Theorem plymulx0 26862

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 3475	. . . . . 6 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> F e. (Poly ` RR ) )
2		ax-resscn 9335	. . . . . . . 8 |- RR C_ CC
3		1re 9381	. . . . . . . 8 |- 1 e. RR
4		plyid 21620	. . . . . . . 8 |- ( ( RR C_ CC /\ 1 e. RR ) -> Xp e. (Poly ` RR ) )
5	2, 3, 4	mp2an 667	. . . . . . 7 |- Xp e. (Poly ` RR )
6	5	a1i 11	. . . . . 6 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> Xp e. (Poly ` RR ) )
7		simprl 750	. . . . . . 7 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ ( x e. RR /\ y e. RR ) ) -> x e. RR )
8		simprr 751	. . . . . . 7 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ ( x e. RR /\ y e. RR ) ) -> y e. RR )
9	7, 8	readdcld 9409	. . . . . 6 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ ( x e. RR /\ y e. RR ) ) -> ( x + y ) e. RR )
10	7, 8	remulcld 9410	. . . . . 6 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ ( x e. RR /\ y e. RR ) ) -> ( x x. y ) e. RR )
11	1, 6, 9, 10	plymul 21629	. . . . 5 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> ( F oF x. Xp ) e. (Poly ` RR ) )
12		0re 9382	. . . . 5 |- 0 e. RR
13		eqid 2441	. . . . . 6 |- (coeff ` ( F oF x. Xp ) ) = (coeff ` ( F oF x. Xp ) )
14	13	coef2 21642	. . . . 5 |- ( ( ( F oF x. Xp ) e. (Poly ` RR ) /\ 0 e. RR ) -> (coeff ` ( F oF x. Xp ) ) : NN0 --> RR )
15	11, 12, 14	sylancl 657	. . . 4 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> (coeff ` ( F oF x. Xp ) ) : NN0 --> RR )
16		ffn 5556	. . . 4 |- ( (coeff ` ( F oF x. Xp ) ) : NN0 --> RR -> (coeff ` ( F oF x. Xp ) ) Fn NN0 )
17	15, 16	syl 16	. . 3 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> (coeff ` ( F oF x. Xp ) ) Fn NN0 )
18		dffn5 5734	. . 3 |- ( (coeff ` ( F oF x. Xp ) ) Fn NN0 <-> (coeff ` ( F oF x. Xp ) ) = ( n e. NN0 |-> ( (coeff ` ( F oF x. Xp ) ) ` n ) ) )
19	17, 18	sylib 196	. 2 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> (coeff ` ( F oF x. Xp ) ) = ( n e. NN0 |-> ( (coeff ` ( F oF x. Xp ) ) ` n ) ) )
20		cnex 9359	. . . . . . . . 9 |- CC e. _V
21	20	a1i 11	. . . . . . . 8 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> CC e. _V )
22		plyf 21609	. . . . . . . . 9 |- ( F e. (Poly ` RR ) -> F : CC --> CC )
23	1, 22	syl 16	. . . . . . . 8 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> F : CC --> CC )
24		plyf 21609	. . . . . . . . . 10 |- ( Xp e. (Poly ` RR ) -> Xp : CC --> CC )
25	5, 24	ax-mp 5	. . . . . . . . 9 |- Xp : CC --> CC
26	25	a1i 11	. . . . . . . 8 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> Xp : CC --> CC )
27		simprl 750	. . . . . . . . 9 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ ( x e. CC /\ y e. CC ) ) -> x e. CC )
28		simprr 751	. . . . . . . . 9 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ ( x e. CC /\ y e. CC ) ) -> y e. CC )
29	27, 28	mulcomd 9403	. . . . . . . 8 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) = ( y x. x ) )
30	21, 23, 26, 29	caofcom 6351	. . . . . . 7 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> ( F oF x. Xp ) = ( Xp oF x. F ) )
31	30	fveq2d 5692	. . . . . 6 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> (coeff ` ( F oF x. Xp ) ) = (coeff ` ( Xp oF x. F ) ) )
32	31	fveq1d 5690	. . . . 5 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> ( (coeff ` ( F oF x. Xp ) ) ` n ) = ( (coeff ` ( Xp oF x. F ) ) ` n ) )
33	32	adantr 462	. . . 4 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) -> ( (coeff ` ( F oF x. Xp ) ) ` n ) = ( (coeff ` ( Xp oF x. F ) ) ` n ) )
34	5	a1i 11	. . . . . 6 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) -> Xp e. (Poly ` RR ) )
35	1	adantr 462	. . . . . 6 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) -> F e. (Poly ` RR ) )
36		simpr 458	. . . . . 6 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) -> n e. NN0 )
37		eqid 2441	. . . . . . 7 |- (coeff ` Xp ) = (coeff ` Xp )
38		eqid 2441	. . . . . . 7 |- (coeff ` F ) = (coeff ` F )
39	37, 38	coemul 21662	. . . . . 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 ) ) ) )
40	34, 35, 36, 39	syl3anc 1213	. . . . 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 ) ) ) )
41		elfznn0 11477	. . . . . . . . . 10 |- ( i e. ( 0 ... n ) -> i e. NN0 )
42		coeidp 21673	. . . . . . . . . 10 |- ( i e. NN0 -> ( (coeff ` Xp ) ` i ) = if ( i = 1 , 1 , 0 ) )
43	41, 42	syl 16	. . . . . . . . 9 |- ( i e. ( 0 ... n ) -> ( (coeff ` Xp ) ` i ) = if ( i = 1 , 1 , 0 ) )
44	43	oveq1d 6105	. . . . . . . 8 |- ( i e. ( 0 ... n ) -> ( ( (coeff ` Xp ) ` i ) x. ( (coeff ` F ) ` ( n - i ) ) ) = ( if ( i = 1 , 1 , 0 ) x. ( (coeff ` F ) ` ( n - i ) ) ) )
45		ovif 6167	. . . . . . . 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 ) ) ) )
46	44, 45	syl6eq 2489	. . . . . . 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 ) ) ) ) )
47	46	adantl 463	. . . . . 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 ) ) ) ) )
48	47	sumeq2dv 13176	. . . . 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 ) ) ) ) )
49		elsn 3888	. . . . . . . . . 10 |- ( i e. { 1 } <-> i = 1 )
50	49	bicomi 202	. . . . . . . . 9 |- ( i = 1 <-> i e. { 1 } )
51	50	a1i 11	. . . . . . . 8 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ i e. ( 0 ... n ) ) -> ( i = 1 <-> i e. { 1 } ) )
52	38	coef2 21642	. . . . . . . . . . . . 13 |- ( ( F e. (Poly ` RR ) /\ 0 e. RR ) -> (coeff ` F ) : NN0 --> RR )
53	1, 12, 52	sylancl 657	. . . . . . . . . . . 12 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> (coeff ` F ) : NN0 --> RR )
54	53	ad2antrr 720	. . . . . . . . . . 11 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ i e. ( 0 ... n ) ) -> (coeff ` F ) : NN0 --> RR )
55		fznn0sub 11483	. . . . . . . . . . . 12 |- ( i e. ( 0 ... n ) -> ( n - i ) e. NN0 )
56	55	adantl 463	. . . . . . . . . . 11 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ i e. ( 0 ... n ) ) -> ( n - i ) e. NN0 )
57	54, 56	ffvelrnd 5841	. . . . . . . . . 10 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ i e. ( 0 ... n ) ) -> ( (coeff ` F ) ` ( n - i ) ) e. RR )
58	2, 57	sseldi 3351	. . . . . . . . 9 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ i e. ( 0 ... n ) ) -> ( (coeff ` F ) ` ( n - i ) ) e. CC )
59	58	mulid2d 9400	. . . . . . . 8 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ i e. ( 0 ... n ) ) -> ( 1 x. ( (coeff ` F ) ` ( n - i ) ) ) = ( (coeff ` F ) ` ( n - i ) ) )
60	58	mul02d 9563	. . . . . . . 8 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ i e. ( 0 ... n ) ) -> ( 0 x. ( (coeff ` F ) ` ( n - i ) ) ) = 0 )
61	51, 59, 60	ifbieq12d 3813	. . . . . . 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 ) )
62	61	sumeq2dv 13176	. . . . . 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 ) )
63		eqeq2 2450	. . . . . . 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 ) ) ) ) )
64		eqeq2 2450	. . . . . . 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 ) ) ) ) )
65		oveq2 6098	. . . . . . . . . . 11 |- ( n = 0 -> ( 0 ... n ) = ( 0 ... 0 ) )
66		0z 10653	. . . . . . . . . . . 12 |- 0 e. ZZ
67		fzsn 11496	. . . . . . . . . . . 12 |- ( 0 e. ZZ -> ( 0 ... 0 ) = { 0 } )
68	66, 67	ax-mp 5	. . . . . . . . . . 11 |- ( 0 ... 0 ) = { 0 }
69	65, 68	syl6eq 2489	. . . . . . . . . 10 |- ( n = 0 -> ( 0 ... n ) = { 0 } )
70		elsni 3899	. . . . . . . . . . . . 13 |- ( i e. { 0 } -> i = 0 )
71	70	adantl 463	. . . . . . . . . . . 12 |- ( ( n = 0 /\ i e. { 0 } ) -> i = 0 )
72		ax-1ne0 9347	. . . . . . . . . . . . . 14 |- 1 =/= 0
73	72	nesymi 2646	. . . . . . . . . . . . 13 |- -. 0 = 1
74		eqeq1 2447	. . . . . . . . . . . . 13 |- ( i = 0 -> ( i = 1 <-> 0 = 1 ) )
75	73, 74	mtbiri 303	. . . . . . . . . . . 12 |- ( i = 0 -> -. i = 1 )
76	71, 75	syl 16	. . . . . . . . . . 11 |- ( ( n = 0 /\ i e. { 0 } ) -> -. i = 1 )
77	50	notbii 296	. . . . . . . . . . . 12 |- ( -. i = 1 <-> -. i e. { 1 } )
78	77	biimpi 194	. . . . . . . . . . 11 |- ( -. i = 1 -> -. i e. { 1 } )
79		iffalse 3796	. . . . . . . . . . 11 |- ( -. i e. { 1 } -> if ( i e. { 1 } , ( (coeff ` F ) ` ( n - i ) ) , 0 ) = 0 )
80	76, 78, 79	3syl 20	. . . . . . . . . 10 |- ( ( n = 0 /\ i e. { 0 } ) -> if ( i e. { 1 } , ( (coeff ` F ) ` ( n - i ) ) , 0 ) = 0 )
81	69, 80	sumeq12rdv 13180	. . . . . . . . 9 |- ( n = 0 -> sum_ i e. ( 0 ... n ) if ( i e. { 1 } , ( (coeff ` F ) ` ( n - i ) ) , 0 ) = sum_ i e. { 0 } 0 )
82		snfi 7386	. . . . . . . . . . 11 |- { 0 } e. Fin
83	82	olci 391	. . . . . . . . . 10 |- ( { 0 } C_ ( ZZ>= ` 0 ) \/ { 0 } e. Fin )
84		sumz 13195	. . . . . . . . . 10 |- ( ( { 0 } C_ ( ZZ>= ` 0 ) \/ { 0 } e. Fin ) -> sum_ i e. { 0 } 0 = 0 )
85	83, 84	ax-mp 5	. . . . . . . . 9 |- sum_ i e. { 0 } 0 = 0
86	81, 85	syl6eq 2489	. . . . . . . 8 |- ( n = 0 -> sum_ i e. ( 0 ... n ) if ( i e. { 1 } , ( (coeff ` F ) ` ( n - i ) ) , 0 ) = 0 )
87	86	adantl 463	. . . . . . 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 )
88		simpll 748	. . . . . . . 8 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ -. n = 0 ) -> F e. ( (Poly ` RR ) \ { 0p } ) )
89	36	adantr 462	. . . . . . . . . 10 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ -. n = 0 ) -> n e. NN0 )
90		simpr 458	. . . . . . . . . . 11 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ -. n = 0 ) -> -. n = 0 )
91	90	neneqad 2679	. . . . . . . . . 10 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ -. n = 0 ) -> n =/= 0 )
92	89, 91	jca 529	. . . . . . . . 9 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ -. n = 0 ) -> ( n e. NN0 /\ n =/= 0 ) )
93		elnnne0 10589	. . . . . . . . 9 |- ( n e. NN <-> ( n e. NN0 /\ n =/= 0 ) )
94	92, 93	sylibr 212	. . . . . . . 8 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ -. n = 0 ) -> n e. NN )
95		1nn0 10591	. . . . . . . . . . . . 13 |- 1 e. NN0
96	95	a1i 11	. . . . . . . . . . . 12 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> 1 e. NN0 )
97		simpr 458	. . . . . . . . . . . . 13 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> n e. NN )
98	97	nnnn0d 10632	. . . . . . . . . . . 12 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> n e. NN0 )
99		nnge1 10344	. . . . . . . . . . . . 13 |- ( n e. NN -> 1 <_ n )
100	97, 99	syl 16	. . . . . . . . . . . 12 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> 1 <_ n )
101		elfz2nn0 11476	. . . . . . . . . . . 12 |- ( 1 e. ( 0 ... n ) <-> ( 1 e. NN0 /\ n e. NN0 /\ 1 <_ n ) )
102	96, 98, 100, 101	syl3anbrc 1167	. . . . . . . . . . 11 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> 1 e. ( 0 ... n ) )
103	3	elexi 2980	. . . . . . . . . . . 12 |- 1 e. _V
104	103	snss 3996	. . . . . . . . . . 11 |- ( 1 e. ( 0 ... n ) <-> { 1 } C_ ( 0 ... n ) )
105	102, 104	sylib 196	. . . . . . . . . 10 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> { 1 } C_ ( 0 ... n ) )
106	53	ad2antrr 720	. . . . . . . . . . . . 13 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) /\ i e. { 1 } ) -> (coeff ` F ) : NN0 --> RR )
107		oveq2 6098	. . . . . . . . . . . . . . . 16 |- ( i = 1 -> ( n - i ) = ( n - 1 ) )
108	49, 107	sylbi 195	. . . . . . . . . . . . . . 15 |- ( i e. { 1 } -> ( n - i ) = ( n - 1 ) )
109	108	adantl 463	. . . . . . . . . . . . . 14 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) /\ i e. { 1 } ) -> ( n - i ) = ( n - 1 ) )
110		nnm1nn0 10617	. . . . . . . . . . . . . . 15 |- ( n e. NN -> ( n - 1 ) e. NN0 )
111	110	ad2antlr 721	. . . . . . . . . . . . . 14 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) /\ i e. { 1 } ) -> ( n - 1 ) e. NN0 )
112	109, 111	eqeltrd 2515	. . . . . . . . . . . . 13 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) /\ i e. { 1 } ) -> ( n - i ) e. NN0 )
113	106, 112	ffvelrnd 5841	. . . . . . . . . . . 12 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) /\ i e. { 1 } ) -> ( (coeff ` F ) ` ( n - i ) ) e. RR )
114	2, 113	sseldi 3351	. . . . . . . . . . 11 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) /\ i e. { 1 } ) -> ( (coeff ` F ) ` ( n - i ) ) e. CC )
115	114	ralrimiva 2797	. . . . . . . . . 10 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> A. i e. { 1 } ( (coeff ` F ) ` ( n - i ) ) e. CC )
116		fzfi 11790	. . . . . . . . . . . 12 |- ( 0 ... n ) e. Fin
117	116	olci 391	. . . . . . . . . . 11 |- ( ( 0 ... n ) C_ ( ZZ>= ` 0 ) \/ ( 0 ... n ) e. Fin )
118	117	a1i 11	. . . . . . . . . 10 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> ( ( 0 ... n ) C_ ( ZZ>= ` 0 ) \/ ( 0 ... n ) e. Fin ) )
119		sumss2 13199	. . . . . . . . . 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 ) )
120	105, 115, 118, 119	syl21anc 1212	. . . . . . . . 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 ) )
121	53	adantr 462	. . . . . . . . . . . 12 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> (coeff ` F ) : NN0 --> RR )
122	97, 110	syl 16	. . . . . . . . . . . 12 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> ( n - 1 ) e. NN0 )
123	121, 122	ffvelrnd 5841	. . . . . . . . . . 11 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> ( (coeff ` F ) ` ( n - 1 ) ) e. RR )
124	2, 123	sseldi 3351	. . . . . . . . . 10 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> ( (coeff ` F ) ` ( n - 1 ) ) e. CC )
125	107	fveq2d 5692	. . . . . . . . . . 11 |- ( i = 1 -> ( (coeff ` F ) ` ( n - i ) ) = ( (coeff ` F ) ` ( n - 1 ) ) )
126	125	sumsn 13213	. . . . . . . . . 10 |- ( ( 1 e. RR /\ ( (coeff ` F ) ` ( n - 1 ) ) e. CC ) -> sum_ i e. { 1 } ( (coeff ` F ) ` ( n - i ) ) = ( (coeff ` F ) ` ( n - 1 ) ) )
127	3, 124, 126	sylancr 658	. . . . . . . . 9 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> sum_ i e. { 1 } ( (coeff ` F ) ` ( n - i ) ) = ( (coeff ` F ) ` ( n - 1 ) ) )
128	120, 127	eqtr3d 2475	. . . . . . . 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 ) ) )
129	88, 94, 128	syl2anc 656	. . . . . . 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 ) ) )
130	63, 64, 87, 129	ifbothda 3821	. . . . . 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 ) ) ) )
131	62, 130	eqtrd 2473	. . . . 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 ) ) ) )
132	40, 48, 131	3eqtrd 2477	. . . 4 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) -> ( (coeff ` ( Xp oF x. F ) ) ` n ) = if ( n = 0 , 0 , ( (coeff ` F ) ` ( n - 1 ) ) ) )
133	33, 132	eqtrd 2473	. . 3 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) -> ( (coeff ` ( F oF x. Xp ) ) ` n ) = if ( n = 0 , 0 , ( (coeff ` F ) ` ( n - 1 ) ) ) )
134	133	mpteq2dva 4375	. 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 ) ) ) ) )
135	19, 134	eqtrd 2473	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 184   \/ wo 368   /\ wa 369   = wceq 1364   e. wcel 1761   =/= wne 2604   A.wral 2713   _Vcvv 2970   \ cdif 3322   C_ wss 3325   ifcif 3788   {csn 3874   class class class wbr 4289   e. cmpt 4347   Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090   oFcof 6317   Fincfn 7306   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279   x. cmul 9283   <_ cle 9415   - cmin 9591   NNcn 10318   NN0cn0 10575   ZZcz 10642   ZZ>=cuz 10857   ...cfz 11433   sum_csu 13159   0pc0p 21047  Polycply 21595   Xpcidp 21596  coeffccoe 21597

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-oi 7720  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-fz 11434  df-fzo 11545  df-fl 11638  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-rlim 12963  df-sum 13160  df-0p 21048  df-ply 21599  df-idp 21600  df-coe 21601  df-dgr 21602

This theorem is referenced by:  plymulx  26863