Theorem coeeu 23981

Description: Uniqueness of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Assertion

Ref	Expression
coeeu	|- ( F e. (Poly ` S ) -> E! a e. ( CC ^m NN0 ) E. n e. NN0 ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) )

Distinct variable groups:   z, k   n, a, F   S, a, n   k, a, z, n

Allowed substitution hints:   S( z, k)   F( z, k)

Proof of Theorem coeeu

Dummy variables b j m w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		plyssc 23956	. . . . 5 |- (Poly ` S ) C_ (Poly ` CC )
2	1	sseli 3599	. . . 4 |- ( F e. (Poly ` S ) -> F e. (Poly ` CC ) )
3		elply2 23952	. . . . . 6 |- ( F e. (Poly ` CC ) <-> ( CC C_ CC /\ E. n e. NN0 E. a e. ( ( CC u. { 0 } ) ^m NN0 ) ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) ) )
4	3	simprbi 480	. . . . 5 |- ( F e. (Poly ` CC ) -> E. n e. NN0 E. a e. ( ( CC u. { 0 } ) ^m NN0 ) ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) )
5		rexcom 3099	. . . . 5 |- ( E. n e. NN0 E. a e. ( ( CC u. { 0 } ) ^m NN0 ) ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) <-> E. a e. ( ( CC u. { 0 } ) ^m NN0 ) E. n e. NN0 ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) )
6	4, 5	sylib 208	. . . 4 |- ( F e. (Poly ` CC ) -> E. a e. ( ( CC u. { 0 } ) ^m NN0 ) E. n e. NN0 ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) )
7	2, 6	syl 17	. . 3 |- ( F e. (Poly ` S ) -> E. a e. ( ( CC u. { 0 } ) ^m NN0 ) E. n e. NN0 ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) )
8		0cn 10032	. . . . . . 7 |- 0 e. CC
9		snssi 4339	. . . . . . 7 |- ( 0 e. CC -> { 0 } C_ CC )
10	8, 9	ax-mp 5	. . . . . 6 |- { 0 } C_ CC
11		ssequn2 3786	. . . . . 6 |- ( { 0 } C_ CC <-> ( CC u. { 0 } ) = CC )
12	10, 11	mpbi 220	. . . . 5 |- ( CC u. { 0 } ) = CC
13	12	oveq1i 6660	. . . 4 |- ( ( CC u. { 0 } ) ^m NN0 ) = ( CC ^m NN0 )
14	13	rexeqi 3143	. . 3 |- ( E. a e. ( ( CC u. { 0 } ) ^m NN0 ) E. n e. NN0 ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) <-> E. a e. ( CC ^m NN0 ) E. n e. NN0 ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) )
15	7, 14	sylib 208	. 2 |- ( F e. (Poly ` S ) -> E. a e. ( CC ^m NN0 ) E. n e. NN0 ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) )
16		reeanv 3107	. . . 4 |- ( E. n e. NN0 E. m e. NN0 ( ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) /\ ( ( b " ( ZZ>= ` ( m + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... m ) ( ( b ` k ) x. ( z ^ k ) ) ) ) ) <-> ( E. n e. NN0 ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) /\ E. m e. NN0 ( ( b " ( ZZ>= ` ( m + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... m ) ( ( b ` k ) x. ( z ^ k ) ) ) ) ) )
17		simp1l 1085	. . . . . . 7 |- ( ( ( F e. (Poly ` S ) /\ ( a e. ( CC ^m NN0 ) /\ b e. ( CC ^m NN0 ) ) ) /\ ( n e. NN0 /\ m e. NN0 ) /\ ( ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) /\ ( ( b " ( ZZ>= ` ( m + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... m ) ( ( b ` k ) x. ( z ^ k ) ) ) ) ) ) -> F e. (Poly ` S ) )
18		simp1rl 1126	. . . . . . 7 |- ( ( ( F e. (Poly ` S ) /\ ( a e. ( CC ^m NN0 ) /\ b e. ( CC ^m NN0 ) ) ) /\ ( n e. NN0 /\ m e. NN0 ) /\ ( ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) /\ ( ( b " ( ZZ>= ` ( m + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... m ) ( ( b ` k ) x. ( z ^ k ) ) ) ) ) ) -> a e. ( CC ^m NN0 ) )
19		simp1rr 1127	. . . . . . 7 |- ( ( ( F e. (Poly ` S ) /\ ( a e. ( CC ^m NN0 ) /\ b e. ( CC ^m NN0 ) ) ) /\ ( n e. NN0 /\ m e. NN0 ) /\ ( ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) /\ ( ( b " ( ZZ>= ` ( m + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... m ) ( ( b ` k ) x. ( z ^ k ) ) ) ) ) ) -> b e. ( CC ^m NN0 ) )
20		simp2l 1087	. . . . . . 7 |- ( ( ( F e. (Poly ` S ) /\ ( a e. ( CC ^m NN0 ) /\ b e. ( CC ^m NN0 ) ) ) /\ ( n e. NN0 /\ m e. NN0 ) /\ ( ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) /\ ( ( b " ( ZZ>= ` ( m + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... m ) ( ( b ` k ) x. ( z ^ k ) ) ) ) ) ) -> n e. NN0 )
21		simp2r 1088	. . . . . . 7 |- ( ( ( F e. (Poly ` S ) /\ ( a e. ( CC ^m NN0 ) /\ b e. ( CC ^m NN0 ) ) ) /\ ( n e. NN0 /\ m e. NN0 ) /\ ( ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) /\ ( ( b " ( ZZ>= ` ( m + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... m ) ( ( b ` k ) x. ( z ^ k ) ) ) ) ) ) -> m e. NN0 )
22		simp3ll 1132	. . . . . . 7 |- ( ( ( F e. (Poly ` S ) /\ ( a e. ( CC ^m NN0 ) /\ b e. ( CC ^m NN0 ) ) ) /\ ( n e. NN0 /\ m e. NN0 ) /\ ( ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) /\ ( ( b " ( ZZ>= ` ( m + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... m ) ( ( b ` k ) x. ( z ^ k ) ) ) ) ) ) -> ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } )
23		simp3rl 1134	. . . . . . 7 |- ( ( ( F e. (Poly ` S ) /\ ( a e. ( CC ^m NN0 ) /\ b e. ( CC ^m NN0 ) ) ) /\ ( n e. NN0 /\ m e. NN0 ) /\ ( ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) /\ ( ( b " ( ZZ>= ` ( m + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... m ) ( ( b ` k ) x. ( z ^ k ) ) ) ) ) ) -> ( b " ( ZZ>= ` ( m + 1 ) ) ) = { 0 } )
24		simp3lr 1133	. . . . . . . 8 |- ( ( ( F e. (Poly ` S ) /\ ( a e. ( CC ^m NN0 ) /\ b e. ( CC ^m NN0 ) ) ) /\ ( n e. NN0 /\ m e. NN0 ) /\ ( ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) /\ ( ( b " ( ZZ>= ` ( m + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... m ) ( ( b ` k ) x. ( z ^ k ) ) ) ) ) ) -> F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) )
25		oveq1 6657	. . . . . . . . . . . 12 |- ( z = w -> ( z ^ k ) = ( w ^ k ) )
26	25	oveq2d 6666	. . . . . . . . . . 11 |- ( z = w -> ( ( a ` k ) x. ( z ^ k ) ) = ( ( a ` k ) x. ( w ^ k ) ) )
27	26	sumeq2sdv 14435	. . . . . . . . . 10 |- ( z = w -> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) = sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( w ^ k ) ) )
28		fveq2 6191	. . . . . . . . . . . 12 |- ( k = j -> ( a ` k ) = ( a ` j ) )
29		oveq2 6658	. . . . . . . . . . . 12 |- ( k = j -> ( w ^ k ) = ( w ^ j ) )
30	28, 29	oveq12d 6668	. . . . . . . . . . 11 |- ( k = j -> ( ( a ` k ) x. ( w ^ k ) ) = ( ( a ` j ) x. ( w ^ j ) ) )
31	30	cbvsumv 14426	. . . . . . . . . 10 |- sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( w ^ k ) ) = sum_ j e. ( 0 ... n ) ( ( a ` j ) x. ( w ^ j ) )
32	27, 31	syl6eq 2672	. . . . . . . . 9 |- ( z = w -> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) = sum_ j e. ( 0 ... n ) ( ( a ` j ) x. ( w ^ j ) ) )
33	32	cbvmptv 4750	. . . . . . . 8 |- ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) = ( w e. CC |-> sum_ j e. ( 0 ... n ) ( ( a ` j ) x. ( w ^ j ) ) )
34	24, 33	syl6eq 2672	. . . . . . 7 |- ( ( ( F e. (Poly ` S ) /\ ( a e. ( CC ^m NN0 ) /\ b e. ( CC ^m NN0 ) ) ) /\ ( n e. NN0 /\ m e. NN0 ) /\ ( ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) /\ ( ( b " ( ZZ>= ` ( m + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... m ) ( ( b ` k ) x. ( z ^ k ) ) ) ) ) ) -> F = ( w e. CC |-> sum_ j e. ( 0 ... n ) ( ( a ` j ) x. ( w ^ j ) ) ) )
35		simp3rr 1135	. . . . . . . 8 |- ( ( ( F e. (Poly ` S ) /\ ( a e. ( CC ^m NN0 ) /\ b e. ( CC ^m NN0 ) ) ) /\ ( n e. NN0 /\ m e. NN0 ) /\ ( ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) /\ ( ( b " ( ZZ>= ` ( m + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... m ) ( ( b ` k ) x. ( z ^ k ) ) ) ) ) ) -> F = ( z e. CC |-> sum_ k e. ( 0 ... m ) ( ( b ` k ) x. ( z ^ k ) ) ) )
36	25	oveq2d 6666	. . . . . . . . . . 11 |- ( z = w -> ( ( b ` k ) x. ( z ^ k ) ) = ( ( b ` k ) x. ( w ^ k ) ) )
37	36	sumeq2sdv 14435	. . . . . . . . . 10 |- ( z = w -> sum_ k e. ( 0 ... m ) ( ( b ` k ) x. ( z ^ k ) ) = sum_ k e. ( 0 ... m ) ( ( b ` k ) x. ( w ^ k ) ) )
38		fveq2 6191	. . . . . . . . . . . 12 |- ( k = j -> ( b ` k ) = ( b ` j ) )
39	38, 29	oveq12d 6668	. . . . . . . . . . 11 |- ( k = j -> ( ( b ` k ) x. ( w ^ k ) ) = ( ( b ` j ) x. ( w ^ j ) ) )
40	39	cbvsumv 14426	. . . . . . . . . 10 |- sum_ k e. ( 0 ... m ) ( ( b ` k ) x. ( w ^ k ) ) = sum_ j e. ( 0 ... m ) ( ( b ` j ) x. ( w ^ j ) )
41	37, 40	syl6eq 2672	. . . . . . . . 9 |- ( z = w -> sum_ k e. ( 0 ... m ) ( ( b ` k ) x. ( z ^ k ) ) = sum_ j e. ( 0 ... m ) ( ( b ` j ) x. ( w ^ j ) ) )
42	41	cbvmptv 4750	. . . . . . . 8 |- ( z e. CC |-> sum_ k e. ( 0 ... m ) ( ( b ` k ) x. ( z ^ k ) ) ) = ( w e. CC |-> sum_ j e. ( 0 ... m ) ( ( b ` j ) x. ( w ^ j ) ) )
43	35, 42	syl6eq 2672	. . . . . . 7 |- ( ( ( F e. (Poly ` S ) /\ ( a e. ( CC ^m NN0 ) /\ b e. ( CC ^m NN0 ) ) ) /\ ( n e. NN0 /\ m e. NN0 ) /\ ( ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) /\ ( ( b " ( ZZ>= ` ( m + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... m ) ( ( b ` k ) x. ( z ^ k ) ) ) ) ) ) -> F = ( w e. CC |-> sum_ j e. ( 0 ... m ) ( ( b ` j ) x. ( w ^ j ) ) ) )
44	17, 18, 19, 20, 21, 22, 23, 34, 43	coeeulem 23980	. . . . . 6 |- ( ( ( F e. (Poly ` S ) /\ ( a e. ( CC ^m NN0 ) /\ b e. ( CC ^m NN0 ) ) ) /\ ( n e. NN0 /\ m e. NN0 ) /\ ( ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) /\ ( ( b " ( ZZ>= ` ( m + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... m ) ( ( b ` k ) x. ( z ^ k ) ) ) ) ) ) -> a = b )
45	44	3expia 1267	. . . . 5 |- ( ( ( F e. (Poly ` S ) /\ ( a e. ( CC ^m NN0 ) /\ b e. ( CC ^m NN0 ) ) ) /\ ( n e. NN0 /\ m e. NN0 ) ) -> ( ( ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) /\ ( ( b " ( ZZ>= ` ( m + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... m ) ( ( b ` k ) x. ( z ^ k ) ) ) ) ) -> a = b ) )
46	45	rexlimdvva 3038	. . . 4 |- ( ( F e. (Poly ` S ) /\ ( a e. ( CC ^m NN0 ) /\ b e. ( CC ^m NN0 ) ) ) -> ( E. n e. NN0 E. m e. NN0 ( ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) /\ ( ( b " ( ZZ>= ` ( m + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... m ) ( ( b ` k ) x. ( z ^ k ) ) ) ) ) -> a = b ) )
47	16, 46	syl5bir 233	. . 3 |- ( ( F e. (Poly ` S ) /\ ( a e. ( CC ^m NN0 ) /\ b e. ( CC ^m NN0 ) ) ) -> ( ( E. n e. NN0 ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) /\ E. m e. NN0 ( ( b " ( ZZ>= ` ( m + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... m ) ( ( b ` k ) x. ( z ^ k ) ) ) ) ) -> a = b ) )
48	47	ralrimivva 2971	. 2 |- ( F e. (Poly ` S ) -> A. a e. ( CC ^m NN0 ) A. b e. ( CC ^m NN0 ) ( ( E. n e. NN0 ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) /\ E. m e. NN0 ( ( b " ( ZZ>= ` ( m + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... m ) ( ( b ` k ) x. ( z ^ k ) ) ) ) ) -> a = b ) )
49		imaeq1 5461	. . . . . . 7 |- ( a = b -> ( a " ( ZZ>= ` ( n + 1 ) ) ) = ( b " ( ZZ>= ` ( n + 1 ) ) ) )
50	49	eqeq1d 2624	. . . . . 6 |- ( a = b -> ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } <-> ( b " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } ) )
51		fveq1 6190	. . . . . . . . . 10 |- ( a = b -> ( a ` k ) = ( b ` k ) )
52	51	oveq1d 6665	. . . . . . . . 9 |- ( a = b -> ( ( a ` k ) x. ( z ^ k ) ) = ( ( b ` k ) x. ( z ^ k ) ) )
53	52	sumeq2sdv 14435	. . . . . . . 8 |- ( a = b -> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) = sum_ k e. ( 0 ... n ) ( ( b ` k ) x. ( z ^ k ) ) )
54	53	mpteq2dv 4745	. . . . . . 7 |- ( a = b -> ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( b ` k ) x. ( z ^ k ) ) ) )
55	54	eqeq2d 2632	. . . . . 6 |- ( a = b -> ( F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) <-> F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( b ` k ) x. ( z ^ k ) ) ) ) )
56	50, 55	anbi12d 747	. . . . 5 |- ( a = b -> ( ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) <-> ( ( b " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( b ` k ) x. ( z ^ k ) ) ) ) ) )
57	56	rexbidv 3052	. . . 4 |- ( a = b -> ( E. n e. NN0 ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) <-> E. n e. NN0 ( ( b " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( b ` k ) x. ( z ^ k ) ) ) ) ) )
58		oveq1 6657	. . . . . . . . 9 |- ( n = m -> ( n + 1 ) = ( m + 1 ) )
59	58	fveq2d 6195	. . . . . . . 8 |- ( n = m -> ( ZZ>= ` ( n + 1 ) ) = ( ZZ>= ` ( m + 1 ) ) )
60	59	imaeq2d 5466	. . . . . . 7 |- ( n = m -> ( b " ( ZZ>= ` ( n + 1 ) ) ) = ( b " ( ZZ>= ` ( m + 1 ) ) ) )
61	60	eqeq1d 2624	. . . . . 6 |- ( n = m -> ( ( b " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } <-> ( b " ( ZZ>= ` ( m + 1 ) ) ) = { 0 } ) )
62		oveq2 6658	. . . . . . . . 9 |- ( n = m -> ( 0 ... n ) = ( 0 ... m ) )
63	62	sumeq1d 14431	. . . . . . . 8 |- ( n = m -> sum_ k e. ( 0 ... n ) ( ( b ` k ) x. ( z ^ k ) ) = sum_ k e. ( 0 ... m ) ( ( b ` k ) x. ( z ^ k ) ) )
64	63	mpteq2dv 4745	. . . . . . 7 |- ( n = m -> ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( b ` k ) x. ( z ^ k ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... m ) ( ( b ` k ) x. ( z ^ k ) ) ) )
65	64	eqeq2d 2632	. . . . . 6 |- ( n = m -> ( F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( b ` k ) x. ( z ^ k ) ) ) <-> F = ( z e. CC |-> sum_ k e. ( 0 ... m ) ( ( b ` k ) x. ( z ^ k ) ) ) ) )
66	61, 65	anbi12d 747	. . . . 5 |- ( n = m -> ( ( ( b " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( b ` k ) x. ( z ^ k ) ) ) ) <-> ( ( b " ( ZZ>= ` ( m + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... m ) ( ( b ` k ) x. ( z ^ k ) ) ) ) ) )
67	66	cbvrexv 3172	. . . 4 |- ( E. n e. NN0 ( ( b " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( b ` k ) x. ( z ^ k ) ) ) ) <-> E. m e. NN0 ( ( b " ( ZZ>= ` ( m + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... m ) ( ( b ` k ) x. ( z ^ k ) ) ) ) )
68	57, 67	syl6bb 276	. . 3 |- ( a = b -> ( E. n e. NN0 ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) <-> E. m e. NN0 ( ( b " ( ZZ>= ` ( m + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... m ) ( ( b ` k ) x. ( z ^ k ) ) ) ) ) )
69	68	reu4 3400	. 2 |- ( E! a e. ( CC ^m NN0 ) E. n e. NN0 ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) <-> ( E. a e. ( CC ^m NN0 ) E. n e. NN0 ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) /\ A. a e. ( CC ^m NN0 ) A. b e. ( CC ^m NN0 ) ( ( E. n e. NN0 ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) /\ E. m e. NN0 ( ( b " ( ZZ>= ` ( m + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... m ) ( ( b ` k ) x. ( z ^ k ) ) ) ) ) -> a = b ) ) )
70	15, 48, 69	sylanbrc 698	1 |- ( F e. (Poly ` S ) -> E! a e. ( CC ^m NN0 ) E. n e. NN0 ( ( a " ( ZZ>= ` ( n + 1 ) ) ) = { 0 } /\ F = ( z e. CC |-> sum_ k e. ( 0 ... n ) ( ( a ` k ) x. ( z ^ k ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   E!wreu 2914   u. cun 3572   C_ wss 3574   {csn 4177   |-> cmpt 4729   "cima 5117   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   NN0cn0 11292   ZZ>=cuz 11687   ...cfz 12326   ^cexp 12860   sum_csu 14416  Polycply 23940

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

This theorem is referenced by:  coelem  23982  coeeq  23983