Theorem coemulhi 24010

Description: The leading coefficient of a product of polynomials. (Contributed by Mario Carneiro, 24-Jul-2014.)

Hypotheses

Ref	Expression
coefv0.1	|- A = (coeff ` F )
coeadd.2	|- B = (coeff ` G )
coemulhi.3	|- M = (deg ` F )
coemulhi.4	|- N = (deg ` G )

Assertion

Ref	Expression
coemulhi	|- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( (coeff ` ( F oF x. G ) ) ` ( M + N ) ) = ( ( A ` M ) x. ( B ` N ) ) )

Proof of Theorem coemulhi

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		coemulhi.3	. . . . 5 |- M = (deg ` F )
2		dgrcl 23989	. . . . 5 |- ( F e. (Poly ` S ) -> (deg ` F ) e. NN0 )
3	1, 2	syl5eqel 2705	. . . 4 |- ( F e. (Poly ` S ) -> M e. NN0 )
4		coemulhi.4	. . . . 5 |- N = (deg ` G )
5		dgrcl 23989	. . . . 5 |- ( G e. (Poly ` S ) -> (deg ` G ) e. NN0 )
6	4, 5	syl5eqel 2705	. . . 4 |- ( G e. (Poly ` S ) -> N e. NN0 )
7		nn0addcl 11328	. . . 4 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( M + N ) e. NN0 )
8	3, 6, 7	syl2an 494	. . 3 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( M + N ) e. NN0 )
9		coefv0.1	. . . 4 |- A = (coeff ` F )
10		coeadd.2	. . . 4 |- B = (coeff ` G )
11	9, 10	coemul 24008	. . 3 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ ( M + N ) e. NN0 ) -> ( (coeff ` ( F oF x. G ) ) ` ( M + N ) ) = sum_ k e. ( 0 ... ( M + N ) ) ( ( A ` k ) x. ( B ` ( ( M + N ) - k ) ) ) )
12	8, 11	mpd3an3 1425	. 2 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( (coeff ` ( F oF x. G ) ) ` ( M + N ) ) = sum_ k e. ( 0 ... ( M + N ) ) ( ( A ` k ) x. ( B ` ( ( M + N ) - k ) ) ) )
13	6	adantl 482	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> N e. NN0 )
14	13	nn0ge0d 11354	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> 0 <_ N )
15	3	adantr 481	. . . . . . . 8 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> M e. NN0 )
16	15	nn0red 11352	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> M e. RR )
17	13	nn0red 11352	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> N e. RR )
18	16, 17	addge01d 10615	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( 0 <_ N <-> M <_ ( M + N ) ) )
19	14, 18	mpbid 222	. . . . 5 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> M <_ ( M + N ) )
20		nn0uz 11722	. . . . . . 7 |- NN0 = ( ZZ>= ` 0 )
21	15, 20	syl6eleq 2711	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> M e. ( ZZ>= ` 0 ) )
22	8	nn0zd 11480	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( M + N ) e. ZZ )
23		elfz5 12334	. . . . . 6 |- ( ( M e. ( ZZ>= ` 0 ) /\ ( M + N ) e. ZZ ) -> ( M e. ( 0 ... ( M + N ) ) <-> M <_ ( M + N ) ) )
24	21, 22, 23	syl2anc 693	. . . . 5 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( M e. ( 0 ... ( M + N ) ) <-> M <_ ( M + N ) ) )
25	19, 24	mpbird 247	. . . 4 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> M e. ( 0 ... ( M + N ) ) )
26	25	snssd 4340	. . 3 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> { M } C_ ( 0 ... ( M + N ) ) )
27		elsni 4194	. . . . . 6 |- ( k e. { M } -> k = M )
28	27	adantl 482	. . . . 5 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. { M } ) -> k = M )
29		fveq2 6191	. . . . . 6 |- ( k = M -> ( A ` k ) = ( A ` M ) )
30		oveq2 6658	. . . . . . 7 |- ( k = M -> ( ( M + N ) - k ) = ( ( M + N ) - M ) )
31	30	fveq2d 6195	. . . . . 6 |- ( k = M -> ( B ` ( ( M + N ) - k ) ) = ( B ` ( ( M + N ) - M ) ) )
32	29, 31	oveq12d 6668	. . . . 5 |- ( k = M -> ( ( A ` k ) x. ( B ` ( ( M + N ) - k ) ) ) = ( ( A ` M ) x. ( B ` ( ( M + N ) - M ) ) ) )
33	28, 32	syl 17	. . . 4 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. { M } ) -> ( ( A ` k ) x. ( B ` ( ( M + N ) - k ) ) ) = ( ( A ` M ) x. ( B ` ( ( M + N ) - M ) ) ) )
34	16	recnd 10068	. . . . . . . . 9 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> M e. CC )
35	17	recnd 10068	. . . . . . . . 9 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> N e. CC )
36	34, 35	pncan2d 10394	. . . . . . . 8 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( ( M + N ) - M ) = N )
37	36	fveq2d 6195	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( B ` ( ( M + N ) - M ) ) = ( B ` N ) )
38	37	oveq2d 6666	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( ( A ` M ) x. ( B ` ( ( M + N ) - M ) ) ) = ( ( A ` M ) x. ( B ` N ) ) )
39	9	coef3 23988	. . . . . . . . 9 |- ( F e. (Poly ` S ) -> A : NN0 --> CC )
40	39	adantr 481	. . . . . . . 8 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> A : NN0 --> CC )
41	40, 15	ffvelrnd 6360	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( A ` M ) e. CC )
42	10	coef3 23988	. . . . . . . . 9 |- ( G e. (Poly ` S ) -> B : NN0 --> CC )
43	42	adantl 482	. . . . . . . 8 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> B : NN0 --> CC )
44	43, 13	ffvelrnd 6360	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( B ` N ) e. CC )
45	41, 44	mulcld 10060	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( ( A ` M ) x. ( B ` N ) ) e. CC )
46	38, 45	eqeltrd 2701	. . . . 5 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( ( A ` M ) x. ( B ` ( ( M + N ) - M ) ) ) e. CC )
47	46	adantr 481	. . . 4 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. { M } ) -> ( ( A ` M ) x. ( B ` ( ( M + N ) - M ) ) ) e. CC )
48	33, 47	eqeltrd 2701	. . 3 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. { M } ) -> ( ( A ` k ) x. ( B ` ( ( M + N ) - k ) ) ) e. CC )
49		simpl 473	. . . . . . . . 9 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> F e. (Poly ` S ) )
50		eldifi 3732	. . . . . . . . . 10 |- ( k e. ( ( 0 ... ( M + N ) ) \ { M } ) -> k e. ( 0 ... ( M + N ) ) )
51		elfznn0 12433	. . . . . . . . . 10 |- ( k e. ( 0 ... ( M + N ) ) -> k e. NN0 )
52	50, 51	syl 17	. . . . . . . . 9 |- ( k e. ( ( 0 ... ( M + N ) ) \ { M } ) -> k e. NN0 )
53	9, 1	dgrub 23990	. . . . . . . . . 10 |- ( ( F e. (Poly ` S ) /\ k e. NN0 /\ ( A ` k ) =/= 0 ) -> k <_ M )
54	53	3expia 1267	. . . . . . . . 9 |- ( ( F e. (Poly ` S ) /\ k e. NN0 ) -> ( ( A ` k ) =/= 0 -> k <_ M ) )
55	49, 52, 54	syl2an 494	. . . . . . . 8 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) -> ( ( A ` k ) =/= 0 -> k <_ M ) )
56	55	necon1bd 2812	. . . . . . 7 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) -> ( -. k <_ M -> ( A ` k ) = 0 ) )
57	56	imp 445	. . . . . 6 |- ( ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) /\ -. k <_ M ) -> ( A ` k ) = 0 )
58	57	oveq1d 6665	. . . . 5 |- ( ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) /\ -. k <_ M ) -> ( ( A ` k ) x. ( B ` ( ( M + N ) - k ) ) ) = ( 0 x. ( B ` ( ( M + N ) - k ) ) ) )
59	43	ad2antrr 762	. . . . . . 7 |- ( ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) /\ -. k <_ M ) -> B : NN0 --> CC )
60	50	ad2antlr 763	. . . . . . . 8 |- ( ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) /\ -. k <_ M ) -> k e. ( 0 ... ( M + N ) ) )
61		fznn0sub 12373	. . . . . . . 8 |- ( k e. ( 0 ... ( M + N ) ) -> ( ( M + N ) - k ) e. NN0 )
62	60, 61	syl 17	. . . . . . 7 |- ( ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) /\ -. k <_ M ) -> ( ( M + N ) - k ) e. NN0 )
63	59, 62	ffvelrnd 6360	. . . . . 6 |- ( ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) /\ -. k <_ M ) -> ( B ` ( ( M + N ) - k ) ) e. CC )
64	63	mul02d 10234	. . . . 5 |- ( ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) /\ -. k <_ M ) -> ( 0 x. ( B ` ( ( M + N ) - k ) ) ) = 0 )
65	58, 64	eqtrd 2656	. . . 4 |- ( ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) /\ -. k <_ M ) -> ( ( A ` k ) x. ( B ` ( ( M + N ) - k ) ) ) = 0 )
66	16	adantr 481	. . . . . . . . . . 11 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) -> M e. RR )
67	50	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) -> k e. ( 0 ... ( M + N ) ) )
68	67, 51	syl 17	. . . . . . . . . . . 12 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) -> k e. NN0 )
69	68	nn0red 11352	. . . . . . . . . . 11 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) -> k e. RR )
70	17	adantr 481	. . . . . . . . . . 11 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) -> N e. RR )
71	66, 69, 70	leadd1d 10621	. . . . . . . . . 10 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) -> ( M <_ k <-> ( M + N ) <_ ( k + N ) ) )
72	8	adantr 481	. . . . . . . . . . . 12 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) -> ( M + N ) e. NN0 )
73	72	nn0red 11352	. . . . . . . . . . 11 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) -> ( M + N ) e. RR )
74	73, 69, 70	lesubadd2d 10626	. . . . . . . . . 10 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) -> ( ( ( M + N ) - k ) <_ N <-> ( M + N ) <_ ( k + N ) ) )
75	71, 74	bitr4d 271	. . . . . . . . 9 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) -> ( M <_ k <-> ( ( M + N ) - k ) <_ N ) )
76	75	notbid 308	. . . . . . . 8 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) -> ( -. M <_ k <-> -. ( ( M + N ) - k ) <_ N ) )
77	76	biimpa 501	. . . . . . 7 |- ( ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) /\ -. M <_ k ) -> -. ( ( M + N ) - k ) <_ N )
78		simpr 477	. . . . . . . . . 10 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> G e. (Poly ` S ) )
79	50, 61	syl 17	. . . . . . . . . 10 |- ( k e. ( ( 0 ... ( M + N ) ) \ { M } ) -> ( ( M + N ) - k ) e. NN0 )
80	10, 4	dgrub 23990	. . . . . . . . . . 11 |- ( ( G e. (Poly ` S ) /\ ( ( M + N ) - k ) e. NN0 /\ ( B ` ( ( M + N ) - k ) ) =/= 0 ) -> ( ( M + N ) - k ) <_ N )
81	80	3expia 1267	. . . . . . . . . 10 |- ( ( G e. (Poly ` S ) /\ ( ( M + N ) - k ) e. NN0 ) -> ( ( B ` ( ( M + N ) - k ) ) =/= 0 -> ( ( M + N ) - k ) <_ N ) )
82	78, 79, 81	syl2an 494	. . . . . . . . 9 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) -> ( ( B ` ( ( M + N ) - k ) ) =/= 0 -> ( ( M + N ) - k ) <_ N ) )
83	82	necon1bd 2812	. . . . . . . 8 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) -> ( -. ( ( M + N ) - k ) <_ N -> ( B ` ( ( M + N ) - k ) ) = 0 ) )
84	83	imp 445	. . . . . . 7 |- ( ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) /\ -. ( ( M + N ) - k ) <_ N ) -> ( B ` ( ( M + N ) - k ) ) = 0 )
85	77, 84	syldan 487	. . . . . 6 |- ( ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) /\ -. M <_ k ) -> ( B ` ( ( M + N ) - k ) ) = 0 )
86	85	oveq2d 6666	. . . . 5 |- ( ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) /\ -. M <_ k ) -> ( ( A ` k ) x. ( B ` ( ( M + N ) - k ) ) ) = ( ( A ` k ) x. 0 ) )
87	40	ad2antrr 762	. . . . . . 7 |- ( ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) /\ -. M <_ k ) -> A : NN0 --> CC )
88	52	ad2antlr 763	. . . . . . 7 |- ( ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) /\ -. M <_ k ) -> k e. NN0 )
89	87, 88	ffvelrnd 6360	. . . . . 6 |- ( ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) /\ -. M <_ k ) -> ( A ` k ) e. CC )
90	89	mul01d 10235	. . . . 5 |- ( ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) /\ -. M <_ k ) -> ( ( A ` k ) x. 0 ) = 0 )
91	86, 90	eqtrd 2656	. . . 4 |- ( ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) /\ -. M <_ k ) -> ( ( A ` k ) x. ( B ` ( ( M + N ) - k ) ) ) = 0 )
92		eldifsni 4320	. . . . . . 7 |- ( k e. ( ( 0 ... ( M + N ) ) \ { M } ) -> k =/= M )
93	92	adantl 482	. . . . . 6 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) -> k =/= M )
94	69, 66	letri3d 10179	. . . . . . 7 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) -> ( k = M <-> ( k <_ M /\ M <_ k ) ) )
95	94	necon3abid 2830	. . . . . 6 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) -> ( k =/= M <-> -. ( k <_ M /\ M <_ k ) ) )
96	93, 95	mpbid 222	. . . . 5 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) -> -. ( k <_ M /\ M <_ k ) )
97		ianor 509	. . . . 5 |- ( -. ( k <_ M /\ M <_ k ) <-> ( -. k <_ M \/ -. M <_ k ) )
98	96, 97	sylib 208	. . . 4 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) -> ( -. k <_ M \/ -. M <_ k ) )
99	65, 91, 98	mpjaodan 827	. . 3 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( ( 0 ... ( M + N ) ) \ { M } ) ) -> ( ( A ` k ) x. ( B ` ( ( M + N ) - k ) ) ) = 0 )
100		fzfid 12772	. . 3 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( 0 ... ( M + N ) ) e. Fin )
101	26, 48, 99, 100	fsumss 14456	. 2 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> sum_ k e. { M } ( ( A ` k ) x. ( B ` ( ( M + N ) - k ) ) ) = sum_ k e. ( 0 ... ( M + N ) ) ( ( A ` k ) x. ( B ` ( ( M + N ) - k ) ) ) )
102	32	sumsn 14475	. . . 4 |- ( ( M e. NN0 /\ ( ( A ` M ) x. ( B ` ( ( M + N ) - M ) ) ) e. CC ) -> sum_ k e. { M } ( ( A ` k ) x. ( B ` ( ( M + N ) - k ) ) ) = ( ( A ` M ) x. ( B ` ( ( M + N ) - M ) ) ) )
103	15, 46, 102	syl2anc 693	. . 3 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> sum_ k e. { M } ( ( A ` k ) x. ( B ` ( ( M + N ) - k ) ) ) = ( ( A ` M ) x. ( B ` ( ( M + N ) - M ) ) ) )
104	103, 38	eqtrd 2656	. 2 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> sum_ k e. { M } ( ( A ` k ) x. ( B ` ( ( M + N ) - k ) ) ) = ( ( A ` M ) x. ( B ` N ) ) )
105	12, 101, 104	3eqtr2d 2662	1 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( (coeff ` ( F oF x. G ) ) ` ( M + N ) ) = ( ( A ` M ) x. ( B ` N ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   {csn 4177   class class class wbr 4653   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   x. cmul 9941   <_ cle 10075   - cmin 10266   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   sum_csu 14416  Polycply 23940  coeffccoe 23942  degcdgr 23943

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  ax-pre-sup 10014  ax-addf 10015

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  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-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-0p 23437  df-ply 23944  df-coe 23946  df-dgr 23947

This theorem is referenced by:  dgrmul  24026  plymul0or  24036  plydivlem4  24051  vieta1lem2  24066