) ) = ( C ` ( n - i ) ) )
73	44	fveq1i 6192	. . . . . . . . . 10 |- ( C ` ( n - i ) ) = ( (coe1 ` L ) ` ( n - i ) )
74	72, 73	syl6eq 2672	. . . . . . . . 9 |- ( l = i -> ( C ` ( n - l ) ) = ( (coe1 ` L ) ` ( n - i ) ) )
75	70, 74	oveq12d 6668	. . . . . . . 8 |- ( l = i -> ( ( A ` l ) .* ( C ` ( n - l ) ) ) = ( ( (coe1 ` K ) ` i ) .* ( (coe1 ` L ) ` ( n - i ) ) ) )
76	75	cbvmptv 4750	. . . . . . 7 |- ( l e. ( 0 ... n ) |-> ( ( A ` l ) .* ( C ` ( n - l ) ) ) ) = ( i e. ( 0 ... n ) |-> ( ( (coe1 ` K ) ` i ) .* ( (coe1 ` L ) ` ( n - i ) ) ) )
77	76	a1i 11	. . . . . 6 |- ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ n e. NN0 ) -> ( l e. ( 0 ... n ) |-> ( ( A ` l ) .* ( C ` ( n - l ) ) ) ) = ( i e. ( 0 ... n ) |-> ( ( (coe1 ` K ) ` i ) .* ( (coe1 ` L ) ` ( n - i ) ) ) ) )
78	77	oveq2d 6666	. . . . 5 |- ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ n e. NN0 ) -> ( R gsumg ( l e. ( 0 ... n ) |-> ( ( A ` l ) .* ( C ` ( n - l ) ) ) ) ) = ( R gsumg ( i e. ( 0 ... n ) |-> ( ( (coe1 ` K ) ` i ) .* ( (coe1 ` L ) ` ( n - i ) ) ) ) ) )
79	54, 67, 78	3eqtrrd 2661	. . . 4 |- ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ n e. NN0 ) -> ( R gsumg ( i e. ( 0 ... n ) |-> ( ( (coe1 ` K ) ` i ) .* ( (coe1 ` L ) ` ( n - i ) ) ) ) ) = ( (coe1 ` ( P gsumg ( k e. NN0 |-> ( ( R gsumg ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) ) .x. ( k .^ X ) ) ) ) ) ` n ) )
80	7, 18, 79	3eqtrd 2660	. . 3 |- ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ n e. NN0 ) -> ( (coe1 ` ( K .X. L ) ) ` n ) = ( (coe1 ` ( P gsumg ( k e. NN0 |-> ( ( R gsumg ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) ) .x. ( k .^ X ) ) ) ) ) ` n ) )
81	80	ralrimiva 2966	. 2 |- ( ( R e. Ring /\ K e. B /\ L e. B ) -> A. n e. NN0 ( (coe1 ` ( K .X. L ) ) ` n ) = ( (coe1 ` ( P gsumg ( k e. NN0 |-> ( ( R gsumg ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) ) .x. ( k .^ X ) ) ) ) ) ` n ) )
82	1	ply1ring 19618	. . . 4 |- ( R e. Ring -> P e. Ring )
83	4, 2	ringcl 18561	. . . 4 |- ( ( P e. Ring /\ K e. B /\ L e. B ) -> ( K .X. L ) e. B )
84	82, 83	syl3an1 1359	. . 3 |- ( ( R e. Ring /\ K e. B /\ L e. B ) -> ( K .X. L ) e. B )
85		eqid 2622	. . . 4 |- ( 0g ` P ) = ( 0g ` P )
86		ringcmn 18581	. . . . . 6 |- ( P e. Ring -> P e. CMnd )
87	82, 86	syl 17	. . . . 5 |- ( R e. Ring -> P e. CMnd )
88	87	3ad2ant1 1082	. . . 4 |- ( ( R e. Ring /\ K e. B /\ L e. B ) -> P e. CMnd )
89		nn0ex 11298	. . . . 5 |- NN0 e. _V
90	89	a1i 11	. . . 4 |- ( ( R e. Ring /\ K e. B /\ L e. B ) -> NN0 e. _V )
91	1	ply1lmod 19622	. . . . . . . 8 |- ( R e. Ring -> P e. LMod )
92	91	3ad2ant1 1082	. . . . . . 7 |- ( ( R e. Ring /\ K e. B /\ L e. B ) -> P e. LMod )
93	92	adantr 481	. . . . . 6 |- ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ k e. NN0 ) -> P e. LMod )
94	30	adantr 481	. . . . . . . 8 |- ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ k e. NN0 ) -> R e. CMnd )
95		fzfid 12772	. . . . . . . 8 |- ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ k e. NN0 ) -> ( 0 ... k ) e. Fin )
96		simpll1 1100	. . . . . . . . . 10 |- ( ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ k e. NN0 ) /\ l e. ( 0 ... k ) ) -> R e. Ring )
97	35	adantr 481	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ k e. NN0 ) -> K e. B )
98	97, 37, 39	syl2an 494	. . . . . . . . . 10 |- ( ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ k e. NN0 ) /\ l e. ( 0 ... k ) ) -> ( A ` l ) e. ( Base ` R ) )
99	41	adantr 481	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ k e. NN0 ) -> L e. B )
100	99, 43, 45	syl2an 494	. . . . . . . . . 10 |- ( ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ k e. NN0 ) /\ l e. ( 0 ... k ) ) -> ( C ` ( k - l ) ) e. ( Base ` R ) )
101	96, 98, 100, 47	syl3anc 1326	. . . . . . . . 9 |- ( ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ k e. NN0 ) /\ l e. ( 0 ... k ) ) -> ( ( A ` l ) .* ( C ` ( k - l ) ) ) e. ( Base ` R ) )
102	101	ralrimiva 2966	. . . . . . . 8 |- ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ k e. NN0 ) -> A. l e. ( 0 ... k ) ( ( A ` l ) .* ( C ` ( k - l ) ) ) e. ( Base ` R ) )
103	26, 94, 95, 102	gsummptcl 18366	. . . . . . 7 |- ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ k e. NN0 ) -> ( R gsumg ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) ) e. ( Base ` R ) )
104	24	adantr 481	. . . . . . . . 9 |- ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ k e. NN0 ) -> R e. Ring )
105	1	ply1sca 19623	. . . . . . . . 9 |- ( R e. Ring -> R = (Scalar ` P ) )
106	104, 105	syl 17	. . . . . . . 8 |- ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ k e. NN0 ) -> R = (Scalar ` P ) )
107	106	fveq2d 6195	. . . . . . 7 |- ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ k e. NN0 ) -> ( Base ` R ) = ( Base ` (Scalar ` P ) ) )
108	103, 107	eleqtrd 2703	. . . . . 6 |- ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ k e. NN0 ) -> ( R gsumg ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) ) e. ( Base ` (Scalar ` P ) ) )
109	21	ringmgp 18553	. . . . . . . . . 10 |- ( P e. Ring -> M e. Mnd )
110	82, 109	syl 17	. . . . . . . . 9 |- ( R e. Ring -> M e. Mnd )
111	110	3ad2ant1 1082	. . . . . . . 8 |- ( ( R e. Ring /\ K e. B /\ L e. B ) -> M e. Mnd )
112	111	adantr 481	. . . . . . 7 |- ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ k e. NN0 ) -> M e. Mnd )
113		simpr 477	. . . . . . 7 |- ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ k e. NN0 ) -> k e. NN0 )
114	19, 1, 4	vr1cl 19587	. . . . . . . . 9 |- ( R e. Ring -> X e. B )
115	114	3ad2ant1 1082	. . . . . . . 8 |- ( ( R e. Ring /\ K e. B /\ L e. B ) -> X e. B )
116	115	adantr 481	. . . . . . 7 |- ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ k e. NN0 ) -> X e. B )
117	21, 4	mgpbas 18495	. . . . . . . 8 |- B = ( Base ` M )
118	117, 20	mulgnn0cl 17558	. . . . . . 7 |- ( ( M e. Mnd /\ k e. NN0 /\ X e. B ) -> ( k .^ X ) e. B )
119	112, 113, 116, 118	syl3anc 1326	. . . . . 6 |- ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ k e. NN0 ) -> ( k .^ X ) e. B )
120		eqid 2622	. . . . . . 7 |- (Scalar ` P ) = (Scalar ` P )
121		eqid 2622	. . . . . . 7 |- ( Base ` (Scalar ` P ) ) = ( Base ` (Scalar ` P ) )
122	4, 120, 27, 121	lmodvscl 18880	. . . . . 6 |- ( ( P e. LMod /\ ( R gsumg ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) ) e. ( Base ` (Scalar ` P ) ) /\ ( k .^ X ) e. B ) -> ( ( R gsumg ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) ) .x. ( k .^ X ) ) e. B )
123	93, 108, 119, 122	syl3anc 1326	. . . . 5 |- ( ( ( R e. Ring /\ K e. B /\ L e. B ) /\ k e. NN0 ) -> ( ( R gsumg ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) ) .x. ( k .^ X ) ) e. B )
124		eqid 2622	. . . . 5 |- ( k e. NN0 |-> ( ( R gsumg ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) ) .x. ( k .^ X ) ) ) = ( k e. NN0 |-> ( ( R gsumg ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) ) .x. ( k .^ X ) ) )
125	123, 124	fmptd 6385	. . . 4 |- ( ( R e. Ring /\ K e. B /\ L e. B ) -> ( k e. NN0 |-> ( ( R gsumg ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) ) .x. ( k .^ X ) ) ) : NN0 --> B )
126	1, 4, 38, 44, 19, 2, 27, 3, 21, 20	ply1mulgsumlem4 42177	. . . 4 |- ( ( R e. Ring /\ K e. B /\ L e. B ) -> ( k e. NN0 |-> ( ( R gsumg ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) ) .x. ( k .^ X ) ) ) finSupp ( 0g ` P ) )
127	4, 85, 88, 90, 125, 126	gsumcl 18316	. . 3 |- ( ( R e. Ring /\ K e. B /\ L e. B ) -> ( P gsumg ( k e. NN0 |-> ( ( R gsumg ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) ) .x. ( k .^ X ) ) ) ) e. B )
128		eqid 2622	. . . 4 |- (coe1 ` ( K .X. L ) ) = (coe1 ` ( K .X. L ) )
129		eqid 2622	. . . 4 |- (coe1 ` ( P gsumg ( k e. NN0 |-> ( ( R gsumg ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) ) .x. ( k .^ X ) ) ) ) ) = (coe1 ` ( P gsumg ( k e. NN0 |-> ( ( R gsumg ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) ) .x. ( k .^ X ) ) ) ) )
130	1, 4, 128, 129	ply1coe1eq 19668	. . 3 |- ( ( R e. Ring /\ ( K .X. L ) e. B /\ ( P gsumg ( k e. NN0 |-> ( ( R gsumg ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) ) .x. ( k .^ X ) ) ) ) e. B ) -> ( A. n e. NN0 ( (coe1 ` ( K .X. L ) ) ` n ) = ( (coe1 ` ( P gsumg ( k e. NN0 |-> ( ( R gsumg ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) ) .x. ( k .^ X ) ) ) ) ) ` n ) <-> ( K .X. L ) = ( P gsumg ( k e. NN0 |-> ( ( R gsumg ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) ) .x. ( k .^ X ) ) ) ) ) )
131	24, 84, 127, 130	syl3anc 1326	. 2 |- ( ( R e. Ring /\ K e. B /\ L e. B ) -> ( A. n e. NN0 ( (coe1 ` ( K .X. L ) ) ` n ) = ( (coe1 ` ( P gsumg ( k e. NN0 |-> ( ( R gsumg ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) ) .x. ( k .^ X ) ) ) ) ) ` n ) <-> ( K .X. L ) = ( P gsumg ( k e. NN0 |-> ( ( R gsumg ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) ) .x. ( k .^ X ) ) ) ) ) )
132	81, 131	mpbid 222	1 |- ( ( R e. Ring /\ K e. B /\ L e. B ) -> ( K .X. L ) = ( P gsumg ( k e. NN0 |-> ( ( R gsumg ( l e. ( 0 ... k ) |-> ( ( A ` l ) .* ( C ` ( k - l ) ) ) ) ) .x. ( k .^ X ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   [_csb 3533   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   finSupp cfsupp 8275   0cc0 9936   - cmin 10266   NN0cn0 11292   ...cfz 12326   Basecbs 15857   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   gsum_g cgsu 16101   Mndcmnd 17294  ._gcmg 17540  CMndccmn 18193  mulGrpcmgp 18489   Ringcrg 18547   LModclmod 18863  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-fal 1489  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-srg 18506  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: (None)