Theorem cply1mul 18461

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 2457	. . . . . . . . . 10 |- ( .r ` R ) = ( .r ` R )
4		cply1mul.b	. . . . . . . . . 10 |- B = ( Base ` P )
5	1, 2, 3, 4	coe1mul 18437	. . . . . . . . 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 1197	. . . . . . . 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 6304	. . . . . . . . 9 |- ( s = n -> ( 0 ... s ) = ( 0 ... n ) )
10		oveq1 6303	. . . . . . . . . . 11 |- ( s = n -> ( s - k ) = ( n - k ) )
11	10	fveq2d 5876	. . . . . . . . . 10 |- ( s = n -> ( (coe1 ` G ) ` ( s - k ) ) = ( (coe1 ` G ) ` ( n - k ) ) )
12	11	oveq2d 6312	. . . . . . . . 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 4535	. . . . . . . 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 6312	. . . . . . 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 10823	. . . . . . 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 6324	. . . . . . 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 5961	. . . . 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 2984	. . . . . . . . . 10 |- ( 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 6304	. . . . . . . . . . . . . . . . . . . 20 |- ( k = 0 -> ( n - k ) = ( n - 0 ) )
23		nncn 10564	. . . . . . . . . . . . . . . . . . . . . 22 |- ( n e. NN -> n e. CC )
24	23	subid1d 9939	. . . . . . . . . . . . . . . . . . . . 21 |- ( n e. NN -> ( n - 0 ) = n )
25	24	adantr 465	. . . . . . . . . . . . . . . . . . . 20 |- ( ( n e. NN /\ k e. ( 0 ... n ) ) -> ( n - 0 ) = n )
26	22, 25	sylan9eqr 2520	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( n e. NN /\ k e. ( 0 ... n ) ) /\ k = 0 ) -> ( n - k ) = n )
27		simpll 753	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( n e. NN /\ k e. ( 0 ... n ) ) /\ k = 0 ) -> n e. NN )
28	26, 27	eqeltrd 2545	. . . . . . . . . . . . . . . . . 18 |- ( ( ( n e. NN /\ k e. ( 0 ... n ) ) /\ k = 0 ) -> ( n - k ) e. NN )
29		fveq2 5872	. . . . . . . . . . . . . . . . . . . 20 |- ( c = ( n - k ) -> ( (coe1 ` G ) ` c ) = ( (coe1 ` G ) ` ( n - k ) ) )
30	29	eqeq1d 2459	. . . . . . . . . . . . . . . . . . 19 |- ( c = ( n - k ) -> ( ( (coe1 ` G ) ` c ) = .0. <-> ( (coe1 ` G ) ` ( n - k ) ) = .0. ) )
31	30	rspcv 3206	. . . . . . . . . . . . . . . . . 18 |- ( ( n - k ) e. NN -> ( A. c e. NN ( (coe1 ` G ) ` c ) = .0. -> ( (coe1 ` G ) ` ( n - k ) ) = .0. ) )
32	28, 31	syl 16	. . . . . . . . . . . . . . . . 17 |- ( ( ( n e. NN /\ k e. ( 0 ... n ) ) /\ k = 0 ) -> ( A. c e. NN ( (coe1 ` G ) ` c ) = .0. -> ( (coe1 ` G ) ` ( n - k ) ) = .0. ) )
33		oveq2 6304	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( (coe1 ` G ) ` ( n - k ) ) = .0. -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = ( ( (coe1 ` F ) ` k ) ( .r ` R ) .0. ) )
34		simpll 753	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ ( ( n e. NN /\ k e. ( 0 ... n ) ) /\ k = 0 ) ) -> R e. Ring )
35		simpl 457	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( F e. B /\ G e. B ) -> F e. B )
36	35	adantl 466	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) -> F e. B )
37		elfznn0 11796	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( k e. ( 0 ... n ) -> k e. NN0 )
38	37	adantl 466	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( n e. NN /\ k e. ( 0 ... n ) ) -> k e. NN0 )
39	38	adantr 465	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( n e. NN /\ k e. ( 0 ... n ) ) /\ k = 0 ) -> k e. NN0 )
40		eqid 2457	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (coe1 ` F ) = (coe1 ` F )
41		eqid 2457	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( Base ` R ) = ( Base ` R )
42	40, 4, 1, 41	coe1fvalcl 18377	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( F e. B /\ k e. NN0 ) -> ( (coe1 ` F ) ` k ) e. ( Base ` R ) )
43	36, 39, 42	syl2an 477	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ ( ( n e. NN /\ k e. ( 0 ... n ) ) /\ k = 0 ) ) -> ( (coe1 ` F ) ` k ) e. ( Base ` R ) )
44		cply1mul.0	. . . . . . . . . . . . . . . . . . . . . . 23 |- .0. = ( 0g ` R )
45	41, 3, 44	ringrz 17362	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( R e. Ring /\ ( (coe1 ` F ) ` k ) e. ( Base ` R ) ) -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) .0. ) = .0. )
46	34, 43, 45	syl2anc 661	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( 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. )
47	33, 46	sylan9eqr 2520	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( 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. )
48	47	ex 434	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( 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. ) )
49	48	expcom 435	. . . . . . . . . . . . . . . . . 18 |- ( ( ( 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. ) ) )
50	49	com23 78	. . . . . . . . . . . . . . . . 17 |- ( ( ( 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. ) ) )
51	32, 50	syld 44	. . . . . . . . . . . . . . . 16 |- ( ( ( 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. ) ) )
52	51	com12 31	. . . . . . . . . . . . . . 15 |- ( 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. ) ) )
53	52	expd 436	. . . . . . . . . . . . . 14 |- ( 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. ) ) ) )
54	53	com24 87	. . . . . . . . . . . . 13 |- ( 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. ) ) ) )
55	54	adantl 466	. . . . . . . . . . . 12 |- ( ( 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. ) ) ) )
56	55	com13 80	. . . . . . . . . . 11 |- ( 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. ) ) ) )
57		df-ne 2654	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( k =/= 0 <-> -. k = 0 )
58	57	biimpri 206	. . . . . . . . . . . . . . . . . . . . . 22 |- ( -. k = 0 -> k =/= 0 )
59	58, 37	anim12ci 567	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( -. k = 0 /\ k e. ( 0 ... n ) ) -> ( k e. NN0 /\ k =/= 0 ) )
60		elnnne0 10830	. . . . . . . . . . . . . . . . . . . . 21 |- ( k e. NN <-> ( k e. NN0 /\ k =/= 0 ) )
61	59, 60	sylibr 212	. . . . . . . . . . . . . . . . . . . 20 |- ( ( -. k = 0 /\ k e. ( 0 ... n ) ) -> k e. NN )
62		fveq2 5872	. . . . . . . . . . . . . . . . . . . . . 22 |- ( c = k -> ( (coe1 ` F ) ` c ) = ( (coe1 ` F ) ` k ) )
63	62	eqeq1d 2459	. . . . . . . . . . . . . . . . . . . . 21 |- ( c = k -> ( ( (coe1 ` F ) ` c ) = .0. <-> ( (coe1 ` F ) ` k ) = .0. ) )
64	63	rspcv 3206	. . . . . . . . . . . . . . . . . . . 20 |- ( k e. NN -> ( A. c e. NN ( (coe1 ` F ) ` c ) = .0. -> ( (coe1 ` F ) ` k ) = .0. ) )
65	61, 64	syl 16	. . . . . . . . . . . . . . . . . . 19 |- ( ( -. k = 0 /\ k e. ( 0 ... n ) ) -> ( A. c e. NN ( (coe1 ` F ) ` c ) = .0. -> ( (coe1 ` F ) ` k ) = .0. ) )
66		oveq1 6303	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( (coe1 ` F ) ` k ) = .0. -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = ( .0. ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) )
67		simpll 753	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ k e. ( 0 ... n ) ) -> R e. Ring )
68	4	eleq2i 2535	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ( G e. B <-> G e. ( Base ` P ) )
69	68	biimpi 194	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( G e. B -> G e. ( Base ` P ) )
70	69	adantl 466	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( ( F e. B /\ G e. B ) -> G e. ( Base ` P ) )
71	70	adantl 466	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) -> G e. ( Base ` P ) )
72		fznn0sub 11741	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( k e. ( 0 ... n ) -> ( n - k ) e. NN0 )
73		eqid 2457	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (coe1 ` G ) = (coe1 ` G )
74		eqid 2457	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( Base ` P ) = ( Base ` P )
75	73, 74, 1, 41	coe1fvalcl 18377	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( G e. ( Base ` P ) /\ ( n - k ) e. NN0 ) -> ( (coe1 ` G ) ` ( n - k ) ) e. ( Base ` R ) )
76	71, 72, 75	syl2an 477	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ k e. ( 0 ... n ) ) -> ( (coe1 ` G ) ` ( n - k ) ) e. ( Base ` R ) )
77	41, 3, 44	ringlz 17361	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( R e. Ring /\ ( (coe1 ` G ) ` ( n - k ) ) e. ( Base ` R ) ) -> ( .0. ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. )
78	67, 76, 77	syl2anc 661	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ k e. ( 0 ... n ) ) -> ( .0. ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. )
79	66, 78	sylan9eqr 2520	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( ( 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. )
80	79	ex 434	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( 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. ) )
81	80	ex 434	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( 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. ) ) )
82	81	com23 78	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( 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. ) ) )
83	82	a1dd 46	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( 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. ) ) ) )
84	83	com14 88	. . . . . . . . . . . . . . . . . . . 20 |- ( 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. ) ) ) )
85	84	adantl 466	. . . . . . . . . . . . . . . . . . 19 |- ( ( -. 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. ) ) ) )
86	65, 85	syld 44	. . . . . . . . . . . . . . . . . 18 |- ( ( -. 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. ) ) ) )
87	86	com24 87	. . . . . . . . . . . . . . . . 17 |- ( ( -. 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. ) ) ) )
88	87	ex 434	. . . . . . . . . . . . . . . 16 |- ( -. 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. ) ) ) ) )
89	88	com14 88	. . . . . . . . . . . . . . 15 |- ( 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. ) ) ) ) )
90	89	imp 429	. . . . . . . . . . . . . 14 |- ( ( 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. ) ) ) )
91	90	com14 88	. . . . . . . . . . . . 13 |- ( 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. ) ) ) )
92	91	adantr 465	. . . . . . . . . . . 12 |- ( ( 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. ) ) ) )
93	92	com13 80	. . . . . . . . . . 11 |- ( -. 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. ) ) ) )
94	56, 93	pm2.61i 164	. . . . . . . . . 10 |- ( ( 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. ) ) )
95	21, 94	syl5bi 217	. . . . . . . . 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. ) ) )
96	95	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. ) )
97	96	impl 620	. . . . . . 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 ) ) ) = .0. )
98	97	mpteq2dva 4543	. . . . . 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 ) -> ( k e. ( 0 ... n ) |-> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) ) = ( k e. ( 0 ... n ) |-> .0. ) )
99	98	oveq2d 6312	. . . . 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 ) ) ) ) ) = ( R gsumg ( k e. ( 0 ... n ) |-> .0. ) ) )
100		ringmnd 17333	. . . . . . . . 9 |- ( R e. Ring -> R e. Mnd )
101		ovex 6324	. . . . . . . . . 10 |- ( 0 ... n ) e. _V
102	101	a1i 11	. . . . . . . . 9 |- ( R e. Ring -> ( 0 ... n ) e. _V )
103	44	gsumz 16131	. . . . . . . . 9 |- ( ( R e. Mnd /\ ( 0 ... n ) e. _V ) -> ( R gsumg ( k e. ( 0 ... n ) |-> .0. ) ) = .0. )
104	100, 102, 103	syl2anc 661	. . . . . . . 8 |- ( R e. Ring -> ( R gsumg ( k e. ( 0 ... n ) |-> .0. ) ) = .0. )
105	104	adantr 465	. . . . . . 7 |- ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) -> ( R gsumg ( k e. ( 0 ... n ) |-> .0. ) ) = .0. )
106	105	adantr 465	. . . . . 6 |- ( ( ( 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. )
107	106	adantr 465	. . . . 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 ) |-> .0. ) ) = .0. )
108	20, 99, 107	3eqtrd 2502	. . . 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. )
109	108	ralrimiva 2871	. . 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. )
110		fveq2 5872	. . . . 5 |- ( c = n -> ( (coe1 ` ( F .X. G ) ) ` c ) = ( (coe1 ` ( F .X. G ) ) ` n ) )
111	110	eqeq1d 2459	. . . 4 |- ( c = n -> ( ( (coe1 ` ( F .X. G ) ) ` c ) = .0. <-> ( (coe1 ` ( F .X. G ) ) ` n ) = .0. ) )
112	111	cbvralv 3084	. . 3 |- ( A. c e. NN ( (coe1 ` ( F .X. G ) ) ` c ) = .0. <-> A. n e. NN ( (coe1 ` ( F .X. G ) ) ` n ) = .0. )
113	109, 112	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. )
114	113	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 1395   e. wcel 1819   =/= wne 2652   A.wral 2807   _Vcvv 3109   |-> cmpt 4515   ` cfv 5594  (class class class)co 6296   0cc0 9509   - cmin 9824   NNcn 10556   NN0cn0 10816   ...cfz 11697   Basecbs 14643   .rcmulr 14712   0gc0g 14856   gsum_g cgsu 14857   Mndcmnd 16045   Ringcrg 17324  Poly₁cpl1 18342  coe₁cco1 18343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-ofr 6540  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11821  df-seq 12110  df-hash 12408  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-mulr 14725  df-sca 14727  df-vsca 14728  df-tset 14730  df-ple 14731  df-0g 14858  df-gsum 14859  df-mre 15002  df-mrc 15003  df-acs 15005  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-mhm 16092  df-submnd 16093  df-grp 16183  df-minusg 16184  df-mulg 16186  df-ghm 16391  df-cntz 16481  df-cmn 16926  df-abl 16927  df-mgp 17268  df-ur 17280  df-ring 17326  df-psr 18131  df-mpl 18133  df-opsr 18135  df-psr1 18345  df-ply1 18347  df-coe1 18348

This theorem is referenced by:  cpmatmcllem  19345