Theorem cply1mul 19664

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 2622	. . . . . . . . . 10 |- ( .r ` R ) = ( .r ` R )
4		cply1mul.b	. . . . . . . . . 10 |- B = ( Base ` P )
5	1, 2, 3, 4	coe1mul 19640	. . . . . . . . 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 1266	. . . . . . . 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 481	. . . . . . 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 481	. . . . . 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 6658	. . . . . . . . 9 |- ( s = n -> ( 0 ... s ) = ( 0 ... n ) )
10		oveq1 6657	. . . . . . . . . . 11 |- ( s = n -> ( s - k ) = ( n - k ) )
11	10	fveq2d 6195	. . . . . . . . . 10 |- ( s = n -> ( (coe1 ` G ) ` ( s - k ) ) = ( (coe1 ` G ) ` ( n - k ) ) )
12	11	oveq2d 6666	. . . . . . . . 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 4733	. . . . . . . 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 6666	. . . . . . 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 482	. . . . . 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 11299	. . . . . . 7 |- ( n e. NN -> n e. NN0 )
17	16	adantl 482	. . . . . 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		ovexd 6680	. . . . . 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 )
19	8, 15, 17, 18	fvmptd 6288	. . . . 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 ) ) ) ) ) )
20		r19.26 3064	. . . . . . . . . 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. ) )
21		oveq2 6658	. . . . . . . . . . . . . . . . . . 19 |- ( k = 0 -> ( n - k ) = ( n - 0 ) )
22		nncn 11028	. . . . . . . . . . . . . . . . . . . . 21 |- ( n e. NN -> n e. CC )
23	22	subid1d 10381	. . . . . . . . . . . . . . . . . . . 20 |- ( n e. NN -> ( n - 0 ) = n )
24	23	adantr 481	. . . . . . . . . . . . . . . . . . 19 |- ( ( n e. NN /\ k e. ( 0 ... n ) ) -> ( n - 0 ) = n )
25	21, 24	sylan9eqr 2678	. . . . . . . . . . . . . . . . . 18 |- ( ( ( n e. NN /\ k e. ( 0 ... n ) ) /\ k = 0 ) -> ( n - k ) = n )
26		simpll 790	. . . . . . . . . . . . . . . . . 18 |- ( ( ( n e. NN /\ k e. ( 0 ... n ) ) /\ k = 0 ) -> n e. NN )
27	25, 26	eqeltrd 2701	. . . . . . . . . . . . . . . . 17 |- ( ( ( n e. NN /\ k e. ( 0 ... n ) ) /\ k = 0 ) -> ( n - k ) e. NN )
28		fveq2 6191	. . . . . . . . . . . . . . . . . . 19 |- ( c = ( n - k ) -> ( (coe1 ` G ) ` c ) = ( (coe1 ` G ) ` ( n - k ) ) )
29	28	eqeq1d 2624	. . . . . . . . . . . . . . . . . 18 |- ( c = ( n - k ) -> ( ( (coe1 ` G ) ` c ) = .0. <-> ( (coe1 ` G ) ` ( n - k ) ) = .0. ) )
30	29	rspcv 3305	. . . . . . . . . . . . . . . . 17 |- ( ( n - k ) e. NN -> ( A. c e. NN ( (coe1 ` G ) ` c ) = .0. -> ( (coe1 ` G ) ` ( n - k ) ) = .0. ) )
31	27, 30	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( ( n e. NN /\ k e. ( 0 ... n ) ) /\ k = 0 ) -> ( A. c e. NN ( (coe1 ` G ) ` c ) = .0. -> ( (coe1 ` G ) ` ( n - k ) ) = .0. ) )
32		oveq2 6658	. . . . . . . . . . . . . . . . . . . 20 |- ( ( (coe1 ` G ) ` ( n - k ) ) = .0. -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = ( ( (coe1 ` F ) ` k ) ( .r ` R ) .0. ) )
33		simpll 790	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ ( ( n e. NN /\ k e. ( 0 ... n ) ) /\ k = 0 ) ) -> R e. Ring )
34		simpl 473	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( F e. B /\ G e. B ) -> F e. B )
35	34	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) -> F e. B )
36		elfznn0 12433	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( k e. ( 0 ... n ) -> k e. NN0 )
37	36	adantl 482	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( n e. NN /\ k e. ( 0 ... n ) ) -> k e. NN0 )
38	37	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( n e. NN /\ k e. ( 0 ... n ) ) /\ k = 0 ) -> k e. NN0 )
39		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . 23 |- (coe1 ` F ) = (coe1 ` F )
40		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( Base ` R ) = ( Base ` R )
41	39, 4, 1, 40	coe1fvalcl 19582	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( F e. B /\ k e. NN0 ) -> ( (coe1 ` F ) ` k ) e. ( Base ` R ) )
42	35, 38, 41	syl2an 494	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ ( ( n e. NN /\ k e. ( 0 ... n ) ) /\ k = 0 ) ) -> ( (coe1 ` F ) ` k ) e. ( Base ` R ) )
43		cply1mul.0	. . . . . . . . . . . . . . . . . . . . . 22 |- .0. = ( 0g ` R )
44	40, 3, 43	ringrz 18588	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( R e. Ring /\ ( (coe1 ` F ) ` k ) e. ( Base ` R ) ) -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) .0. ) = .0. )
45	33, 42, 44	syl2anc 693	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( 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. )
46	32, 45	sylan9eqr 2678	. . . . . . . . . . . . . . . . . . 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. )
47	46	ex 450	. . . . . . . . . . . . . . . . . 18 |- ( ( ( 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	expcom 451	. . . . . . . . . . . . . . . . 17 |- ( ( ( 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. ) ) )
49	48	com23 86	. . . . . . . . . . . . . . . 16 |- ( ( ( 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. ) ) )
50	31, 49	syldc 48	. . . . . . . . . . . . . . 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. ) ) )
51	50	expd 452	. . . . . . . . . . . . . 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. ) ) ) )
52	51	com24 95	. . . . . . . . . . . . 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. ) ) ) )
53	52	adantl 482	. . . . . . . . . . . 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. ) ) ) )
54	53	com13 88	. . . . . . . . . . 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. ) ) ) )
55		df-ne 2795	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( k =/= 0 <-> -. k = 0 )
56	55	biimpri 218	. . . . . . . . . . . . . . . . . . . . . 22 |- ( -. k = 0 -> k =/= 0 )
57	56, 36	anim12ci 591	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( -. k = 0 /\ k e. ( 0 ... n ) ) -> ( k e. NN0 /\ k =/= 0 ) )
58		elnnne0 11306	. . . . . . . . . . . . . . . . . . . . 21 |- ( k e. NN <-> ( k e. NN0 /\ k =/= 0 ) )
59	57, 58	sylibr 224	. . . . . . . . . . . . . . . . . . . 20 |- ( ( -. k = 0 /\ k e. ( 0 ... n ) ) -> k e. NN )
60		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . 22 |- ( c = k -> ( (coe1 ` F ) ` c ) = ( (coe1 ` F ) ` k ) )
61	60	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . . 21 |- ( c = k -> ( ( (coe1 ` F ) ` c ) = .0. <-> ( (coe1 ` F ) ` k ) = .0. ) )
62	61	rspcv 3305	. . . . . . . . . . . . . . . . . . . 20 |- ( k e. NN -> ( A. c e. NN ( (coe1 ` F ) ` c ) = .0. -> ( (coe1 ` F ) ` k ) = .0. ) )
63	59, 62	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ( -. k = 0 /\ k e. ( 0 ... n ) ) -> ( A. c e. NN ( (coe1 ` F ) ` c ) = .0. -> ( (coe1 ` F ) ` k ) = .0. ) )
64		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( (coe1 ` F ) ` k ) = .0. -> ( ( (coe1 ` F ) ` k ) ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = ( .0. ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) )
65		simpll 790	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ k e. ( 0 ... n ) ) -> R e. Ring )
66	4	eleq2i 2693	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ( G e. B <-> G e. ( Base ` P ) )
67	66	biimpi 206	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( G e. B -> G e. ( Base ` P ) )
68	67	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( ( F e. B /\ G e. B ) -> G e. ( Base ` P ) )
69	68	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) -> G e. ( Base ` P ) )
70		fznn0sub 12373	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( k e. ( 0 ... n ) -> ( n - k ) e. NN0 )
71		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (coe1 ` G ) = (coe1 ` G )
72		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( Base ` P ) = ( Base ` P )
73	71, 72, 1, 40	coe1fvalcl 19582	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( G e. ( Base ` P ) /\ ( n - k ) e. NN0 ) -> ( (coe1 ` G ) ` ( n - k ) ) e. ( Base ` R ) )
74	69, 70, 73	syl2an 494	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ k e. ( 0 ... n ) ) -> ( (coe1 ` G ) ` ( n - k ) ) e. ( Base ` R ) )
75	40, 3, 43	ringlz 18587	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( R e. Ring /\ ( (coe1 ` G ) ` ( n - k ) ) e. ( Base ` R ) ) -> ( .0. ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. )
76	65, 74, 75	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) /\ k e. ( 0 ... n ) ) -> ( .0. ( .r ` R ) ( (coe1 ` G ) ` ( n - k ) ) ) = .0. )
77	64, 76	sylan9eqr 2678	. . . . . . . . . . . . . . . . . . . . . . . . 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. )
78	77	ex 450	. . . . . . . . . . . . . . . . . . . . . . . 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. ) )
79	78	ex 450	. . . . . . . . . . . . . . . . . . . . . . 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. ) ) )
80	79	com23 86	. . . . . . . . . . . . . . . . . . . . . 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. ) ) )
81	80	a1dd 50	. . . . . . . . . . . . . . . . . . . . 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. ) ) ) )
82	81	com14 96	. . . . . . . . . . . . . . . . . . . 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. ) ) ) )
83	82	adantl 482	. . . . . . . . . . . . . . . . . . 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. ) ) ) )
84	63, 83	syld 47	. . . . . . . . . . . . . . . . . 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. ) ) ) )
85	84	com24 95	. . . . . . . . . . . . . . . . 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. ) ) ) )
86	85	ex 450	. . . . . . . . . . . . . . . 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. ) ) ) ) )
87	86	com14 96	. . . . . . . . . . . . . . 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. ) ) ) ) )
88	87	imp 445	. . . . . . . . . . . . . 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. ) ) ) )
89	88	com14 96	. . . . . . . . . . . . 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. ) ) ) )
90	89	adantr 481	. . . . . . . . . . . 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. ) ) ) )
91	90	com13 88	. . . . . . . . . . 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. ) ) ) )
92	54, 91	pm2.61i 176	. . . . . . . . . 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. ) ) )
93	20, 92	syl5bi 232	. . . . . . . . 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. ) ) )
94	93	imp 445	. . . . . . . 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. ) )
95	94	impl 650	. . . . . . 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. )
96	95	mpteq2dva 4744	. . . . . 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. ) )
97	96	oveq2d 6666	. . . . 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. ) ) )
98		ringmnd 18556	. . . . . . . . 9 |- ( R e. Ring -> R e. Mnd )
99		ovexd 6680	. . . . . . . . 9 |- ( R e. Ring -> ( 0 ... n ) e. _V )
100	43	gsumz 17374	. . . . . . . . 9 |- ( ( R e. Mnd /\ ( 0 ... n ) e. _V ) -> ( R gsumg ( k e. ( 0 ... n ) |-> .0. ) ) = .0. )
101	98, 99, 100	syl2anc 693	. . . . . . . 8 |- ( R e. Ring -> ( R gsumg ( k e. ( 0 ... n ) |-> .0. ) ) = .0. )
102	101	adantr 481	. . . . . . 7 |- ( ( R e. Ring /\ ( F e. B /\ G e. B ) ) -> ( R gsumg ( k e. ( 0 ... n ) |-> .0. ) ) = .0. )
103	102	adantr 481	. . . . . 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. )
104	103	adantr 481	. . . . 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. )
105	19, 97, 104	3eqtrd 2660	. . . 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. )
106	105	ralrimiva 2966	. . 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. )
107		fveq2 6191	. . . . 5 |- ( c = n -> ( (coe1 ` ( F .X. G ) ) ` c ) = ( (coe1 ` ( F .X. G ) ) ` n ) )
108	107	eqeq1d 2624	. . . 4 |- ( c = n -> ( ( (coe1 ` ( F .X. G ) ) ` c ) = .0. <-> ( (coe1 ` ( F .X. G ) ) ` n ) = .0. ) )
109	108	cbvralv 3171	. . 3 |- ( A. c e. NN ( (coe1 ` ( F .X. G ) ) ` c ) = .0. <-> A. n e. NN ( (coe1 ` ( F .X. G ) ) ` n ) = .0. )
110	106, 109	sylibr 224	. 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. )
111	110	ex 450	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 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   0cc0 9936   - cmin 10266   NNcn 11020   NN0cn0 11292   ...cfz 12326   Basecbs 15857   .rcmulr 15942   0gc0g 16100   gsum_g cgsu 16101   Mndcmnd 17294   Ringcrg 18547  Poly₁cpl1 19547  coe₁cco1 19548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-tset 15960  df-ple 15961  df-0g 16102  df-gsum 16103  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-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-mpl 19358  df-opsr 19360  df-psr1 19550  df-ply1 19552  df-coe1 19553

This theorem is referenced by:  cpmatmcllem  20523