Theorem cply1mul 31006

Description: The product of two constant polynomials is a constant polynomial. (Contributed by AV, 18-Nov-2019.)

Hypotheses

Ref	Expression
cply1mul.p	|- P = (Poly1 ` R )
cply1mul.b	|- B = ( Base ` P )
cply1mul.0	|- .0. = ( 0g ` R )
cply1mul.m	|- .X. = ( .r ` P )

Assertion

Ref	Expression
cply1mul	|- ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) -> ( A. c e. NN ( ( (coe1 ` F ) ` c ) = .0. /\ ( (coe1 ` G ) ` c ) = .0. ) -> A. c e. NN ( (coe1 ` ( F .X. G ) ) ` c ) = .0. ) )

Distinct variable groups:   F, c   G, c   .X. , c   .0. , c

Allowed substitution hints:   B( c)   P( c)   R( c)

Proof of Theorem cply1mul

Dummy variables k n s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cply1mul.p	. . . . . . . . . 10 |- P = (Poly1 ` R )
2		cply1mul.m	. . . . . . . . . 10 |- .X. = ( .r ` P )
3		eqid 2454	. . . . . . . . . 10 |- ( .r ` R ) = ( .r ` R )
4		cply1mul.b	. . . . . . . . . 10 |- B = ( Base ` P )
5	1, 2, 3, 4	coe1mul 17848	. . . . . . . . 9 |- ( ( R e. Ring /\ F e. B /\ G e. B ) -> (coe1 ` ( F .X. G ) ) = ( s e. NN0 |-> ( R gsumg ( k e. ( 0 ... s ) |-> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( s - k ) ) ) ) ) ) )
6	5	3expb 1189	. . . . . . . 8 |- ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) -> (coe1 ` ( F .X. G ) ) = ( s e. NN0 |-> ( R gsumg ( k e. ( 0 ... s ) |-> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( s - k ) ) ) ) ) ) )
7	6	adantr 465	. . . . . . 7 |- ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ A. c e. NN ( ( (coe1 ` F ) ` c ) = .0. /\ ( (coe1 ` G ) ` c ) = .0. ) ) -> (coe1 ` ( F .X. G ) ) = ( s e. NN0 |-> ( R gsumg ( k e. ( 0 ... s ) |-> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( s - k ) ) ) ) ) ) )
8	7	adantr 465	. . . . . 6 |- ( ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ A. c e. NN ( ( (coe1 ` F ) ` c ) = .0. /\ ( (coe1 ` G ) ` c ) = .0. ) ) /\ n e. NN ) -> (coe1 ` ( F .X. G ) ) = ( s e. NN0 |-> ( R gsumg ( k e. ( 0 ... s ) |-> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( s - k ) ) ) ) ) ) )
9		oveq2 6209	. . . . . . . . 9 |- ( s = n -> ( 0 ... s ) = ( 0 ... n ) )
10		oveq1 6208	. . . . . . . . . . 11 |- ( s = n -> ( s - k ) = ( n - k ) )
11	10	fveq2d 5804	. . . . . . . . . 10 |- ( s = n -> ( (coe1 ` G ) ` ( s - k ) ) = ( (coe1 ` G ) ` ( n - k ) ) )
12	11	oveq2d 6217	. . . . . . . . 9 |- ( s = n -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( s - k ) ) ) = ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) )
13	9, 12	mpteq12dv 4479	. . . . . . . 8 |- ( s = n -> ( k e. ( 0 ... s ) |-> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( s - k ) ) ) ) = ( k e. ( 0 ... n ) |-> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) ) )
14	13	oveq2d 6217	. . . . . . 7 |- ( s = n -> ( R gsumg ( k e. ( 0 ... s ) |-> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( s - k ) ) ) ) ) = ( R gsumg ( k e. ( 0 ... n ) |-> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) ) ) )
15	14	adantl 466	. . . . . 6 |- ( ( ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ A. c e. NN ( ( (coe1 ` F ) ` c ) = .0. /\ ( (coe1 ` G ) ` c ) = .0. ) ) /\ n e. NN ) /\ s = n ) -> ( R gsumg ( k e. ( 0 ... s ) |-> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( s - k ) ) ) ) ) = ( R gsumg ( k e. ( 0 ... n ) |-> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) ) ) )
16		nnnn0 10698	. . . . . . 7 |- ( n e. NN -> n e. NN0 )
17	16	adantl 466	. . . . . 6 |- ( ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ A. c e. NN ( ( (coe1 ` F ) ` c ) = .0. /\ ( (coe1 ` G ) ` c ) = .0. ) ) /\ n e. NN ) -> n e. NN0 )
18		ovex 6226	. . . . . . 7 |- ( R gsumg ( k e. ( 0 ... n ) |-> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) ) ) e. _V
19	18	a1i 11	. . . . . 6 |- ( ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ A. c e. NN ( ( (coe1 ` F ) ` c ) = .0. /\ ( (coe1 ` G ) ` c ) = .0. ) ) /\ n e. NN ) -> ( R gsumg ( k e. ( 0 ... n ) |-> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) ) ) e. _V )
20	8, 15, 17, 19	fvmptd 5889	. . . . 5 |- ( ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ A. c e. NN ( ( (coe1 ` F ) ` c ) = .0. /\ ( (coe1 ` G ) ` c ) = .0. ) ) /\ n e. NN ) -> ( (coe1 ` ( F .X. G ) ) ` n ) = ( R gsumg ( k e. ( 0 ... n ) |-> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) ) ) )
21		r19.26 2955	. . . . . . . . . . . . 13 |- ( A. c e. NN ( ( (coe1 ` F ) ` c ) = .0. /\ ( (coe1 ` G ) ` c ) = .0. ) <-> ( A. c e. NN ( (coe1 ` F ) ` c ) = .0. /\ A. c e. NN ( (coe1 ` G ) ` c ) = .0. ) )
22		oveq2 6209	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( k = 0 -> ( n - k ) = ( n - 0 ) )
23		nncn 10442	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( n e. NN -> n e. CC )
24	23	subid1d 9820	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( n e. NN -> ( n - 0 ) = n )
25	24	adantr 465	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( n e. NN /\ k e. ( 0 ... n ) ) -> ( n - 0 ) = n )
26	22, 25	sylan9eqr 2517	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( n e. NN /\ k e. ( 0 ... n ) ) /\ k = 0 ) -> ( n - k ) = n )
27		simpl 457	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( n e. NN /\ k e. ( 0 ... n ) ) -> n e. NN )
28	27	adantr 465	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( n e. NN /\ k e. ( 0 ... n ) ) /\ k = 0 ) -> n e. NN )
29	26, 28	eqeltrd 2542	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( n e. NN /\ k e. ( 0 ... n ) ) /\ k = 0 ) -> ( n - k ) e. NN )
30		fveq2 5800	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( c = ( n - k ) -> ( (coe1 ` G ) ` c ) = ( (coe1 ` G ) ` ( n - k ) ) )
31	30	eqeq1d 2456	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( c = ( n - k ) -> ( ( (coe1 ` G ) ` c ) = .0. <-> ( (coe1 ` G ) ` ( n - k ) ) = .0. ) )
32	31	rspcva 3177	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( n - k ) e. NN /\ A. c e. NN ( (coe1 ` G ) ` c ) = .0. ) -> ( (coe1 ` G ) ` ( n - k ) ) = .0. )
33	32	ex 434	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( n - k ) e. NN -> ( A. c e. NN ( (coe1 ` G ) ` c ) = .0. -> ( (coe1 ` G ) ` ( n - k ) ) = .0. ) )
34	29, 33	syl 16	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( n e. NN /\ k e. ( 0 ... n ) ) /\ k = 0 ) -> ( A. c e. NN ( (coe1 ` G ) ` c ) = .0. -> ( (coe1 ` G ) ` ( n - k ) ) = .0. ) )
35		oveq2 6209	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( (coe1 ` G ) ` ( n - k ) ) = .0. -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = ( ( (coe1 ` F ) ` k ) ( .r ` R ) .0. ) )
36		simpl 457	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) -> R e. Ring )
37	36	adantr 465	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ ( ( n e. NN /\ k e. ( 0 ... n ) ) /\ k = 0 ) ) -> R e. Ring )
38		simpl 457	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( F e. B /\ G e. B ) -> F e. B )
39	38	adantl 466	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) -> F e. B )
40		elfznn0 11599	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( k e. ( 0 ... n ) -> k e. NN0 )
41	40	adantl 466	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( n e. NN /\ k e. ( 0 ... n ) ) -> k e. NN0 )
42	41	adantr 465	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( ( n e. NN /\ k e. ( 0 ... n ) ) /\ k = 0 ) -> k e. NN0 )
43		eqid 2454	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (coe1 ` F ) = (coe1 ` F )
44		eqid 2454	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( Base ` R ) = ( Base ` R )
45	43, 4, 1, 44	coe1fvalcl 30984	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( F e. B /\ k e. NN0 ) -> ( (coe1 ` F ) ` k ) e. ( Base ` R ) )
46	39, 42, 45	syl2an 477	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ ( ( n e. NN /\ k e. ( 0 ... n ) ) /\ k = 0 ) ) -> ( (coe1 ` F ) ` k ) e. ( Base ` R ) )
47		cply1mul.0	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- .0. = ( 0g ` R )
48	44, 3, 47	rngrz 16806	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( R e. Ring /\ ( (coe1 ` F ) ` k ) e. ( Base ` R ) ) -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) .0. ) = .0. )
49	37, 46, 48	syl2anc 661	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ ( ( n e. NN /\ k e. ( 0 ... n ) ) /\ k = 0 ) ) -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) .0. ) = .0. )
50	35, 49	sylan9eqr 2517	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ ( ( n e. NN /\ k e. ( 0 ... n ) ) /\ k = 0 ) ) /\ ( (coe1 ` G ) ` ( n - k ) ) = .0. ) -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. )
51	50	ex 434	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ ( ( n e. NN /\ k e. ( 0 ... n ) ) /\ k = 0 ) ) -> ( ( (coe1 ` G ) ` ( n - k ) ) = .0. -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. ) )
52	51	expcom 435	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( n e. NN /\ k e. ( 0 ... n ) ) /\ k = 0 ) -> ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) -> ( ( (coe1 ` G ) ` ( n - k ) ) = .0. -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. ) ) )
53	52	com23 78	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( n e. NN /\ k e. ( 0 ... n ) ) /\ k = 0 ) -> ( ( (coe1 ` G ) ` ( n - k ) ) = .0. -> ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. ) ) )
54	34, 53	syld 44	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( n e. NN /\ k e. ( 0 ... n ) ) /\ k = 0 ) -> ( A. c e. NN ( (coe1 ` G ) ` c ) = .0. -> ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. ) ) )
55	54	com12 31	. . . . . . . . . . . . . . . . . 18 |- ( A. c e. NN ( (coe1 ` G ) ` c ) = .0. -> ( ( ( n e. NN /\ k e. ( 0 ... n ) ) /\ k = 0 ) -> ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. ) ) )
56	55	expd 436	. . . . . . . . . . . . . . . . 17 |- ( A. c e. NN ( (coe1 ` G ) ` c ) = .0. -> ( ( n e. NN /\ k e. ( 0 ... n ) ) -> ( k = 0 -> ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. ) ) ) )
57	56	com24 87	. . . . . . . . . . . . . . . 16 |- ( A. c e. NN ( (coe1 ` G ) ` c ) = .0. -> ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) -> ( k = 0 -> ( ( n e. NN /\ k e. ( 0 ... n ) ) -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. ) ) ) )
58	57	adantl 466	. . . . . . . . . . . . . . 15 |- ( ( A. c e. NN ( (coe1 ` F ) ` c ) = .0. /\ A. c e. NN ( (coe1 ` G ) ` c ) = .0. ) -> ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) -> ( k = 0 -> ( ( n e. NN /\ k e. ( 0 ... n ) ) -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. ) ) ) )
59	58	com13 80	. . . . . . . . . . . . . 14 |- ( k = 0 -> ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) -> ( ( A. c e. NN ( (coe1 ` F ) ` c ) = .0. /\ A. c e. NN ( (coe1 ` G ) ` c ) = .0. ) -> ( ( n e. NN /\ k e. ( 0 ... n ) ) -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. ) ) ) )
60		df-ne 2650	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( k =/= 0 <-> -. k = 0 )
61	60	biimpri 206	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( -. k = 0 -> k =/= 0 )
62	61, 40	anim12ci 567	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( -. k = 0 /\ k e. ( 0 ... n ) ) -> ( k e. NN0 /\ k =/= 0 ) )
63		elnnne0 10705	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( k e. NN <-> ( k e. NN0 /\ k =/= 0 ) )
64	62, 63	sylibr 212	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( -. k = 0 /\ k e. ( 0 ... n ) ) -> k e. NN )
65		fveq2 5800	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( c = k -> ( (coe1 ` F ) ` c ) = ( (coe1 ` F ) ` k ) )
66	65	eqeq1d 2456	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( c = k -> ( ( (coe1 ` F ) ` c ) = .0. <-> ( (coe1 ` F ) ` k ) = .0. ) )
67	66	rspcva 3177	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( k e. NN /\ A. c e. NN ( (coe1 ` F ) ` c ) = .0. ) -> ( (coe1 ` F ) ` k ) = .0. )
68	67	ex 434	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( k e. NN -> ( A. c e. NN ( (coe1 ` F ) ` c ) = .0. -> ( (coe1 ` F ) ` k ) = .0. ) )
69	64, 68	syl 16	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( -. k = 0 /\ k e. ( 0 ... n ) ) -> ( A. c e. NN ( (coe1 ` F ) ` c ) = .0. -> ( (coe1 ` F ) ` k ) = .0. ) )
70		oveq1 6208	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( ( (coe1 ` F ) ` k ) = .0. -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = ( .0. ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) )
71	36	adantr 465	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ k e. ( 0 ... n ) ) -> R e. Ring )
72	4	eleq2i 2532	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- ( G e. B <-> G e. ( Base ` P ) )
73	72	biimpi 194	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ( G e. B -> G e. ( Base ` P ) )
74	73	adantl 466	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( ( F e. B /\ G e. B ) -> G e. ( Base ` P ) )
75	74	adantl 466	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) -> G e. ( Base ` P ) )
76		fznn0sub 11605	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ( k e. ( 0 ... n ) -> ( n - k ) e. NN0 )
77		eqid 2454	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- (coe1 ` G ) = (coe1 ` G )
78		eqid 2454	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( Base ` P ) = ( Base ` P )
79	77, 78, 1, 44	coe1fvalcl 30984	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ( ( G e. ( Base ` P ) /\ ( n - k ) e. NN0 ) -> ( (coe1 ` G ) ` ( n - k ) ) e. ( Base ` R ) )
80	75, 76, 79	syl2an 477	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ k e. ( 0 ... n ) ) -> ( (coe1 ` G ) ` ( n - k ) ) e. ( Base ` R ) )
81	44, 3, 47	rnglz 16805	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( ( R e. Ring /\ ( (coe1 ` G ) ` ( n - k ) ) e. ( Base ` R ) ) -> ( .0. ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. )
82	71, 80, 81	syl2anc 661	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ k e. ( 0 ... n ) ) -> ( .0. ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. )
83	70, 82	sylan9eqr 2517	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ k e. ( 0 ... n ) ) /\ ( (coe1 ` F ) ` k ) = .0. ) -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. )
84	83	ex 434	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ k e. ( 0 ... n ) ) -> ( ( (coe1 ` F ) ` k ) = .0. -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. ) )
85	84	ex 434	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) -> ( k e. ( 0 ... n ) -> ( ( (coe1 ` F ) ` k ) = .0. -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. ) ) )
86	85	com23 78	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) -> ( ( (coe1 ` F ) ` k ) = .0. -> ( k e. ( 0 ... n ) -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. ) ) )
87	86	a1dd 46	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) -> ( ( (coe1 ` F ) ` k ) = .0. -> ( n e. NN -> ( k e. ( 0 ... n ) -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. ) ) ) )
88	87	com14 88	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( k e. ( 0 ... n ) -> ( ( (coe1 ` F ) ` k ) = .0. -> ( n e. NN -> ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. ) ) ) )
89	88	adantl 466	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( -. k = 0 /\ k e. ( 0 ... n ) ) -> ( ( (coe1 ` F ) ` k ) = .0. -> ( n e. NN -> ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. ) ) ) )
90	69, 89	syld 44	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( -. k = 0 /\ k e. ( 0 ... n ) ) -> ( A. c e. NN ( (coe1 ` F ) ` c ) = .0. -> ( n e. NN -> ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. ) ) ) )
91	90	com24 87	. . . . . . . . . . . . . . . . . . . 20 |- ( ( -. k = 0 /\ k e. ( 0 ... n ) ) -> ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) -> ( n e. NN -> ( A. c e. NN ( (coe1 ` F ) ` c ) = .0. -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. ) ) ) )
92	91	ex 434	. . . . . . . . . . . . . . . . . . 19 |- ( -. k = 0 -> ( k e. ( 0 ... n ) -> ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) -> ( n e. NN -> ( A. c e. NN ( (coe1 ` F ) ` c ) = .0. -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. ) ) ) ) )
93	92	com14 88	. . . . . . . . . . . . . . . . . 18 |- ( n e. NN -> ( k e. ( 0 ... n ) -> ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) -> ( -. k = 0 -> ( A. c e. NN ( (coe1 ` F ) ` c ) = .0. -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. ) ) ) ) )
94	93	imp 429	. . . . . . . . . . . . . . . . 17 |- ( ( n e. NN /\ k e. ( 0 ... n ) ) -> ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) -> ( -. k = 0 -> ( A. c e. NN ( (coe1 ` F ) ` c ) = .0. -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. ) ) ) )
95	94	com14 88	. . . . . . . . . . . . . . . 16 |- ( A. c e. NN ( (coe1 ` F ) ` c ) = .0. -> ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) -> ( -. k = 0 -> ( ( n e. NN /\ k e. ( 0 ... n ) ) -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. ) ) ) )
96	95	adantr 465	. . . . . . . . . . . . . . 15 |- ( ( A. c e. NN ( (coe1 ` F ) ` c ) = .0. /\ A. c e. NN ( (coe1 ` G ) ` c ) = .0. ) -> ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) -> ( -. k = 0 -> ( ( n e. NN /\ k e. ( 0 ... n ) ) -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. ) ) ) )
97	96	com13 80	. . . . . . . . . . . . . 14 |- ( -. k = 0 -> ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) -> ( ( A. c e. NN ( (coe1 ` F ) ` c ) = .0. /\ A. c e. NN ( (coe1 ` G ) ` c ) = .0. ) -> ( ( n e. NN /\ k e. ( 0 ... n ) ) -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. ) ) ) )
98	59, 97	pm2.61i 164	. . . . . . . . . . . . 13 |- ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) -> ( ( A. c e. NN ( (coe1 ` F ) ` c ) = .0. /\ A. c e. NN ( (coe1 ` G ) ` c ) = .0. ) -> ( ( n e. NN /\ k e. ( 0 ... n ) ) -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. ) ) )
99	21, 98	syl5bi 217	. . . . . . . . . . . 12 |- ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) -> ( A. c e. NN ( ( (coe1 ` F ) ` c ) = .0. /\ ( (coe1 ` G ) ` c ) = .0. ) -> ( ( n e. NN /\ k e. ( 0 ... n ) ) -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. ) ) )
100	99	imp 429	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ A. c e. NN ( ( (coe1 ` F ) ` c ) = .0. /\ ( (coe1 ` G ) ` c ) = .0. ) ) -> ( ( n e. NN /\ k e. ( 0 ... n ) ) -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. ) )
101	100	expd 436	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ A. c e. NN ( ( (coe1 ` F ) ` c ) = .0. /\ ( (coe1 ` G ) ` c ) = .0. ) ) -> ( n e. NN -> ( k e. ( 0 ... n ) -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. ) ) )
102	101	imp 429	. . . . . . . . 9 |- ( ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ A. c e. NN ( ( (coe1 ` F ) ` c ) = .0. /\ ( (coe1 ` G ) ` c ) = .0. ) ) /\ n e. NN ) -> ( k e. ( 0 ... n ) -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. ) )
103	102	imp 429	. . . . . . . 8 |- ( ( ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ A. c e. NN ( ( (coe1 ` F ) ` c ) = .0. /\ ( (coe1 ` G ) ` c ) = .0. ) ) /\ n e. NN ) /\ k e. ( 0 ... n ) ) -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. )
104	103	mpteq2dva 4487	. . . . . . 7 |- ( ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ A. c e. NN ( ( (coe1 ` F ) ` c ) = .0. /\ ( (coe1 ` G ) ` c ) = .0. ) ) /\ n e. NN ) -> ( k e. ( 0 ... n ) |-> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) ) = ( k e. ( 0 ... n ) |-> .0. ) )
105	104	oveq2d 6217	. . . . . 6 |- ( ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ A. c e. NN ( ( (coe1 ` F ) ` c ) = .0. /\ ( (coe1 ` G ) ` c ) = .0. ) ) /\ n e. NN ) -> ( R gsumg ( k e. ( 0 ... n ) |-> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) ) ) = ( R gsumg ( k e. ( 0 ... n ) |-> .0. ) ) )
106		rngmnd 16778	. . . . . . . . . 10 |- ( R e. Ring -> R e. Mnd )
107		ovex 6226	. . . . . . . . . . 11 |- ( 0 ... n ) e. _V
108	107	a1i 11	. . . . . . . . . 10 |- ( R e. Ring -> ( 0 ... n ) e. _V )
109	47	gsumz 15631	. . . . . . . . . 10 |- ( ( R e. Mnd /\ ( 0 ... n ) e. _V ) -> ( R gsumg ( k e. ( 0 ... n ) |-> .0. ) ) = .0. )
110	106, 108, 109	syl2anc 661	. . . . . . . . 9 |- ( R e. Ring -> ( R gsumg ( k e. ( 0 ... n ) |-> .0. ) ) = .0. )
111	110	adantr 465	. . . . . . . 8 |- ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) -> ( R gsumg ( k e. ( 0 ... n ) |-> .0. ) ) = .0. )
112	111	adantr 465	. . . . . . 7 |- ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ A. c e. NN ( ( (coe1 ` F ) ` c ) = .0. /\ ( (coe1 ` G ) ` c ) = .0. ) ) -> ( R gsumg ( k e. ( 0 ... n ) |-> .0. ) ) = .0. )
113	112	adantr 465	. . . . . 6 |- ( ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ A. c e. NN ( ( (coe1 ` F ) ` c ) = .0. /\ ( (coe1 ` G ) ` c ) = .0. ) ) /\ n e. NN ) -> ( R gsumg ( k e. ( 0 ... n ) |-> .0. ) ) = .0. )
114	105, 113	eqtrd 2495	. . . . 5 |- ( ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ A. c e. NN ( ( (coe1 ` F ) ` c ) = .0. /\ ( (coe1 ` G ) ` c ) = .0. ) ) /\ n e. NN ) -> ( R gsumg ( k e. ( 0 ... n ) |-> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) ) ) = .0. )
115	20, 114	eqtrd 2495	. . . 4 |- ( ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ A. c e. NN ( ( (coe1 ` F ) ` c ) = .0. /\ ( (coe1 ` G ) ` c ) = .0. ) ) /\ n e. NN ) -> ( (coe1 ` ( F .X. G ) ) ` n ) = .0. )
116	115	ralrimiva 2830	. . 3 |- ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ A. c e. NN ( ( (coe1 ` F ) ` c ) = .0. /\ ( (coe1 ` G ) ` c ) = .0. ) ) -> A. n e. NN ( (coe1 ` ( F .X. G ) ) ` n ) = .0. )
117		fveq2 5800	. . . . 5 |- ( c = n -> ( (coe1 ` ( F .X. G ) ) ` c ) = ( (coe1 ` ( F .X. G ) ) ` n ) )
118	117	eqeq1d 2456	. . . 4 |- ( c = n -> ( ( (coe1 ` ( F .X. G ) ) ` c ) = .0. <-> ( (coe1 ` ( F .X. G ) ) ` n ) = .0. ) )
119	118	cbvralv 3053	. . 3 |- ( A. c e. NN ( (coe1 ` ( F .X. G ) ) ` c ) = .0. <-> A. n e. NN ( (coe1 ` ( F .X. G ) ) ` n ) = .0. )
120	116, 119	sylibr 212	. 2 |- ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ A. c e. NN ( ( (coe1 ` F ) ` c ) = .0. /\ ( (coe1 ` G ) ` c ) = .0. ) ) -> A. c e. NN ( (coe1 ` ( F .X. G ) ) ` c ) = .0. )
121	120	ex 434	1 |- ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) -> ( A. c e. NN ( ( (coe1 ` F ) ` c ) = .0. /\ ( (coe1 ` G ) ` c ) = .0. ) -> A. c e. NN ( (coe1 ` ( F .X. G ) ) ` c ) = .0. ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1370   e. wcel 1758   =/= wne 2648   A.wral 2799   _Vcvv 3078   |-> cmpt 4459   ` cfv 5527  (class class class)co 6201   0cc0 9394   - cmin 9707   NNcn 10434   NN0cn0 10691   ...cfz 11555   Basecbs 14293   .rcmulr 14359   0gc0g 14498   gsum_g cgsu 14499   Mndcmnd 15529   Ringcrg 16769  Poly₁cpl1 17758  coe₁cco1 17759

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-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-inf2 7959  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  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-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-iin 4283  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-se 4789  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-isom 5536  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-of 6431  df-ofr 6432  df-om 6588  df-1st 6688  df-2nd 6689  df-supp 6802  df-recs 6943  df-rdg 6977  df-1o 7031  df-2o 7032  df-oadd 7035  df-er 7212  df-map 7327  df-pm 7328  df-ixp 7375  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-fsupp 7733  df-oi 7836  df-card 8221  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-3 10493  df-4 10494  df-5 10495  df-6 10496  df-7 10497  df-8 10498  df-9 10499  df-10 10500  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-fzo 11667  df-seq 11925  df-hash 12222  df-struct 14295  df-ndx 14296  df-slot 14297  df-base 14298  df-sets 14299  df-ress 14300  df-plusg 14371  df-mulr 14372  df-sca 14374  df-vsca 14375  df-tset 14377  df-ple 14378  df-0g 14500  df-gsum 14501  df-mre 14644  df-mrc 14645  df-acs 14647  df-mnd 15535  df-mhm 15584  df-submnd 15585  df-grp 15665  df-minusg 15666  df-mulg 15668  df-ghm 15865  df-cntz 15955  df-cmn 16401  df-abl 16402  df-mgp 16715  df-ur 16727  df-rng 16771  df-psr 17547  df-mpl 17549  df-opsr 17551  df-psr1 17761  df-ply1 17763  df-coe1 17764

This theorem is referenced by:  cpmatmcllem  31208