Theorem plymulx0 29436

Description: Coefficients of a polynomial multiplyed by Xp. (Contributed by Thierry Arnoux, 25-Sep-2018.)

Assertion

Ref	Expression
plymulx0	|- ( F e. ( (Poly ` RR ) \ { 0p } ) -> (coeff ` ( F oF x. Xp ) ) = ( n e. NN0 |-> if ( n = 0 , 0 , ( (coeff ` F ) ` ( n - 1 ) ) ) ) )

Distinct variable group:   n, F

Proof of Theorem plymulx0

Dummy variables i x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldifi 3555	. . . . 5 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> F e. (Poly ` RR ) )
2		ax-resscn 9596	. . . . . . 7 |- RR C_ CC
3		1re 9642	. . . . . . 7 |- 1 e. RR
4		plyid 23163	. . . . . . 7 |- ( ( RR C_ CC /\ 1 e. RR ) -> Xp e. (Poly ` RR ) )
5	2, 3, 4	mp2an 678	. . . . . 6 |- Xp e. (Poly ` RR )
6	5	a1i 11	. . . . 5 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> Xp e. (Poly ` RR ) )
7		simprl 764	. . . . . 6 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ ( x e. RR /\ y e. RR ) ) -> x e. RR )
8		simprr 766	. . . . . 6 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ ( x e. RR /\ y e. RR ) ) -> y e. RR )
9	7, 8	readdcld 9670	. . . . 5 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ ( x e. RR /\ y e. RR ) ) -> ( x + y ) e. RR )
10	7, 8	remulcld 9671	. . . . 5 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ ( x e. RR /\ y e. RR ) ) -> ( x x. y ) e. RR )
11	1, 6, 9, 10	plymul 23172	. . . 4 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> ( F oF x. Xp ) e. (Poly ` RR ) )
12		0re 9643	. . . 4 |- 0 e. RR
13		eqid 2451	. . . . 5 |- (coeff ` ( F oF x. Xp ) ) = (coeff ` ( F oF x. Xp ) )
14	13	coef2 23185	. . . 4 |- ( ( ( F oF x. Xp ) e. (Poly ` RR ) /\ 0 e. RR ) -> (coeff ` ( F oF x. Xp ) ) : NN0 --> RR )
15	11, 12, 14	sylancl 668	. . 3 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> (coeff ` ( F oF x. Xp ) ) : NN0 --> RR )
16	15	feqmptd 5918	. 2 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> (coeff ` ( F oF x. Xp ) ) = ( n e. NN0 |-> ( (coeff ` ( F oF x. Xp ) ) ` n ) ) )
17		cnex 9620	. . . . . . . . 9 |- CC e. _V
18	17	a1i 11	. . . . . . . 8 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> CC e. _V )
19		plyf 23152	. . . . . . . . 9 |- ( F e. (Poly ` RR ) -> F : CC --> CC )
20	1, 19	syl 17	. . . . . . . 8 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> F : CC --> CC )
21		plyf 23152	. . . . . . . . . 10 |- ( Xp e. (Poly ` RR ) -> Xp : CC --> CC )
22	5, 21	ax-mp 5	. . . . . . . . 9 |- Xp : CC --> CC
23	22	a1i 11	. . . . . . . 8 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> Xp : CC --> CC )
24		simprl 764	. . . . . . . . 9 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ ( x e. CC /\ y e. CC ) ) -> x e. CC )
25		simprr 766	. . . . . . . . 9 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ ( x e. CC /\ y e. CC ) ) -> y e. CC )
26	24, 25	mulcomd 9664	. . . . . . . 8 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) = ( y x. x ) )
27	18, 20, 23, 26	caofcom 6563	. . . . . . 7 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> ( F oF x. Xp ) = ( Xp oF x. F ) )
28	27	fveq2d 5869	. . . . . 6 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> (coeff ` ( F oF x. Xp ) ) = (coeff ` ( Xp oF x. F ) ) )
29	28	fveq1d 5867	. . . . 5 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> ( (coeff ` ( F oF x. Xp ) ) ` n ) = ( (coeff ` ( Xp oF x. F ) ) ` n ) )
30	29	adantr 467	. . . 4 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) -> ( (coeff ` ( F oF x. Xp ) ) ` n ) = ( (coeff ` ( Xp oF x. F ) ) ` n ) )
31	5	a1i 11	. . . . . 6 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) -> Xp e. (Poly ` RR ) )
32	1	adantr 467	. . . . . 6 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) -> F e. (Poly ` RR ) )
33		simpr 463	. . . . . 6 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) -> n e. NN0 )
34		eqid 2451	. . . . . . 7 |- (coeff ` Xp ) = (coeff ` Xp )
35		eqid 2451	. . . . . . 7 |- (coeff ` F ) = (coeff ` F )
36	34, 35	coemul 23206	. . . . . 6 |- ( ( Xp e. (Poly ` RR ) /\ F e. (Poly ` RR ) /\ n e. NN0 ) -> ( (coeff ` ( Xp oF x. F ) ) ` n ) = sum_ i e. ( 0 ... n ) ( ( (coeff ` Xp ) ` i ) x. ( (coeff ` F ) ` ( n - i ) ) ) )
37	31, 32, 33, 36	syl3anc 1268	. . . . 5 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) -> ( (coeff ` ( Xp oF x. F ) ) ` n ) = sum_ i e. ( 0 ... n ) ( ( (coeff ` Xp ) ` i ) x. ( (coeff ` F ) ` ( n - i ) ) ) )
38		elfznn0 11887	. . . . . . . . . 10 |- ( i e. ( 0 ... n ) -> i e. NN0 )
39		coeidp 23217	. . . . . . . . . 10 |- ( i e. NN0 -> ( (coeff ` Xp ) ` i ) = if ( i = 1 , 1 , 0 ) )
40	38, 39	syl 17	. . . . . . . . 9 |- ( i e. ( 0 ... n ) -> ( (coeff ` Xp ) ` i ) = if ( i = 1 , 1 , 0 ) )
41	40	oveq1d 6305	. . . . . . . 8 |- ( i e. ( 0 ... n ) -> ( ( (coeff ` Xp ) ` i ) x. ( (coeff ` F ) ` ( n - i ) ) ) = ( if ( i = 1 , 1 , 0 ) x. ( (coeff ` F ) ` ( n - i ) ) ) )
42		ovif 6373	. . . . . . . 8 |- ( if ( i = 1 , 1 , 0 ) x. ( (coeff ` F ) ` ( n - i ) ) ) = if ( i = 1 , ( 1 x. ( (coeff ` F ) ` ( n - i ) ) ) , ( 0 x. ( (coeff ` F ) ` ( n - i ) ) ) )
43	41, 42	syl6eq 2501	. . . . . . 7 |- ( i e. ( 0 ... n ) -> ( ( (coeff ` Xp ) ` i ) x. ( (coeff ` F ) ` ( n - i ) ) ) = if ( i = 1 , ( 1 x. ( (coeff ` F ) ` ( n - i ) ) ) , ( 0 x. ( (coeff ` F ) ` ( n - i ) ) ) ) )
44	43	adantl 468	. . . . . 6 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ i e. ( 0 ... n ) ) -> ( ( (coeff ` Xp ) ` i ) x. ( (coeff ` F ) ` ( n - i ) ) ) = if ( i = 1 , ( 1 x. ( (coeff ` F ) ` ( n - i ) ) ) , ( 0 x. ( (coeff ` F ) ` ( n - i ) ) ) ) )
45	44	sumeq2dv 13769	. . . . 5 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) -> sum_ i e. ( 0 ... n ) ( ( (coeff ` Xp ) ` i ) x. ( (coeff ` F ) ` ( n - i ) ) ) = sum_ i e. ( 0 ... n ) if ( i = 1 , ( 1 x. ( (coeff ` F ) ` ( n - i ) ) ) , ( 0 x. ( (coeff ` F ) ` ( n - i ) ) ) ) )
46		elsn 3982	. . . . . . . . . 10 |- ( i e. { 1 } <-> i = 1 )
47	46	bicomi 206	. . . . . . . . 9 |- ( i = 1 <-> i e. { 1 } )
48	47	a1i 11	. . . . . . . 8 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ i e. ( 0 ... n ) ) -> ( i = 1 <-> i e. { 1 } ) )
49	35	coef2 23185	. . . . . . . . . . . . 13 |- ( ( F e. (Poly ` RR ) /\ 0 e. RR ) -> (coeff ` F ) : NN0 --> RR )
50	1, 12, 49	sylancl 668	. . . . . . . . . . . 12 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> (coeff ` F ) : NN0 --> RR )
51	50	ad2antrr 732	. . . . . . . . . . 11 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ i e. ( 0 ... n ) ) -> (coeff ` F ) : NN0 --> RR )
52		fznn0sub 11831	. . . . . . . . . . . 12 |- ( i e. ( 0 ... n ) -> ( n - i ) e. NN0 )
53	52	adantl 468	. . . . . . . . . . 11 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ i e. ( 0 ... n ) ) -> ( n - i ) e. NN0 )
54	51, 53	ffvelrnd 6023	. . . . . . . . . 10 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ i e. ( 0 ... n ) ) -> ( (coeff ` F ) ` ( n - i ) ) e. RR )
55	54	recnd 9669	. . . . . . . . 9 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ i e. ( 0 ... n ) ) -> ( (coeff ` F ) ` ( n - i ) ) e. CC )
56	55	mulid2d 9661	. . . . . . . 8 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ i e. ( 0 ... n ) ) -> ( 1 x. ( (coeff ` F ) ` ( n - i ) ) ) = ( (coeff ` F ) ` ( n - i ) ) )
57	55	mul02d 9831	. . . . . . . 8 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ i e. ( 0 ... n ) ) -> ( 0 x. ( (coeff ` F ) ` ( n - i ) ) ) = 0 )
58	48, 56, 57	ifbieq12d 3908	. . . . . . 7 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ i e. ( 0 ... n ) ) -> if ( i = 1 , ( 1 x. ( (coeff ` F ) ` ( n - i ) ) ) , ( 0 x. ( (coeff ` F ) ` ( n - i ) ) ) ) = if ( i e. { 1 } , ( (coeff ` F ) ` ( n - i ) ) , 0 ) )
59	58	sumeq2dv 13769	. . . . . 6 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) -> sum_ i e. ( 0 ... n ) if ( i = 1 , ( 1 x. ( (coeff ` F ) ` ( n - i ) ) ) , ( 0 x. ( (coeff ` F ) ` ( n - i ) ) ) ) = sum_ i e. ( 0 ... n ) if ( i e. { 1 } , ( (coeff ` F ) ` ( n - i ) ) , 0 ) )
60		eqeq2 2462	. . . . . . 7 |- ( 0 = if ( n = 0 , 0 , ( (coeff ` F ) ` ( n - 1 ) ) ) -> ( sum_ i e. ( 0 ... n ) if ( i e. { 1 } , ( (coeff ` F ) ` ( n - i ) ) , 0 ) = 0 <-> sum_ i e. ( 0 ... n ) if ( i e. { 1 } , ( (coeff ` F ) ` ( n - i ) ) , 0 ) = if ( n = 0 , 0 , ( (coeff ` F ) ` ( n - 1 ) ) ) ) )
61		eqeq2 2462	. . . . . . 7 |- ( ( (coeff ` F ) ` ( n - 1 ) ) = if ( n = 0 , 0 , ( (coeff ` F ) ` ( n - 1 ) ) ) -> ( sum_ i e. ( 0 ... n ) if ( i e. { 1 } , ( (coeff ` F ) ` ( n - i ) ) , 0 ) = ( (coeff ` F ) ` ( n - 1 ) ) <-> sum_ i e. ( 0 ... n ) if ( i e. { 1 } , ( (coeff ` F ) ` ( n - i ) ) , 0 ) = if ( n = 0 , 0 , ( (coeff ` F ) ` ( n - 1 ) ) ) ) )
62		oveq2 6298	. . . . . . . . . . 11 |- ( n = 0 -> ( 0 ... n ) = ( 0 ... 0 ) )
63		0z 10948	. . . . . . . . . . . 12 |- 0 e. ZZ
64		fzsn 11840	. . . . . . . . . . . 12 |- ( 0 e. ZZ -> ( 0 ... 0 ) = { 0 } )
65	63, 64	ax-mp 5	. . . . . . . . . . 11 |- ( 0 ... 0 ) = { 0 }
66	62, 65	syl6eq 2501	. . . . . . . . . 10 |- ( n = 0 -> ( 0 ... n ) = { 0 } )
67		elsni 3993	. . . . . . . . . . . . 13 |- ( i e. { 0 } -> i = 0 )
68	67	adantl 468	. . . . . . . . . . . 12 |- ( ( n = 0 /\ i e. { 0 } ) -> i = 0 )
69		ax-1ne0 9608	. . . . . . . . . . . . . 14 |- 1 =/= 0
70	69	nesymi 2681	. . . . . . . . . . . . 13 |- -. 0 = 1
71		eqeq1 2455	. . . . . . . . . . . . 13 |- ( i = 0 -> ( i = 1 <-> 0 = 1 ) )
72	70, 71	mtbiri 305	. . . . . . . . . . . 12 |- ( i = 0 -> -. i = 1 )
73	68, 72	syl 17	. . . . . . . . . . 11 |- ( ( n = 0 /\ i e. { 0 } ) -> -. i = 1 )
74	47	notbii 298	. . . . . . . . . . . 12 |- ( -. i = 1 <-> -. i e. { 1 } )
75	74	biimpi 198	. . . . . . . . . . 11 |- ( -. i = 1 -> -. i e. { 1 } )
76		iffalse 3890	. . . . . . . . . . 11 |- ( -. i e. { 1 } -> if ( i e. { 1 } , ( (coeff ` F ) ` ( n - i ) ) , 0 ) = 0 )
77	73, 75, 76	3syl 18	. . . . . . . . . 10 |- ( ( n = 0 /\ i e. { 0 } ) -> if ( i e. { 1 } , ( (coeff ` F ) ` ( n - i ) ) , 0 ) = 0 )
78	66, 77	sumeq12rdv 13773	. . . . . . . . 9 |- ( n = 0 -> sum_ i e. ( 0 ... n ) if ( i e. { 1 } , ( (coeff ` F ) ` ( n - i ) ) , 0 ) = sum_ i e. { 0 } 0 )
79		snfi 7650	. . . . . . . . . . 11 |- { 0 } e. Fin
80	79	olci 393	. . . . . . . . . 10 |- ( { 0 } C_ ( ZZ>= ` 0 ) \/ { 0 } e. Fin )
81		sumz 13788	. . . . . . . . . 10 |- ( ( { 0 } C_ ( ZZ>= ` 0 ) \/ { 0 } e. Fin ) -> sum_ i e. { 0 } 0 = 0 )
82	80, 81	ax-mp 5	. . . . . . . . 9 |- sum_ i e. { 0 } 0 = 0
83	78, 82	syl6eq 2501	. . . . . . . 8 |- ( n = 0 -> sum_ i e. ( 0 ... n ) if ( i e. { 1 } , ( (coeff ` F ) ` ( n - i ) ) , 0 ) = 0 )
84	83	adantl 468	. . . . . . 7 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ n = 0 ) -> sum_ i e. ( 0 ... n ) if ( i e. { 1 } , ( (coeff ` F ) ` ( n - i ) ) , 0 ) = 0 )
85		simpll 760	. . . . . . . 8 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ -. n = 0 ) -> F e. ( (Poly ` RR ) \ { 0p } ) )
86	33	adantr 467	. . . . . . . . 9 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ -. n = 0 ) -> n e. NN0 )
87		simpr 463	. . . . . . . . . 10 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ -. n = 0 ) -> -. n = 0 )
88	87	neqned 2631	. . . . . . . . 9 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ -. n = 0 ) -> n =/= 0 )
89		elnnne0 10883	. . . . . . . . 9 |- ( n e. NN <-> ( n e. NN0 /\ n =/= 0 ) )
90	86, 88, 89	sylanbrc 670	. . . . . . . 8 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ -. n = 0 ) -> n e. NN )
91		1nn0 10885	. . . . . . . . . . . . 13 |- 1 e. NN0
92	91	a1i 11	. . . . . . . . . . . 12 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> 1 e. NN0 )
93		simpr 463	. . . . . . . . . . . . 13 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> n e. NN )
94	93	nnnn0d 10925	. . . . . . . . . . . 12 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> n e. NN0 )
95	93	nnge1d 10652	. . . . . . . . . . . 12 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> 1 <_ n )
96		elfz2nn0 11885	. . . . . . . . . . . 12 |- ( 1 e. ( 0 ... n ) <-> ( 1 e. NN0 /\ n e. NN0 /\ 1 <_ n ) )
97	92, 94, 95, 96	syl3anbrc 1192	. . . . . . . . . . 11 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> 1 e. ( 0 ... n ) )
98	97	snssd 4117	. . . . . . . . . 10 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> { 1 } C_ ( 0 ... n ) )
99	50	ad2antrr 732	. . . . . . . . . . . . 13 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) /\ i e. { 1 } ) -> (coeff ` F ) : NN0 --> RR )
100		oveq2 6298	. . . . . . . . . . . . . . . 16 |- ( i = 1 -> ( n - i ) = ( n - 1 ) )
101	46, 100	sylbi 199	. . . . . . . . . . . . . . 15 |- ( i e. { 1 } -> ( n - i ) = ( n - 1 ) )
102	101	adantl 468	. . . . . . . . . . . . . 14 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) /\ i e. { 1 } ) -> ( n - i ) = ( n - 1 ) )
103		nnm1nn0 10911	. . . . . . . . . . . . . . 15 |- ( n e. NN -> ( n - 1 ) e. NN0 )
104	103	ad2antlr 733	. . . . . . . . . . . . . 14 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) /\ i e. { 1 } ) -> ( n - 1 ) e. NN0 )
105	102, 104	eqeltrd 2529	. . . . . . . . . . . . 13 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) /\ i e. { 1 } ) -> ( n - i ) e. NN0 )
106	99, 105	ffvelrnd 6023	. . . . . . . . . . . 12 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) /\ i e. { 1 } ) -> ( (coeff ` F ) ` ( n - i ) ) e. RR )
107	106	recnd 9669	. . . . . . . . . . 11 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) /\ i e. { 1 } ) -> ( (coeff ` F ) ` ( n - i ) ) e. CC )
108	107	ralrimiva 2802	. . . . . . . . . 10 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> A. i e. { 1 } ( (coeff ` F ) ` ( n - i ) ) e. CC )
109		fzfi 12185	. . . . . . . . . . . 12 |- ( 0 ... n ) e. Fin
110	109	olci 393	. . . . . . . . . . 11 |- ( ( 0 ... n ) C_ ( ZZ>= ` 0 ) \/ ( 0 ... n ) e. Fin )
111	110	a1i 11	. . . . . . . . . 10 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> ( ( 0 ... n ) C_ ( ZZ>= ` 0 ) \/ ( 0 ... n ) e. Fin ) )
112		sumss2 13792	. . . . . . . . . 10 |- ( ( ( { 1 } C_ ( 0 ... n ) /\ A. i e. { 1 } ( (coeff ` F ) ` ( n - i ) ) e. CC ) /\ ( ( 0 ... n ) C_ ( ZZ>= ` 0 ) \/ ( 0 ... n ) e. Fin ) ) -> sum_ i e. { 1 } ( (coeff ` F ) ` ( n - i ) ) = sum_ i e. ( 0 ... n ) if ( i e. { 1 } , ( (coeff ` F ) ` ( n - i ) ) , 0 ) )
113	98, 108, 111, 112	syl21anc 1267	. . . . . . . . 9 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> sum_ i e. { 1 } ( (coeff ` F ) ` ( n - i ) ) = sum_ i e. ( 0 ... n ) if ( i e. { 1 } , ( (coeff ` F ) ` ( n - i ) ) , 0 ) )
114	50	adantr 467	. . . . . . . . . . . 12 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> (coeff ` F ) : NN0 --> RR )
115	103	adantl 468	. . . . . . . . . . . 12 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> ( n - 1 ) e. NN0 )
116	114, 115	ffvelrnd 6023	. . . . . . . . . . 11 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> ( (coeff ` F ) ` ( n - 1 ) ) e. RR )
117	116	recnd 9669	. . . . . . . . . 10 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> ( (coeff ` F ) ` ( n - 1 ) ) e. CC )
118	100	fveq2d 5869	. . . . . . . . . . 11 |- ( i = 1 -> ( (coeff ` F ) ` ( n - i ) ) = ( (coeff ` F ) ` ( n - 1 ) ) )
119	118	sumsn 13807	. . . . . . . . . 10 |- ( ( 1 e. RR /\ ( (coeff ` F ) ` ( n - 1 ) ) e. CC ) -> sum_ i e. { 1 } ( (coeff ` F ) ` ( n - i ) ) = ( (coeff ` F ) ` ( n - 1 ) ) )
120	3, 117, 119	sylancr 669	. . . . . . . . 9 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> sum_ i e. { 1 } ( (coeff ` F ) ` ( n - i ) ) = ( (coeff ` F ) ` ( n - 1 ) ) )
121	113, 120	eqtr3d 2487	. . . . . . . 8 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> sum_ i e. ( 0 ... n ) if ( i e. { 1 } , ( (coeff ` F ) ` ( n - i ) ) , 0 ) = ( (coeff ` F ) ` ( n - 1 ) ) )
122	85, 90, 121	syl2anc 667	. . . . . . 7 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ -. n = 0 ) -> sum_ i e. ( 0 ... n ) if ( i e. { 1 } , ( (coeff ` F ) ` ( n - i ) ) , 0 ) = ( (coeff ` F ) ` ( n - 1 ) ) )
123	60, 61, 84, 122	ifbothda 3916	. . . . . 6 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) -> sum_ i e. ( 0 ... n ) if ( i e. { 1 } , ( (coeff ` F ) ` ( n - i ) ) , 0 ) = if ( n = 0 , 0 , ( (coeff ` F ) ` ( n - 1 ) ) ) )
124	59, 123	eqtrd 2485	. . . . 5 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) -> sum_ i e. ( 0 ... n ) if ( i = 1 , ( 1 x. ( (coeff ` F ) ` ( n - i ) ) ) , ( 0 x. ( (coeff ` F ) ` ( n - i ) ) ) ) = if ( n = 0 , 0 , ( (coeff ` F ) ` ( n - 1 ) ) ) )
125	37, 45, 124	3eqtrd 2489	. . . 4 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) -> ( (coeff ` ( Xp oF x. F ) ) ` n ) = if ( n = 0 , 0 , ( (coeff ` F ) ` ( n - 1 ) ) ) )
126	30, 125	eqtrd 2485	. . 3 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) -> ( (coeff ` ( F oF x. Xp ) ) ` n ) = if ( n = 0 , 0 , ( (coeff ` F ) ` ( n - 1 ) ) ) )
127	126	mpteq2dva 4489	. 2 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> ( n e. NN0 |-> ( (coeff ` ( F oF x. Xp ) ) ` n ) ) = ( n e. NN0 |-> if ( n = 0 , 0 , ( (coeff ` F ) ` ( n - 1 ) ) ) ) )
128	16, 127	eqtrd 2485	1 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> (coeff ` ( F oF x. Xp ) ) = ( n e. NN0 |-> if ( n = 0 , 0 , ( (coeff ` F ) ` ( n - 1 ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   \/ wo 370   /\ wa 371   = wceq 1444   e. wcel 1887   =/= wne 2622   A.wral 2737   _Vcvv 3045   \ cdif 3401   C_ wss 3404   ifcif 3881   {csn 3968   class class class wbr 4402   |-> cmpt 4461   -->wf 5578   ` cfv 5582  (class class class)co 6290   oFcof 6529   Fincfn 7569   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540   x. cmul 9544   <_ cle 9676   - cmin 9860   NNcn 10609   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159   ...cfz 11784   sum_csu 13752   0pc0p 22627  Polycply 23138   Xpcidp 23139  coeffccoe 23140

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11785  df-fzo 11916  df-fl 12028  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-rlim 13553  df-sum 13753  df-0p 22628  df-ply 23142  df-idp 23143  df-coe 23144  df-dgr 23145

This theorem is referenced by:  plymulx  29437