Theorem plymulx0 29508

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 3544	. . . . 5 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> F e. (Poly ` RR ) )
2		ax-resscn 9614	. . . . . . 7 |- RR C_ CC
3		1re 9660	. . . . . . 7 |- 1 e. RR
4		plyid 23242	. . . . . . 7 |- ( ( RR C_ CC /\ 1 e. RR ) -> Xp e. (Poly ` RR ) )
5	2, 3, 4	mp2an 686	. . . . . 6 |- Xp e. (Poly ` RR )
6	5	a1i 11	. . . . 5 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> Xp e. (Poly ` RR ) )
7		simprl 772	. . . . . 6 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ ( x e. RR /\ y e. RR ) ) -> x e. RR )
8		simprr 774	. . . . . 6 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ ( x e. RR /\ y e. RR ) ) -> y e. RR )
9	7, 8	readdcld 9688	. . . . 5 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ ( x e. RR /\ y e. RR ) ) -> ( x + y ) e. RR )
10	7, 8	remulcld 9689	. . . . 5 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ ( x e. RR /\ y e. RR ) ) -> ( x x. y ) e. RR )
11	1, 6, 9, 10	plymul 23251	. . . 4 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> ( F oF x. Xp ) e. (Poly ` RR ) )
12		0re 9661	. . . 4 |- 0 e. RR
13		eqid 2471	. . . . 5 |- (coeff ` ( F oF x. Xp ) ) = (coeff ` ( F oF x. Xp ) )
14	13	coef2 23264	. . . 4 |- ( ( ( F oF x. Xp ) e. (Poly ` RR ) /\ 0 e. RR ) -> (coeff ` ( F oF x. Xp ) ) : NN0 --> RR )
15	11, 12, 14	sylancl 675	. . 3 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> (coeff ` ( F oF x. Xp ) ) : NN0 --> RR )
16	15	feqmptd 5932	. 2 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> (coeff ` ( F oF x. Xp ) ) = ( n e. NN0 |-> ( (coeff ` ( F oF x. Xp ) ) ` n ) ) )
17		cnex 9638	. . . . . . . . 9 |- CC e. _V
18	17	a1i 11	. . . . . . . 8 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> CC e. _V )
19		plyf 23231	. . . . . . . . 9 |- ( F e. (Poly ` RR ) -> F : CC --> CC )
20	1, 19	syl 17	. . . . . . . 8 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> F : CC --> CC )
21		plyf 23231	. . . . . . . . . 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 772	. . . . . . . . 9 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ ( x e. CC /\ y e. CC ) ) -> x e. CC )
25		simprr 774	. . . . . . . . 9 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ ( x e. CC /\ y e. CC ) ) -> y e. CC )
26	24, 25	mulcomd 9682	. . . . . . . 8 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) = ( y x. x ) )
27	18, 20, 23, 26	caofcom 6582	. . . . . . 7 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> ( F oF x. Xp ) = ( Xp oF x. F ) )
28	27	fveq2d 5883	. . . . . 6 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> (coeff ` ( F oF x. Xp ) ) = (coeff ` ( Xp oF x. F ) ) )
29	28	fveq1d 5881	. . . . 5 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> ( (coeff ` ( F oF x. Xp ) ) ` n ) = ( (coeff ` ( Xp oF x. F ) ) ` n ) )
30	29	adantr 472	. . . 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 472	. . . . . 6 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) -> F e. (Poly ` RR ) )
33		simpr 468	. . . . . 6 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) -> n e. NN0 )
34		eqid 2471	. . . . . . 7 |- (coeff ` Xp ) = (coeff ` Xp )
35		eqid 2471	. . . . . . 7 |- (coeff ` F ) = (coeff ` F )
36	34, 35	coemul 23285	. . . . . 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 1292	. . . . 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 11913	. . . . . . . . . 10 |- ( i e. ( 0 ... n ) -> i e. NN0 )
39		coeidp 23296	. . . . . . . . . 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 6323	. . . . . . . 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 6392	. . . . . . . 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 2521	. . . . . . 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 473	. . . . . 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 13846	. . . . 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 3973	. . . . . . . . . 10 |- ( i e. { 1 } <-> i = 1 )
47	46	bicomi 207	. . . . . . . . 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 23264	. . . . . . . . . . . . 13 |- ( ( F e. (Poly ` RR ) /\ 0 e. RR ) -> (coeff ` F ) : NN0 --> RR )
50	1, 12, 49	sylancl 675	. . . . . . . . . . . 12 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> (coeff ` F ) : NN0 --> RR )
51	50	ad2antrr 740	. . . . . . . . . . 11 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ i e. ( 0 ... n ) ) -> (coeff ` F ) : NN0 --> RR )
52		fznn0sub 11857	. . . . . . . . . . . 12 |- ( i e. ( 0 ... n ) -> ( n - i ) e. NN0 )
53	52	adantl 473	. . . . . . . . . . 11 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ i e. ( 0 ... n ) ) -> ( n - i ) e. NN0 )
54	51, 53	ffvelrnd 6038	. . . . . . . . . 10 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ i e. ( 0 ... n ) ) -> ( (coeff ` F ) ` ( n - i ) ) e. RR )
55	54	recnd 9687	. . . . . . . . 9 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ i e. ( 0 ... n ) ) -> ( (coeff ` F ) ` ( n - i ) ) e. CC )
56	55	mulid2d 9679	. . . . . . . 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 9849	. . . . . . . 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 3899	. . . . . . 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 13846	. . . . . 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 2482	. . . . . . 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 2482	. . . . . . 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 6316	. . . . . . . . . . 11 |- ( n = 0 -> ( 0 ... n ) = ( 0 ... 0 ) )
63		0z 10972	. . . . . . . . . . . 12 |- 0 e. ZZ
64		fzsn 11866	. . . . . . . . . . . 12 |- ( 0 e. ZZ -> ( 0 ... 0 ) = { 0 } )
65	63, 64	ax-mp 5	. . . . . . . . . . 11 |- ( 0 ... 0 ) = { 0 }
66	62, 65	syl6eq 2521	. . . . . . . . . 10 |- ( n = 0 -> ( 0 ... n ) = { 0 } )
67		elsni 3985	. . . . . . . . . . . . 13 |- ( i e. { 0 } -> i = 0 )
68	67	adantl 473	. . . . . . . . . . . 12 |- ( ( n = 0 /\ i e. { 0 } ) -> i = 0 )
69		ax-1ne0 9626	. . . . . . . . . . . . . 14 |- 1 =/= 0
70	69	nesymi 2700	. . . . . . . . . . . . 13 |- -. 0 = 1
71		eqeq1 2475	. . . . . . . . . . . . 13 |- ( i = 0 -> ( i = 1 <-> 0 = 1 ) )
72	70, 71	mtbiri 310	. . . . . . . . . . . 12 |- ( i = 0 -> -. i = 1 )
73	68, 72	syl 17	. . . . . . . . . . 11 |- ( ( n = 0 /\ i e. { 0 } ) -> -. i = 1 )
74	47	notbii 303	. . . . . . . . . . . 12 |- ( -. i = 1 <-> -. i e. { 1 } )
75	74	biimpi 199	. . . . . . . . . . 11 |- ( -. i = 1 -> -. i e. { 1 } )
76		iffalse 3881	. . . . . . . . . . 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 13850	. . . . . . . . 9 |- ( n = 0 -> sum_ i e. ( 0 ... n ) if ( i e. { 1 } , ( (coeff ` F ) ` ( n - i ) ) , 0 ) = sum_ i e. { 0 } 0 )
79		snfi 7668	. . . . . . . . . . 11 |- { 0 } e. Fin
80	79	olci 398	. . . . . . . . . 10 |- ( { 0 } C_ ( ZZ>= ` 0 ) \/ { 0 } e. Fin )
81		sumz 13865	. . . . . . . . . 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 2521	. . . . . . . 8 |- ( n = 0 -> sum_ i e. ( 0 ... n ) if ( i e. { 1 } , ( (coeff ` F ) ` ( n - i ) ) , 0 ) = 0 )
84	83	adantl 473	. . . . . . 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 768	. . . . . . . 8 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ -. n = 0 ) -> F e. ( (Poly ` RR ) \ { 0p } ) )
86	33	adantr 472	. . . . . . . . 9 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ -. n = 0 ) -> n e. NN0 )
87		simpr 468	. . . . . . . . . 10 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ -. n = 0 ) -> -. n = 0 )
88	87	neqned 2650	. . . . . . . . 9 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ -. n = 0 ) -> n =/= 0 )
89		elnnne0 10907	. . . . . . . . 9 |- ( n e. NN <-> ( n e. NN0 /\ n =/= 0 ) )
90	86, 88, 89	sylanbrc 677	. . . . . . . 8 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ -. n = 0 ) -> n e. NN )
91		1nn0 10909	. . . . . . . . . . . . 13 |- 1 e. NN0
92	91	a1i 11	. . . . . . . . . . . 12 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> 1 e. NN0 )
93		simpr 468	. . . . . . . . . . . . 13 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> n e. NN )
94	93	nnnn0d 10949	. . . . . . . . . . . 12 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> n e. NN0 )
95	93	nnge1d 10674	. . . . . . . . . . . 12 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> 1 <_ n )
96		elfz2nn0 11911	. . . . . . . . . . . 12 |- ( 1 e. ( 0 ... n ) <-> ( 1 e. NN0 /\ n e. NN0 /\ 1 <_ n ) )
97	92, 94, 95, 96	syl3anbrc 1214	. . . . . . . . . . 11 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> 1 e. ( 0 ... n ) )
98	97	snssd 4108	. . . . . . . . . 10 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> { 1 } C_ ( 0 ... n ) )
99	50	ad2antrr 740	. . . . . . . . . . . . 13 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) /\ i e. { 1 } ) -> (coeff ` F ) : NN0 --> RR )
100		oveq2 6316	. . . . . . . . . . . . . . . 16 |- ( i = 1 -> ( n - i ) = ( n - 1 ) )
101	46, 100	sylbi 200	. . . . . . . . . . . . . . 15 |- ( i e. { 1 } -> ( n - i ) = ( n - 1 ) )
102	101	adantl 473	. . . . . . . . . . . . . 14 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) /\ i e. { 1 } ) -> ( n - i ) = ( n - 1 ) )
103		nnm1nn0 10935	. . . . . . . . . . . . . . 15 |- ( n e. NN -> ( n - 1 ) e. NN0 )
104	103	ad2antlr 741	. . . . . . . . . . . . . 14 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) /\ i e. { 1 } ) -> ( n - 1 ) e. NN0 )
105	102, 104	eqeltrd 2549	. . . . . . . . . . . . 13 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) /\ i e. { 1 } ) -> ( n - i ) e. NN0 )
106	99, 105	ffvelrnd 6038	. . . . . . . . . . . 12 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) /\ i e. { 1 } ) -> ( (coeff ` F ) ` ( n - i ) ) e. RR )
107	106	recnd 9687	. . . . . . . . . . 11 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) /\ i e. { 1 } ) -> ( (coeff ` F ) ` ( n - i ) ) e. CC )
108	107	ralrimiva 2809	. . . . . . . . . 10 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> A. i e. { 1 } ( (coeff ` F ) ` ( n - i ) ) e. CC )
109		fzfi 12223	. . . . . . . . . . . 12 |- ( 0 ... n ) e. Fin
110	109	olci 398	. . . . . . . . . . 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 13869	. . . . . . . . . 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 1291	. . . . . . . . 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 472	. . . . . . . . . . . 12 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> (coeff ` F ) : NN0 --> RR )
115	103	adantl 473	. . . . . . . . . . . 12 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> ( n - 1 ) e. NN0 )
116	114, 115	ffvelrnd 6038	. . . . . . . . . . 11 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> ( (coeff ` F ) ` ( n - 1 ) ) e. RR )
117	116	recnd 9687	. . . . . . . . . 10 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> ( (coeff ` F ) ` ( n - 1 ) ) e. CC )
118	100	fveq2d 5883	. . . . . . . . . . 11 |- ( i = 1 -> ( (coeff ` F ) ` ( n - i ) ) = ( (coeff ` F ) ` ( n - 1 ) ) )
119	118	sumsn 13884	. . . . . . . . . 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 676	. . . . . . . . 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 2507	. . . . . . . 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 673	. . . . . . 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 3907	. . . . . 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 2505	. . . . 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 2509	. . . 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 2505	. . 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 4482	. 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 2505	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 189   \/ wo 375   /\ wa 376   = wceq 1452   e. wcel 1904   =/= wne 2641   A.wral 2756   _Vcvv 3031   \ cdif 3387   C_ wss 3390   ifcif 3872   {csn 3959   class class class wbr 4395   |-> cmpt 4454   -->wf 5585   ` cfv 5589  (class class class)co 6308   oFcof 6548   Fincfn 7587   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558   x. cmul 9562   <_ cle 9694   - cmin 9880   NNcn 10631   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   ...cfz 11810   sum_csu 13829   0pc0p 22706  Polycply 23217   Xpcidp 23218  coeffccoe 23219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-rlim 13630  df-sum 13830  df-0p 22707  df-ply 23221  df-idp 23222  df-coe 23223  df-dgr 23224

This theorem is referenced by:  plymulx  29509