Theorem coe1mul2 19639

Description: The coefficient vector of multiplication in the univariate power series ring. (Contributed by Stefan O'Rear, 25-Mar-2015.)

Hypotheses

Ref	Expression
coe1mul2.s	|- S = (PwSer1 ` R )
coe1mul2.t	|- .xb = ( .r ` S )
coe1mul2.u	|- .x. = ( .r ` R )
coe1mul2.b	|- B = ( Base ` S )

Assertion

Ref	Expression
coe1mul2	|- ( ( R e. Ring /\ F e. B /\ G e. B ) -> (coe1 ` ( F .xb G ) ) = ( k e. NN0 |-> ( R gsumg ( x e. ( 0 ... k ) |-> ( ( (coe1 ` F ) ` x ) .x. ( (coe1 ` G ) ` ( k - x ) ) ) ) ) ) )

Distinct variable groups:   x, k, B   k, F, x   .x. , k, x   k, G, x   R, k, x   .xb , k

Allowed substitution hints:   S( x, k)   .xb ( x)

Proof of Theorem coe1mul2

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

Step	Hyp	Ref	Expression
1		fconst6g 6094	. . . . 5 |- ( k e. NN0 -> ( 1o X. { k } ) : 1o --> NN0 )
2		nn0ex 11298	. . . . . 6 |- NN0 e. _V
3		1on 7567	. . . . . . 7 |- 1o e. On
4	3	elexi 3213	. . . . . 6 |- 1o e. _V
5	2, 4	elmap 7886	. . . . 5 |- ( ( 1o X. { k } ) e. ( NN0 ^m 1o ) <-> ( 1o X. { k } ) : 1o --> NN0 )
6	1, 5	sylibr 224	. . . 4 |- ( k e. NN0 -> ( 1o X. { k } ) e. ( NN0 ^m 1o ) )
7	6	adantl 482	. . 3 |- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) -> ( 1o X. { k } ) e. ( NN0 ^m 1o ) )
8		eqidd 2623	. . 3 |- ( ( R e. Ring /\ F e. B /\ G e. B ) -> ( k e. NN0 |-> ( 1o X. { k } ) ) = ( k e. NN0 |-> ( 1o X. { k } ) ) )
9		eqid 2622	. . . 4 |- ( 1o mPwSer R ) = ( 1o mPwSer R )
10		coe1mul2.s	. . . . 5 |- S = (PwSer1 ` R )
11		coe1mul2.b	. . . . 5 |- B = ( Base ` S )
12	10, 11, 9	psr1bas2 19560	. . . 4 |- B = ( Base ` ( 1o mPwSer R ) )
13		coe1mul2.u	. . . 4 |- .x. = ( .r ` R )
14		coe1mul2.t	. . . . 5 |- .xb = ( .r ` S )
15	10, 9, 14	psr1mulr 19594	. . . 4 |- .xb = ( .r ` ( 1o mPwSer R ) )
16		psr1baslem 19555	. . . 4 |- ( NN0 ^m 1o ) = { a e. ( NN0 ^m 1o ) | ( `' a " NN ) e. Fin }
17		simp2 1062	. . . 4 |- ( ( R e. Ring /\ F e. B /\ G e. B ) -> F e. B )
18		simp3 1063	. . . 4 |- ( ( R e. Ring /\ F e. B /\ G e. B ) -> G e. B )
19	9, 12, 13, 15, 16, 17, 18	psrmulfval 19385	. . 3 |- ( ( R e. Ring /\ F e. B /\ G e. B ) -> ( F .xb G ) = ( b e. ( NN0 ^m 1o ) |-> ( R gsumg ( c e. { d e. ( NN0 ^m 1o ) | d oR <_ b } |-> ( ( F ` c ) .x. ( G ` ( b oF - c ) ) ) ) ) ) )
20		breq2 4657	. . . . . 6 |- ( b = ( 1o X. { k } ) -> ( d oR <_ b <-> d oR <_ ( 1o X. { k } ) ) )
21	20	rabbidv 3189	. . . . 5 |- ( b = ( 1o X. { k } ) -> { d e. ( NN0 ^m 1o ) | d oR <_ b } = { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } )
22		oveq1 6657	. . . . . . 7 |- ( b = ( 1o X. { k } ) -> ( b oF - c ) = ( ( 1o X. { k } ) oF - c ) )
23	22	fveq2d 6195	. . . . . 6 |- ( b = ( 1o X. { k } ) -> ( G ` ( b oF - c ) ) = ( G ` ( ( 1o X. { k } ) oF - c ) ) )
24	23	oveq2d 6666	. . . . 5 |- ( b = ( 1o X. { k } ) -> ( ( F ` c ) .x. ( G ` ( b oF - c ) ) ) = ( ( F ` c ) .x. ( G ` ( ( 1o X. { k } ) oF - c ) ) ) )
25	21, 24	mpteq12dv 4733	. . . 4 |- ( b = ( 1o X. { k } ) -> ( c e. { d e. ( NN0 ^m 1o ) | d oR <_ b } |-> ( ( F ` c ) .x. ( G ` ( b oF - c ) ) ) ) = ( c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } |-> ( ( F ` c ) .x. ( G ` ( ( 1o X. { k } ) oF - c ) ) ) ) )
26	25	oveq2d 6666	. . 3 |- ( b = ( 1o X. { k } ) -> ( R gsumg ( c e. { d e. ( NN0 ^m 1o ) | d oR <_ b } |-> ( ( F ` c ) .x. ( G ` ( b oF - c ) ) ) ) ) = ( R gsumg ( c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } |-> ( ( F ` c ) .x. ( G ` ( ( 1o X. { k } ) oF - c ) ) ) ) ) )
27	7, 8, 19, 26	fmptco 6396	. 2 |- ( ( R e. Ring /\ F e. B /\ G e. B ) -> ( ( F .xb G ) o. ( k e. NN0 |-> ( 1o X. { k } ) ) ) = ( k e. NN0 |-> ( R gsumg ( c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } |-> ( ( F ` c ) .x. ( G ` ( ( 1o X. { k } ) oF - c ) ) ) ) ) ) )
28	10	psr1ring 19617	. . . 4 |- ( R e. Ring -> S e. Ring )
29	11, 14	ringcl 18561	. . . 4 |- ( ( S e. Ring /\ F e. B /\ G e. B ) -> ( F .xb G ) e. B )
30	28, 29	syl3an1 1359	. . 3 |- ( ( R e. Ring /\ F e. B /\ G e. B ) -> ( F .xb G ) e. B )
31		eqid 2622	. . . 4 |- (coe1 ` ( F .xb G ) ) = (coe1 ` ( F .xb G ) )
32		eqid 2622	. . . 4 |- ( k e. NN0 |-> ( 1o X. { k } ) ) = ( k e. NN0 |-> ( 1o X. { k } ) )
33	31, 11, 10, 32	coe1fval3 19578	. . 3 |- ( ( F .xb G ) e. B -> (coe1 ` ( F .xb G ) ) = ( ( F .xb G ) o. ( k e. NN0 |-> ( 1o X. { k } ) ) ) )
34	30, 33	syl 17	. 2 |- ( ( R e. Ring /\ F e. B /\ G e. B ) -> (coe1 ` ( F .xb G ) ) = ( ( F .xb G ) o. ( k e. NN0 |-> ( 1o X. { k } ) ) ) )
35		eqid 2622	. . . . 5 |- ( Base ` R ) = ( Base ` R )
36		eqid 2622	. . . . 5 |- ( 0g ` R ) = ( 0g ` R )
37		simpl1 1064	. . . . . 6 |- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) -> R e. Ring )
38		ringcmn 18581	. . . . . 6 |- ( R e. Ring -> R e. CMnd )
39	37, 38	syl 17	. . . . 5 |- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) -> R e. CMnd )
40		fzfi 12771	. . . . . 6 |- ( 0 ... k ) e. Fin
41	40	a1i 11	. . . . 5 |- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) -> ( 0 ... k ) e. Fin )
42		simpll1 1100	. . . . . . 7 |- ( ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) /\ x e. ( 0 ... k ) ) -> R e. Ring )
43		simpll2 1101	. . . . . . . . 9 |- ( ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) /\ x e. ( 0 ... k ) ) -> F e. B )
44		eqid 2622	. . . . . . . . . 10 |- (coe1 ` F ) = (coe1 ` F )
45	44, 11, 10, 35	coe1f2 19579	. . . . . . . . 9 |- ( F e. B -> (coe1 ` F ) : NN0 --> ( Base ` R ) )
46	43, 45	syl 17	. . . . . . . 8 |- ( ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) /\ x e. ( 0 ... k ) ) -> (coe1 ` F ) : NN0 --> ( Base ` R ) )
47		elfznn0 12433	. . . . . . . . 9 |- ( x e. ( 0 ... k ) -> x e. NN0 )
48	47	adantl 482	. . . . . . . 8 |- ( ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) /\ x e. ( 0 ... k ) ) -> x e. NN0 )
49	46, 48	ffvelrnd 6360	. . . . . . 7 |- ( ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) /\ x e. ( 0 ... k ) ) -> ( (coe1 ` F ) ` x ) e. ( Base ` R ) )
50		simpll3 1102	. . . . . . . . 9 |- ( ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) /\ x e. ( 0 ... k ) ) -> G e. B )
51		eqid 2622	. . . . . . . . . 10 |- (coe1 ` G ) = (coe1 ` G )
52	51, 11, 10, 35	coe1f2 19579	. . . . . . . . 9 |- ( G e. B -> (coe1 ` G ) : NN0 --> ( Base ` R ) )
53	50, 52	syl 17	. . . . . . . 8 |- ( ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) /\ x e. ( 0 ... k ) ) -> (coe1 ` G ) : NN0 --> ( Base ` R ) )
54		fznn0sub 12373	. . . . . . . . 9 |- ( x e. ( 0 ... k ) -> ( k - x ) e. NN0 )
55	54	adantl 482	. . . . . . . 8 |- ( ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) /\ x e. ( 0 ... k ) ) -> ( k - x ) e. NN0 )
56	53, 55	ffvelrnd 6360	. . . . . . 7 |- ( ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) /\ x e. ( 0 ... k ) ) -> ( (coe1 ` G ) ` ( k - x ) ) e. ( Base ` R ) )
57	35, 13	ringcl 18561	. . . . . . 7 |- ( ( R e. Ring /\ ( (coe1 ` F ) ` x ) e. ( Base ` R ) /\ ( (coe1 ` G ) ` ( k - x ) ) e. ( Base ` R ) ) -> ( ( (coe1 ` F ) ` x ) .x. ( (coe1 ` G ) ` ( k - x ) ) ) e. ( Base ` R ) )
58	42, 49, 56, 57	syl3anc 1326	. . . . . 6 |- ( ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) /\ x e. ( 0 ... k ) ) -> ( ( (coe1 ` F ) ` x ) .x. ( (coe1 ` G ) ` ( k - x ) ) ) e. ( Base ` R ) )
59		eqid 2622	. . . . . 6 |- ( x e. ( 0 ... k ) |-> ( ( (coe1 ` F ) ` x ) .x. ( (coe1 ` G ) ` ( k - x ) ) ) ) = ( x e. ( 0 ... k ) |-> ( ( (coe1 ` F ) ` x ) .x. ( (coe1 ` G ) ` ( k - x ) ) ) )
60	58, 59	fmptd 6385	. . . . 5 |- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) -> ( x e. ( 0 ... k ) |-> ( ( (coe1 ` F ) ` x ) .x. ( (coe1 ` G ) ` ( k - x ) ) ) ) : ( 0 ... k ) --> ( Base ` R ) )
61	40	elexi 3213	. . . . . . . . 9 |- ( 0 ... k ) e. _V
62	61	mptex 6486	. . . . . . . 8 |- ( x e. ( 0 ... k ) |-> ( ( (coe1 ` F ) ` x ) .x. ( (coe1 ` G ) ` ( k - x ) ) ) ) e. _V
63		funmpt 5926	. . . . . . . 8 |- Fun ( x e. ( 0 ... k ) |-> ( ( (coe1 ` F ) ` x ) .x. ( (coe1 ` G ) ` ( k - x ) ) ) )
64		fvex 6201	. . . . . . . 8 |- ( 0g ` R ) e. _V
65	62, 63, 64	3pm3.2i 1239	. . . . . . 7 |- ( ( x e. ( 0 ... k ) |-> ( ( (coe1 ` F ) ` x ) .x. ( (coe1 ` G ) ` ( k - x ) ) ) ) e. _V /\ Fun ( x e. ( 0 ... k ) |-> ( ( (coe1 ` F ) ` x ) .x. ( (coe1 ` G ) ` ( k - x ) ) ) ) /\ ( 0g ` R ) e. _V )
66		suppssdm 7308	. . . . . . . . 9 |- ( ( x e. ( 0 ... k ) |-> ( ( (coe1 ` F ) ` x ) .x. ( (coe1 ` G ) ` ( k - x ) ) ) ) supp ( 0g ` R ) ) C_ dom ( x e. ( 0 ... k ) |-> ( ( (coe1 ` F ) ` x ) .x. ( (coe1 ` G ) ` ( k - x ) ) ) )
67	59	dmmptss 5631	. . . . . . . . 9 |- dom ( x e. ( 0 ... k ) |-> ( ( (coe1 ` F ) ` x ) .x. ( (coe1 ` G ) ` ( k - x ) ) ) ) C_ ( 0 ... k )
68	66, 67	sstri 3612	. . . . . . . 8 |- ( ( x e. ( 0 ... k ) |-> ( ( (coe1 ` F ) ` x ) .x. ( (coe1 ` G ) ` ( k - x ) ) ) ) supp ( 0g ` R ) ) C_ ( 0 ... k )
69	40, 68	pm3.2i 471	. . . . . . 7 |- ( ( 0 ... k ) e. Fin /\ ( ( x e. ( 0 ... k ) |-> ( ( (coe1 ` F ) ` x ) .x. ( (coe1 ` G ) ` ( k - x ) ) ) ) supp ( 0g ` R ) ) C_ ( 0 ... k ) )
70		suppssfifsupp 8290	. . . . . . 7 |- ( ( ( ( x e. ( 0 ... k ) |-> ( ( (coe1 ` F ) ` x ) .x. ( (coe1 ` G ) ` ( k - x ) ) ) ) e. _V /\ Fun ( x e. ( 0 ... k ) |-> ( ( (coe1 ` F ) ` x ) .x. ( (coe1 ` G ) ` ( k - x ) ) ) ) /\ ( 0g ` R ) e. _V ) /\ ( ( 0 ... k ) e. Fin /\ ( ( x e. ( 0 ... k ) |-> ( ( (coe1 ` F ) ` x ) .x. ( (coe1 ` G ) ` ( k - x ) ) ) ) supp ( 0g ` R ) ) C_ ( 0 ... k ) ) ) -> ( x e. ( 0 ... k ) |-> ( ( (coe1 ` F ) ` x ) .x. ( (coe1 ` G ) ` ( k - x ) ) ) ) finSupp ( 0g ` R ) )
71	65, 69, 70	mp2an 708	. . . . . 6 |- ( x e. ( 0 ... k ) |-> ( ( (coe1 ` F ) ` x ) .x. ( (coe1 ` G ) ` ( k - x ) ) ) ) finSupp ( 0g ` R )
72	71	a1i 11	. . . . 5 |- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) -> ( x e. ( 0 ... k ) |-> ( ( (coe1 ` F ) ` x ) .x. ( (coe1 ` G ) ` ( k - x ) ) ) ) finSupp ( 0g ` R ) )
73		eqid 2622	. . . . . . 7 |- { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } = { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) }
74	73	coe1mul2lem2 19638	. . . . . 6 |- ( k e. NN0 -> ( c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } |-> ( c ` (/) ) ) : { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } -1-1-onto-> ( 0 ... k ) )
75	74	adantl 482	. . . . 5 |- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) -> ( c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } |-> ( c ` (/) ) ) : { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } -1-1-onto-> ( 0 ... k ) )
76	35, 36, 39, 41, 60, 72, 75	gsumf1o 18317	. . . 4 |- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) -> ( R gsumg ( x e. ( 0 ... k ) |-> ( ( (coe1 ` F ) ` x ) .x. ( (coe1 ` G ) ` ( k - x ) ) ) ) ) = ( R gsumg ( ( x e. ( 0 ... k ) |-> ( ( (coe1 ` F ) ` x ) .x. ( (coe1 ` G ) ` ( k - x ) ) ) ) o. ( c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } |-> ( c ` (/) ) ) ) ) )
77		breq1 4656	. . . . . . . . . . 11 |- ( d = c -> ( d oR <_ ( 1o X. { k } ) <-> c oR <_ ( 1o X. { k } ) ) )
78	77	elrab 3363	. . . . . . . . . 10 |- ( c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } <-> ( c e. ( NN0 ^m 1o ) /\ c oR <_ ( 1o X. { k } ) ) )
79	78	simprbi 480	. . . . . . . . 9 |- ( c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } -> c oR <_ ( 1o X. { k } ) )
80	79	adantl 482	. . . . . . . 8 |- ( ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) /\ c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } ) -> c oR <_ ( 1o X. { k } ) )
81		simplr 792	. . . . . . . . 9 |- ( ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) /\ c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } ) -> k e. NN0 )
82		elrabi 3359	. . . . . . . . . 10 |- ( c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } -> c e. ( NN0 ^m 1o ) )
83	82	adantl 482	. . . . . . . . 9 |- ( ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) /\ c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } ) -> c e. ( NN0 ^m 1o ) )
84		coe1mul2lem1 19637	. . . . . . . . 9 |- ( ( k e. NN0 /\ c e. ( NN0 ^m 1o ) ) -> ( c oR <_ ( 1o X. { k } ) <-> ( c ` (/) ) e. ( 0 ... k ) ) )
85	81, 83, 84	syl2anc 693	. . . . . . . 8 |- ( ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) /\ c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } ) -> ( c oR <_ ( 1o X. { k } ) <-> ( c ` (/) ) e. ( 0 ... k ) ) )
86	80, 85	mpbid 222	. . . . . . 7 |- ( ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) /\ c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } ) -> ( c ` (/) ) e. ( 0 ... k ) )
87		eqidd 2623	. . . . . . 7 |- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) -> ( c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } |-> ( c ` (/) ) ) = ( c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } |-> ( c ` (/) ) ) )
88		eqidd 2623	. . . . . . 7 |- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) -> ( x e. ( 0 ... k ) |-> ( ( (coe1 ` F ) ` x ) .x. ( (coe1 ` G ) ` ( k - x ) ) ) ) = ( x e. ( 0 ... k ) |-> ( ( (coe1 ` F ) ` x ) .x. ( (coe1 ` G ) ` ( k - x ) ) ) ) )
89		fveq2 6191	. . . . . . . 8 |- ( x = ( c ` (/) ) -> ( (coe1 ` F ) ` x ) = ( (coe1 ` F ) ` ( c ` (/) ) ) )
90		oveq2 6658	. . . . . . . . 9 |- ( x = ( c ` (/) ) -> ( k - x ) = ( k - ( c ` (/) ) ) )
91	90	fveq2d 6195	. . . . . . . 8 |- ( x = ( c ` (/) ) -> ( (coe1 ` G ) ` ( k - x ) ) = ( (coe1 ` G ) ` ( k - ( c ` (/) ) ) ) )
92	89, 91	oveq12d 6668	. . . . . . 7 |- ( x = ( c ` (/) ) -> ( ( (coe1 ` F ) ` x ) .x. ( (coe1 ` G ) ` ( k - x ) ) ) = ( ( (coe1 ` F ) ` ( c ` (/) ) ) .x. ( (coe1 ` G ) ` ( k - ( c ` (/) ) ) ) ) )
93	86, 87, 88, 92	fmptco 6396	. . . . . 6 |- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) -> ( ( x e. ( 0 ... k ) |-> ( ( (coe1 ` F ) ` x ) .x. ( (coe1 ` G ) ` ( k - x ) ) ) ) o. ( c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } |-> ( c ` (/) ) ) ) = ( c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } |-> ( ( (coe1 ` F ) ` ( c ` (/) ) ) .x. ( (coe1 ` G ) ` ( k - ( c ` (/) ) ) ) ) ) )
94		simpll2 1101	. . . . . . . . 9 |- ( ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) /\ c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } ) -> F e. B )
95	44	fvcoe1 19577	. . . . . . . . 9 |- ( ( F e. B /\ c e. ( NN0 ^m 1o ) ) -> ( F ` c ) = ( (coe1 ` F ) ` ( c ` (/) ) ) )
96	94, 83, 95	syl2anc 693	. . . . . . . 8 |- ( ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) /\ c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } ) -> ( F ` c ) = ( (coe1 ` F ) ` ( c ` (/) ) ) )
97		df1o2 7572	. . . . . . . . . . . . . 14 |- 1o = { (/) }
98		0ex 4790	. . . . . . . . . . . . . 14 |- (/) e. _V
99	97, 2, 98	mapsnconst 7903	. . . . . . . . . . . . 13 |- ( c e. ( NN0 ^m 1o ) -> c = ( 1o X. { ( c ` (/) ) } ) )
100	83, 99	syl 17	. . . . . . . . . . . 12 |- ( ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) /\ c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } ) -> c = ( 1o X. { ( c ` (/) ) } ) )
101	100	oveq2d 6666	. . . . . . . . . . 11 |- ( ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) /\ c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } ) -> ( ( 1o X. { k } ) oF - c ) = ( ( 1o X. { k } ) oF - ( 1o X. { ( c ` (/) ) } ) ) )
102	3	a1i 11	. . . . . . . . . . . 12 |- ( ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) /\ c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } ) -> 1o e. On )
103		vex 3203	. . . . . . . . . . . . 13 |- k e. _V
104	103	a1i 11	. . . . . . . . . . . 12 |- ( ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) /\ c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } ) -> k e. _V )
105		fvexd 6203	. . . . . . . . . . . 12 |- ( ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) /\ c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } ) -> ( c ` (/) ) e. _V )
106	102, 104, 105	ofc12 6922	. . . . . . . . . . 11 |- ( ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) /\ c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } ) -> ( ( 1o X. { k } ) oF - ( 1o X. { ( c ` (/) ) } ) ) = ( 1o X. { ( k - ( c ` (/) ) ) } ) )
107	101, 106	eqtrd 2656	. . . . . . . . . 10 |- ( ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) /\ c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } ) -> ( ( 1o X. { k } ) oF - c ) = ( 1o X. { ( k - ( c ` (/) ) ) } ) )
108	107	fveq2d 6195	. . . . . . . . 9 |- ( ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) /\ c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } ) -> ( G ` ( ( 1o X. { k } ) oF - c ) ) = ( G ` ( 1o X. { ( k - ( c ` (/) ) ) } ) ) )
109		simpll3 1102	. . . . . . . . . 10 |- ( ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) /\ c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } ) -> G e. B )
110		fznn0sub 12373	. . . . . . . . . . 11 |- ( ( c ` (/) ) e. ( 0 ... k ) -> ( k - ( c ` (/) ) ) e. NN0 )
111	86, 110	syl 17	. . . . . . . . . 10 |- ( ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) /\ c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } ) -> ( k - ( c ` (/) ) ) e. NN0 )
112	51	coe1fv 19576	. . . . . . . . . 10 |- ( ( G e. B /\ ( k - ( c ` (/) ) ) e. NN0 ) -> ( (coe1 ` G ) ` ( k - ( c ` (/) ) ) ) = ( G ` ( 1o X. { ( k - ( c ` (/) ) ) } ) ) )
113	109, 111, 112	syl2anc 693	. . . . . . . . 9 |- ( ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) /\ c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } ) -> ( (coe1 ` G ) ` ( k - ( c ` (/) ) ) ) = ( G ` ( 1o X. { ( k - ( c ` (/) ) ) } ) ) )
114	108, 113	eqtr4d 2659	. . . . . . . 8 |- ( ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) /\ c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } ) -> ( G ` ( ( 1o X. { k } ) oF - c ) ) = ( (coe1 ` G ) ` ( k - ( c ` (/) ) ) ) )
115	96, 114	oveq12d 6668	. . . . . . 7 |- ( ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) /\ c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } ) -> ( ( F ` c ) .x. ( G ` ( ( 1o X. { k } ) oF - c ) ) ) = ( ( (coe1 ` F ) ` ( c ` (/) ) ) .x. ( (coe1 ` G ) ` ( k - ( c ` (/) ) ) ) ) )
116	115	mpteq2dva 4744	. . . . . 6 |- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) -> ( c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } |-> ( ( F ` c ) .x. ( G ` ( ( 1o X. { k } ) oF - c ) ) ) ) = ( c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } |-> ( ( (coe1 ` F ) ` ( c ` (/) ) ) .x. ( (coe1 ` G ) ` ( k - ( c ` (/) ) ) ) ) ) )
117	93, 116	eqtr4d 2659	. . . . 5 |- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) -> ( ( x e. ( 0 ... k ) |-> ( ( (coe1 ` F ) ` x ) .x. ( (coe1 ` G ) ` ( k - x ) ) ) ) o. ( c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } |-> ( c ` (/) ) ) ) = ( c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } |-> ( ( F ` c ) .x. ( G ` ( ( 1o X. { k } ) oF - c ) ) ) ) )
118	117	oveq2d 6666	. . . 4 |- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) -> ( R gsumg ( ( x e. ( 0 ... k ) |-> ( ( (coe1 ` F ) ` x ) .x. ( (coe1 ` G ) ` ( k - x ) ) ) ) o. ( c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } |-> ( c ` (/) ) ) ) ) = ( R gsumg ( c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } |-> ( ( F ` c ) .x. ( G ` ( ( 1o X. { k } ) oF - c ) ) ) ) ) )
119	76, 118	eqtrd 2656	. . 3 |- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ k e. NN0 ) -> ( R gsumg ( x e. ( 0 ... k ) |-> ( ( (coe1 ` F ) ` x ) .x. ( (coe1 ` G ) ` ( k - x ) ) ) ) ) = ( R gsumg ( c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } |-> ( ( F ` c ) .x. ( G ` ( ( 1o X. { k } ) oF - c ) ) ) ) ) )
120	119	mpteq2dva 4744	. 2 |- ( ( R e. Ring /\ F e. B /\ G e. B ) -> ( k e. NN0 |-> ( R gsumg ( x e. ( 0 ... k ) |-> ( ( (coe1 ` F ) ` x ) .x. ( (coe1 ` G ) ` ( k - x ) ) ) ) ) ) = ( k e. NN0 |-> ( R gsumg ( c e. { d e. ( NN0 ^m 1o ) | d oR <_ ( 1o X. { k } ) } |-> ( ( F ` c ) .x. ( G ` ( ( 1o X. { k } ) oF - c ) ) ) ) ) ) )
121	27, 34, 120	3eqtr4d 2666	1 |- ( ( R e. Ring /\ F e. B /\ G e. B ) -> (coe1 ` ( F .xb G ) ) = ( k e. NN0 |-> ( R gsumg ( x e. ( 0 ... k ) |-> ( ( (coe1 ` F ) ` x ) .x. ( (coe1 ` G ) ` ( k - x ) ) ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   {csn 4177   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   o. ccom 5118   Oncon0 5723   Fun wfun 5882   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   oFcof 6895   oRcofr 6896   supp csupp 7295   1oc1o 7553   ^m cmap 7857   Fincfn 7955   finSupp cfsupp 8275   0cc0 9936   <_ cle 10075   - cmin 10266   NN0cn0 11292   ...cfz 12326   Basecbs 15857   .rcmulr 15942   0gc0g 16100   gsum_g cgsu 16101  CMndccmn 18193   Ringcrg 18547   mPwSer cmps 19351  PwSer₁cps1 19545  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-mulg 17541  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-psr 19356  df-opsr 19360  df-psr1 19550  df-coe1 19553

This theorem is referenced by:  coe1mul  19640