Theorem plymulx0 26960

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 3490	. . . . . 6 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> F e. (Poly ` RR ) )
2		ax-resscn 9351	. . . . . . . 8 |- RR C_ CC
3		1re 9397	. . . . . . . 8 |- 1 e. RR
4		plyid 21689	. . . . . . . 8 |- ( ( RR C_ CC /\ 1 e. RR ) -> Xp e. (Poly ` RR ) )
5	2, 3, 4	mp2an 672	. . . . . . 7 |- Xp e. (Poly ` RR )
6	5	a1i 11	. . . . . 6 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> Xp e. (Poly ` RR ) )
7		simprl 755	. . . . . . 7 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ ( x e. RR /\ y e. RR ) ) -> x e. RR )
8		simprr 756	. . . . . . 7 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ ( x e. RR /\ y e. RR ) ) -> y e. RR )
9	7, 8	readdcld 9425	. . . . . 6 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ ( x e. RR /\ y e. RR ) ) -> ( x + y ) e. RR )
10	7, 8	remulcld 9426	. . . . . 6 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ ( x e. RR /\ y e. RR ) ) -> ( x x. y ) e. RR )
11	1, 6, 9, 10	plymul 21698	. . . . 5 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> ( F oF x. Xp ) e. (Poly ` RR ) )
12		0re 9398	. . . . 5 |- 0 e. RR
13		eqid 2443	. . . . . 6 |- (coeff ` ( F oF x. Xp ) ) = (coeff ` ( F oF x. Xp ) )
14	13	coef2 21711	. . . . 5 |- ( ( ( F oF x. Xp ) e. (Poly ` RR ) /\ 0 e. RR ) -> (coeff ` ( F oF x. Xp ) ) : NN0 --> RR )
15	11, 12, 14	sylancl 662	. . . 4 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> (coeff ` ( F oF x. Xp ) ) : NN0 --> RR )
16		ffn 5571	. . . 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 5749	. . 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 9375	. . . . . . . . 9 |- CC e. _V
21	20	a1i 11	. . . . . . . 8 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> CC e. _V )
22		plyf 21678	. . . . . . . . 9 |- ( F e. (Poly ` RR ) -> F : CC --> CC )
23	1, 22	syl 16	. . . . . . . 8 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> F : CC --> CC )
24		plyf 21678	. . . . . . . . . 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 755	. . . . . . . . 9 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ ( x e. CC /\ y e. CC ) ) -> x e. CC )
28		simprr 756	. . . . . . . . 9 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ ( x e. CC /\ y e. CC ) ) -> y e. CC )
29	27, 28	mulcomd 9419	. . . . . . . 8 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) = ( y x. x ) )
30	21, 23, 26, 29	caofcom 6364	. . . . . . 7 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> ( F oF x. Xp ) = ( Xp oF x. F ) )
31	30	fveq2d 5707	. . . . . 6 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> (coeff ` ( F oF x. Xp ) ) = (coeff ` ( Xp oF x. F ) ) )
32	31	fveq1d 5705	. . . . 5 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> ( (coeff ` ( F oF x. Xp ) ) ` n ) = ( (coeff ` ( Xp oF x. F ) ) ` n ) )
33	32	adantr 465	. . . 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 465	. . . . . 6 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) -> F e. (Poly ` RR ) )
36		simpr 461	. . . . . 6 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) -> n e. NN0 )
37		eqid 2443	. . . . . . 7 |- (coeff ` Xp ) = (coeff ` Xp )
38		eqid 2443	. . . . . . 7 |- (coeff ` F ) = (coeff ` F )
39	37, 38	coemul 21731	. . . . . 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 1218	. . . . 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 11493	. . . . . . . . . 10 |- ( i e. ( 0 ... n ) -> i e. NN0 )
42		coeidp 21742	. . . . . . . . . 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 6118	. . . . . . . 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 6180	. . . . . . . 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 2491	. . . . . . 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 466	. . . . . 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 13192	. . . . 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 3903	. . . . . . . . . 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 21711	. . . . . . . . . . . . 13 |- ( ( F e. (Poly ` RR ) /\ 0 e. RR ) -> (coeff ` F ) : NN0 --> RR )
53	1, 12, 52	sylancl 662	. . . . . . . . . . . 12 |- ( F e. ( (Poly ` RR ) \ { 0p } ) -> (coeff ` F ) : NN0 --> RR )
54	53	ad2antrr 725	. . . . . . . . . . 11 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ i e. ( 0 ... n ) ) -> (coeff ` F ) : NN0 --> RR )
55		fznn0sub 11499	. . . . . . . . . . . 12 |- ( i e. ( 0 ... n ) -> ( n - i ) e. NN0 )
56	55	adantl 466	. . . . . . . . . . 11 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ i e. ( 0 ... n ) ) -> ( n - i ) e. NN0 )
57	54, 56	ffvelrnd 5856	. . . . . . . . . 10 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ i e. ( 0 ... n ) ) -> ( (coeff ` F ) ` ( n - i ) ) e. RR )
58	2, 57	sseldi 3366	. . . . . . . . 9 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ i e. ( 0 ... n ) ) -> ( (coeff ` F ) ` ( n - i ) ) e. CC )
59	58	mulid2d 9416	. . . . . . . 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 9579	. . . . . . . 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 3828	. . . . . . 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 13192	. . . . . 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 2452	. . . . . . 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 2452	. . . . . . 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 6111	. . . . . . . . . . 11 |- ( n = 0 -> ( 0 ... n ) = ( 0 ... 0 ) )
66		0z 10669	. . . . . . . . . . . 12 |- 0 e. ZZ
67		fzsn 11512	. . . . . . . . . . . 12 |- ( 0 e. ZZ -> ( 0 ... 0 ) = { 0 } )
68	66, 67	ax-mp 5	. . . . . . . . . . 11 |- ( 0 ... 0 ) = { 0 }
69	65, 68	syl6eq 2491	. . . . . . . . . 10 |- ( n = 0 -> ( 0 ... n ) = { 0 } )
70		elsni 3914	. . . . . . . . . . . . 13 |- ( i e. { 0 } -> i = 0 )
71	70	adantl 466	. . . . . . . . . . . 12 |- ( ( n = 0 /\ i e. { 0 } ) -> i = 0 )
72		ax-1ne0 9363	. . . . . . . . . . . . . 14 |- 1 =/= 0
73	72	nesymi 2660	. . . . . . . . . . . . 13 |- -. 0 = 1
74		eqeq1 2449	. . . . . . . . . . . . 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 3811	. . . . . . . . . . 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 13196	. . . . . . . . 9 |- ( n = 0 -> sum_ i e. ( 0 ... n ) if ( i e. { 1 } , ( (coeff ` F ) ` ( n - i ) ) , 0 ) = sum_ i e. { 0 } 0 )
82		snfi 7402	. . . . . . . . . . 11 |- { 0 } e. Fin
83	82	olci 391	. . . . . . . . . 10 |- ( { 0 } C_ ( ZZ>= ` 0 ) \/ { 0 } e. Fin )
84		sumz 13211	. . . . . . . . . 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 2491	. . . . . . . 8 |- ( n = 0 -> sum_ i e. ( 0 ... n ) if ( i e. { 1 } , ( (coeff ` F ) ` ( n - i ) ) , 0 ) = 0 )
87	86	adantl 466	. . . . . . 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 753	. . . . . . . 8 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ -. n = 0 ) -> F e. ( (Poly ` RR ) \ { 0p } ) )
89	36	adantr 465	. . . . . . . . . 10 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ -. n = 0 ) -> n e. NN0 )
90		simpr 461	. . . . . . . . . . 11 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ -. n = 0 ) -> -. n = 0 )
91	90	neneqad 2693	. . . . . . . . . 10 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ -. n = 0 ) -> n =/= 0 )
92	89, 91	jca 532	. . . . . . . . 9 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN0 ) /\ -. n = 0 ) -> ( n e. NN0 /\ n =/= 0 ) )
93		elnnne0 10605	. . . . . . . . 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 10607	. . . . . . . . . . . . 13 |- 1 e. NN0
96	95	a1i 11	. . . . . . . . . . . 12 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> 1 e. NN0 )
97		simpr 461	. . . . . . . . . . . . 13 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> n e. NN )
98	97	nnnn0d 10648	. . . . . . . . . . . 12 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> n e. NN0 )
99		nnge1 10360	. . . . . . . . . . . . 13 |- ( n e. NN -> 1 <_ n )
100	97, 99	syl 16	. . . . . . . . . . . 12 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> 1 <_ n )
101		elfz2nn0 11492	. . . . . . . . . . . 12 |- ( 1 e. ( 0 ... n ) <-> ( 1 e. NN0 /\ n e. NN0 /\ 1 <_ n ) )
102	96, 98, 100, 101	syl3anbrc 1172	. . . . . . . . . . 11 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> 1 e. ( 0 ... n ) )
103	3	elexi 2994	. . . . . . . . . . . 12 |- 1 e. _V
104	103	snss 4011	. . . . . . . . . . 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 725	. . . . . . . . . . . . 13 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) /\ i e. { 1 } ) -> (coeff ` F ) : NN0 --> RR )
107		oveq2 6111	. . . . . . . . . . . . . . . 16 |- ( i = 1 -> ( n - i ) = ( n - 1 ) )
108	49, 107	sylbi 195	. . . . . . . . . . . . . . 15 |- ( i e. { 1 } -> ( n - i ) = ( n - 1 ) )
109	108	adantl 466	. . . . . . . . . . . . . 14 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) /\ i e. { 1 } ) -> ( n - i ) = ( n - 1 ) )
110		nnm1nn0 10633	. . . . . . . . . . . . . . 15 |- ( n e. NN -> ( n - 1 ) e. NN0 )
111	110	ad2antlr 726	. . . . . . . . . . . . . 14 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) /\ i e. { 1 } ) -> ( n - 1 ) e. NN0 )
112	109, 111	eqeltrd 2517	. . . . . . . . . . . . 13 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) /\ i e. { 1 } ) -> ( n - i ) e. NN0 )
113	106, 112	ffvelrnd 5856	. . . . . . . . . . . 12 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) /\ i e. { 1 } ) -> ( (coeff ` F ) ` ( n - i ) ) e. RR )
114	2, 113	sseldi 3366	. . . . . . . . . . 11 |- ( ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) /\ i e. { 1 } ) -> ( (coeff ` F ) ` ( n - i ) ) e. CC )
115	114	ralrimiva 2811	. . . . . . . . . 10 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> A. i e. { 1 } ( (coeff ` F ) ` ( n - i ) ) e. CC )
116		fzfi 11806	. . . . . . . . . . . 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 13215	. . . . . . . . . 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 1217	. . . . . . . . 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 465	. . . . . . . . . . . 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 5856	. . . . . . . . . . 11 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> ( (coeff ` F ) ` ( n - 1 ) ) e. RR )
124	2, 123	sseldi 3366	. . . . . . . . . 10 |- ( ( F e. ( (Poly ` RR ) \ { 0p } ) /\ n e. NN ) -> ( (coeff ` F ) ` ( n - 1 ) ) e. CC )
125	107	fveq2d 5707	. . . . . . . . . . 11 |- ( i = 1 -> ( (coeff ` F ) ` ( n - i ) ) = ( (coeff ` F ) ` ( n - 1 ) ) )
126	125	sumsn 13229	. . . . . . . . . 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 663	. . . . . . . . 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 2477	. . . . . . . 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 661	. . . . . . 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 3836	. . . . . 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 2475	. . . . 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 2479	. . . 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 2475	. . 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 4390	. 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 2475	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 1369   e. wcel 1756   =/= wne 2618   A.wral 2727   _Vcvv 2984   \ cdif 3337   C_ wss 3340   ifcif 3803   {csn 3889   class class class wbr 4304   e. cmpt 4362   Fn wfn 5425   -->wf 5426   ` cfv 5430  (class class class)co 6103   oFcof 6330   Fincfn 7322   CCcc 9292   RRcr 9293   0cc0 9294   1c1 9295   x. cmul 9299   <_ cle 9431   - cmin 9607   NNcn 10334   NN0cn0 10591   ZZcz 10658   ZZ>=cuz 10873   ...cfz 11449   sum_csu 13175   0pc0p 21159  Polycply 21664   Xpcidp 21665  coeffccoe 21666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-inf2 7859  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372  ax-addf 9373

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-se 4692  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-isom 5439  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-of 6332  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-oadd 6936  df-er 7113  df-map 7228  df-pm 7229  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-sup 7703  df-oi 7736  df-card 8121  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-3 10393  df-n0 10592  df-z 10659  df-uz 10874  df-rp 11004  df-fz 11450  df-fzo 11561  df-fl 11654  df-seq 11819  df-exp 11878  df-hash 12116  df-cj 12600  df-re 12601  df-im 12602  df-sqr 12736  df-abs 12737  df-clim 12978  df-rlim 12979  df-sum 13176  df-0p 21160  df-ply 21668  df-idp 21669  df-coe 21670  df-dgr 21671

This theorem is referenced by:  plymulx  26961