Theorem signsvfn 28179

Description: Number of changes in a word compared to a shorter word. (Contributed by Thierry Arnoux, 12-Oct-2018.)

Hypotheses

Ref	Expression
signsv.p	|- .+^ = ( a e. { -u 1 , 0 , 1 } , b e. { -u 1 , 0 , 1 } |-> if ( b = 0 , a , b ) )
signsv.w	|- W = { <. ( Base ` ndx ) , { -u 1 , 0 , 1 } >. , <. ( +g ` ndx ) , .+^ >. }
signsv.t	|- T = ( f e. Word RR |-> ( n e. ( 0..^ ( # ` f ) ) |-> ( W gsumg ( i e. ( 0 ... n ) |-> (sgn ` ( f ` i ) ) ) ) ) )
signsv.v	|- V = ( f e. Word RR |-> sum_ j e. ( 1..^ ( # ` f ) ) if ( ( ( T ` f ) ` j ) =/= ( ( T ` f ) ` ( j - 1 ) ) , 1 , 0 ) )

Assertion

Ref	Expression
signsvfn	|- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( V ` ( F concat <" K "> ) ) = ( ( V ` F ) + if ( ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) x. K ) < 0 , 1 , 0 ) ) )

Distinct variable groups:   a, b, .+^   f, i, n, F   f, K, i, n   f, W, i, n   i, a, j, n, F, b   K, a, b, j, f   T, a   f, b, T, j, n

Allowed substitution hints:   .+^ ( f, i, j, n)   T( i)   V( f, i, j, n, a, b)   W( j, a, b)

Proof of Theorem signsvfn

Step	Hyp	Ref	Expression
1		simpl 457	. . . . . . 7 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> F e. (Word RR \ { (/) } ) )
2	1	eldifad 3488	. . . . . 6 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> F e. Word RR )
3		simpr 461	. . . . . . 7 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> K e. RR )
4	3	s1cld 12574	. . . . . 6 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> <" K "> e. Word RR )
5		ccatcl 12554	. . . . . 6 |- ( ( F e. Word RR /\ <" K "> e. Word RR ) -> ( F concat <" K "> ) e. Word RR )
6	2, 4, 5	syl2anc 661	. . . . 5 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( F concat <" K "> ) e. Word RR )
7		signsv.p	. . . . . 6 |- .+^ = ( a e. { -u 1 , 0 , 1 } , b e. { -u 1 , 0 , 1 } |-> if ( b = 0 , a , b ) )
8		signsv.w	. . . . . 6 |- W = { <. ( Base ` ndx ) , { -u 1 , 0 , 1 } >. , <. ( +g ` ndx ) , .+^ >. }
9		signsv.t	. . . . . 6 |- T = ( f e. Word RR |-> ( n e. ( 0..^ ( # ` f ) ) |-> ( W gsumg ( i e. ( 0 ... n ) |-> (sgn ` ( f ` i ) ) ) ) ) )
10		signsv.v	. . . . . 6 |- V = ( f e. Word RR |-> sum_ j e. ( 1..^ ( # ` f ) ) if ( ( ( T ` f ) ` j ) =/= ( ( T ` f ) ` ( j - 1 ) ) , 1 , 0 ) )
11	7, 8, 9, 10	signsvvfval 28175	. . . . 5 |- ( ( F concat <" K "> ) e. Word RR -> ( V ` ( F concat <" K "> ) ) = sum_ j e. ( 1..^ ( # ` ( F concat <" K "> ) ) ) if ( ( ( T ` ( F concat <" K "> ) ) ` j ) =/= ( ( T ` ( F concat <" K "> ) ) ` ( j - 1 ) ) , 1 , 0 ) )
12	6, 11	syl 16	. . . 4 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( V ` ( F concat <" K "> ) ) = sum_ j e. ( 1..^ ( # ` ( F concat <" K "> ) ) ) if ( ( ( T ` ( F concat <" K "> ) ) ` j ) =/= ( ( T ` ( F concat <" K "> ) ) ` ( j - 1 ) ) , 1 , 0 ) )
13		ccatlen 12555	. . . . . . . 8 |- ( ( F e. Word RR /\ <" K "> e. Word RR ) -> ( # ` ( F concat <" K "> ) ) = ( ( # ` F ) + ( # ` <" K "> ) ) )
14	2, 4, 13	syl2anc 661	. . . . . . 7 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( # ` ( F concat <" K "> ) ) = ( ( # ` F ) + ( # ` <" K "> ) ) )
15		s1len 12576	. . . . . . . 8 |- ( # ` <" K "> ) = 1
16	15	oveq2i 6293	. . . . . . 7 |- ( ( # ` F ) + ( # ` <" K "> ) ) = ( ( # ` F ) + 1 )
17	14, 16	syl6eq 2524	. . . . . 6 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( # ` ( F concat <" K "> ) ) = ( ( # ` F ) + 1 ) )
18	17	oveq2d 6298	. . . . 5 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( 1..^ ( # ` ( F concat <" K "> ) ) ) = ( 1..^ ( ( # ` F ) + 1 ) ) )
19	18	sumeq1d 13482	. . . 4 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> sum_ j e. ( 1..^ ( # ` ( F concat <" K "> ) ) ) if ( ( ( T ` ( F concat <" K "> ) ) ` j ) =/= ( ( T ` ( F concat <" K "> ) ) ` ( j - 1 ) ) , 1 , 0 ) = sum_ j e. ( 1..^ ( ( # ` F ) + 1 ) ) if ( ( ( T ` ( F concat <" K "> ) ) ` j ) =/= ( ( T ` ( F concat <" K "> ) ) ` ( j - 1 ) ) , 1 , 0 ) )
20		eldifsn 4152	. . . . . . . 8 |- ( F e. (Word RR \ { (/) } ) <-> ( F e. Word RR /\ F =/= (/) ) )
21		lennncl 12525	. . . . . . . 8 |- ( ( F e. Word RR /\ F =/= (/) ) -> ( # ` F ) e. NN )
22	20, 21	sylbi 195	. . . . . . 7 |- ( F e. (Word RR \ { (/) } ) -> ( # ` F ) e. NN )
23		nnuz 11113	. . . . . . 7 |- NN = ( ZZ>= ` 1 )
24	22, 23	syl6eleq 2565	. . . . . 6 |- ( F e. (Word RR \ { (/) } ) -> ( # ` F ) e. ( ZZ>= ` 1 ) )
25	24	adantr 465	. . . . 5 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( # ` F ) e. ( ZZ>= ` 1 ) )
26		ax-1cn 9546	. . . . . . 7 |- 1 e. CC
27	26	a1i 11	. . . . . 6 |- ( ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1 ... ( # ` F ) ) ) /\ ( ( T ` ( F concat <" K "> ) ) ` j ) =/= ( ( T ` ( F concat <" K "> ) ) ` ( j - 1 ) ) ) -> 1 e. CC )
28		0cnd 9585	. . . . . 6 |- ( ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1 ... ( # ` F ) ) ) /\ -. ( ( T ` ( F concat <" K "> ) ) ` j ) =/= ( ( T ` ( F concat <" K "> ) ) ` ( j - 1 ) ) ) -> 0 e. CC )
29	27, 28	ifclda 3971	. . . . 5 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1 ... ( # ` F ) ) ) -> if ( ( ( T ` ( F concat <" K "> ) ) ` j ) =/= ( ( T ` ( F concat <" K "> ) ) ` ( j - 1 ) ) , 1 , 0 ) e. CC )
30		fveq2 5864	. . . . . . 7 |- ( j = ( # ` F ) -> ( ( T ` ( F concat <" K "> ) ) ` j ) = ( ( T ` ( F concat <" K "> ) ) ` ( # ` F ) ) )
31		oveq1 6289	. . . . . . . 8 |- ( j = ( # ` F ) -> ( j - 1 ) = ( ( # ` F ) - 1 ) )
32	31	fveq2d 5868	. . . . . . 7 |- ( j = ( # ` F ) -> ( ( T ` ( F concat <" K "> ) ) ` ( j - 1 ) ) = ( ( T ` ( F concat <" K "> ) ) ` ( ( # ` F ) - 1 ) ) )
33	30, 32	neeq12d 2746	. . . . . 6 |- ( j = ( # ` F ) -> ( ( ( T ` ( F concat <" K "> ) ) ` j ) =/= ( ( T ` ( F concat <" K "> ) ) ` ( j - 1 ) ) <-> ( ( T ` ( F concat <" K "> ) ) ` ( # ` F ) ) =/= ( ( T ` ( F concat <" K "> ) ) ` ( ( # ` F ) - 1 ) ) ) )
34	33	ifbid 3961	. . . . 5 |- ( j = ( # ` F ) -> if ( ( ( T ` ( F concat <" K "> ) ) ` j ) =/= ( ( T ` ( F concat <" K "> ) ) ` ( j - 1 ) ) , 1 , 0 ) = if ( ( ( T ` ( F concat <" K "> ) ) ` ( # ` F ) ) =/= ( ( T ` ( F concat <" K "> ) ) ` ( ( # ` F ) - 1 ) ) , 1 , 0 ) )
35	25, 29, 34	fzosump1 13526	. . . 4 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> sum_ j e. ( 1..^ ( ( # ` F ) + 1 ) ) if ( ( ( T ` ( F concat <" K "> ) ) ` j ) =/= ( ( T ` ( F concat <" K "> ) ) ` ( j - 1 ) ) , 1 , 0 ) = ( sum_ j e. ( 1..^ ( # ` F ) ) if ( ( ( T ` ( F concat <" K "> ) ) ` j ) =/= ( ( T ` ( F concat <" K "> ) ) ` ( j - 1 ) ) , 1 , 0 ) + if ( ( ( T ` ( F concat <" K "> ) ) ` ( # ` F ) ) =/= ( ( T ` ( F concat <" K "> ) ) ` ( ( # ` F ) - 1 ) ) , 1 , 0 ) ) )
36	12, 19, 35	3eqtrd 2512	. . 3 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( V ` ( F concat <" K "> ) ) = ( sum_ j e. ( 1..^ ( # ` F ) ) if ( ( ( T ` ( F concat <" K "> ) ) ` j ) =/= ( ( T ` ( F concat <" K "> ) ) ` ( j - 1 ) ) , 1 , 0 ) + if ( ( ( T ` ( F concat <" K "> ) ) ` ( # ` F ) ) =/= ( ( T ` ( F concat <" K "> ) ) ` ( ( # ` F ) - 1 ) ) , 1 , 0 ) ) )
37	36	adantlr 714	. 2 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( V ` ( F concat <" K "> ) ) = ( sum_ j e. ( 1..^ ( # ` F ) ) if ( ( ( T ` ( F concat <" K "> ) ) ` j ) =/= ( ( T ` ( F concat <" K "> ) ) ` ( j - 1 ) ) , 1 , 0 ) + if ( ( ( T ` ( F concat <" K "> ) ) ` ( # ` F ) ) =/= ( ( T ` ( F concat <" K "> ) ) ` ( ( # ` F ) - 1 ) ) , 1 , 0 ) ) )
38	2	adantr 465	. . . . . . . . 9 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> F e. Word RR )
39	3	adantr 465	. . . . . . . . 9 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> K e. RR )
40		1nn0 10807	. . . . . . . . . . . . 13 |- 1 e. NN0
41		nn0uz 11112	. . . . . . . . . . . . 13 |- NN0 = ( ZZ>= ` 0 )
42	40, 41	eleqtri 2553	. . . . . . . . . . . 12 |- 1 e. ( ZZ>= ` 0 )
43		fzoss1 11816	. . . . . . . . . . . 12 |- ( 1 e. ( ZZ>= ` 0 ) -> ( 1..^ ( # ` F ) ) C_ ( 0..^ ( # ` F ) ) )
44	42, 43	ax-mp 5	. . . . . . . . . . 11 |- ( 1..^ ( # ` F ) ) C_ ( 0..^ ( # ` F ) )
45	44	a1i 11	. . . . . . . . . 10 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( 1..^ ( # ` F ) ) C_ ( 0..^ ( # ` F ) ) )
46	45	sselda 3504	. . . . . . . . 9 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> j e. ( 0..^ ( # ` F ) ) )
47	7, 8, 9, 10	signstfvp 28168	. . . . . . . . 9 |- ( ( F e. Word RR /\ K e. RR /\ j e. ( 0..^ ( # ` F ) ) ) -> ( ( T ` ( F concat <" K "> ) ) ` j ) = ( ( T ` F ) ` j ) )
48	38, 39, 46, 47	syl3anc 1228	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> ( ( T ` ( F concat <" K "> ) ) ` j ) = ( ( T ` F ) ` j ) )
49		elfzoel2 11792	. . . . . . . . . . . . 13 |- ( j e. ( 1..^ ( # ` F ) ) -> ( # ` F ) e. ZZ )
50	49	adantl 466	. . . . . . . . . . . 12 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> ( # ` F ) e. ZZ )
51		eluzmn 28131	. . . . . . . . . . . 12 |- ( ( ( # ` F ) e. ZZ /\ 1 e. NN0 ) -> ( # ` F ) e. ( ZZ>= ` ( ( # ` F ) - 1 ) ) )
52	50, 40, 51	sylancl 662	. . . . . . . . . . 11 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> ( # ` F ) e. ( ZZ>= ` ( ( # ` F ) - 1 ) ) )
53		fzoss2 11817	. . . . . . . . . . 11 |- ( ( # ` F ) e. ( ZZ>= ` ( ( # ` F ) - 1 ) ) -> ( 0..^ ( ( # ` F ) - 1 ) ) C_ ( 0..^ ( # ` F ) ) )
54	52, 53	syl 16	. . . . . . . . . 10 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> ( 0..^ ( ( # ` F ) - 1 ) ) C_ ( 0..^ ( # ` F ) ) )
55		simpr 461	. . . . . . . . . . 11 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> j e. ( 1..^ ( # ` F ) ) )
56		elfzoelz 11793	. . . . . . . . . . . . 13 |- ( j e. ( 1..^ ( # ` F ) ) -> j e. ZZ )
57	56	adantl 466	. . . . . . . . . . . 12 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> j e. ZZ )
58		elfzom1b 11875	. . . . . . . . . . . 12 |- ( ( j e. ZZ /\ ( # ` F ) e. ZZ ) -> ( j e. ( 1..^ ( # ` F ) ) <-> ( j - 1 ) e. ( 0..^ ( ( # ` F ) - 1 ) ) ) )
59	57, 50, 58	syl2anc 661	. . . . . . . . . . 11 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> ( j e. ( 1..^ ( # ` F ) ) <-> ( j - 1 ) e. ( 0..^ ( ( # ` F ) - 1 ) ) ) )
60	55, 59	mpbid 210	. . . . . . . . . 10 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> ( j - 1 ) e. ( 0..^ ( ( # ` F ) - 1 ) ) )
61	54, 60	sseldd 3505	. . . . . . . . 9 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> ( j - 1 ) e. ( 0..^ ( # ` F ) ) )
62	7, 8, 9, 10	signstfvp 28168	. . . . . . . . 9 |- ( ( F e. Word RR /\ K e. RR /\ ( j - 1 ) e. ( 0..^ ( # ` F ) ) ) -> ( ( T ` ( F concat <" K "> ) ) ` ( j - 1 ) ) = ( ( T ` F ) ` ( j - 1 ) ) )
63	38, 39, 61, 62	syl3anc 1228	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> ( ( T ` ( F concat <" K "> ) ) ` ( j - 1 ) ) = ( ( T ` F ) ` ( j - 1 ) ) )
64	48, 63	neeq12d 2746	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> ( ( ( T ` ( F concat <" K "> ) ) ` j ) =/= ( ( T ` ( F concat <" K "> ) ) ` ( j - 1 ) ) <-> ( ( T ` F ) ` j ) =/= ( ( T ` F ) ` ( j - 1 ) ) ) )
65	64	ifbid 3961	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> if ( ( ( T ` ( F concat <" K "> ) ) ` j ) =/= ( ( T ` ( F concat <" K "> ) ) ` ( j - 1 ) ) , 1 , 0 ) = if ( ( ( T ` F ) ` j ) =/= ( ( T ` F ) ` ( j - 1 ) ) , 1 , 0 ) )
66	65	sumeq2dv 13484	. . . . 5 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> sum_ j e. ( 1..^ ( # ` F ) ) if ( ( ( T ` ( F concat <" K "> ) ) ` j ) =/= ( ( T ` ( F concat <" K "> ) ) ` ( j - 1 ) ) , 1 , 0 ) = sum_ j e. ( 1..^ ( # ` F ) ) if ( ( ( T ` F ) ` j ) =/= ( ( T ` F ) ` ( j - 1 ) ) , 1 , 0 ) )
67	7, 8, 9, 10	signsvvfval 28175	. . . . . 6 |- ( F e. Word RR -> ( V ` F ) = sum_ j e. ( 1..^ ( # ` F ) ) if ( ( ( T ` F ) ` j ) =/= ( ( T ` F ) ` ( j - 1 ) ) , 1 , 0 ) )
68	2, 67	syl 16	. . . . 5 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( V ` F ) = sum_ j e. ( 1..^ ( # ` F ) ) if ( ( ( T ` F ) ` j ) =/= ( ( T ` F ) ` ( j - 1 ) ) , 1 , 0 ) )
69	66, 68	eqtr4d 2511	. . . 4 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> sum_ j e. ( 1..^ ( # ` F ) ) if ( ( ( T ` ( F concat <" K "> ) ) ` j ) =/= ( ( T ` ( F concat <" K "> ) ) ` ( j - 1 ) ) , 1 , 0 ) = ( V ` F ) )
70	69	adantlr 714	. . 3 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> sum_ j e. ( 1..^ ( # ` F ) ) if ( ( ( T ` ( F concat <" K "> ) ) ` j ) =/= ( ( T ` ( F concat <" K "> ) ) ` ( j - 1 ) ) , 1 , 0 ) = ( V ` F ) )
71	7, 8, 9, 10	signstfvn 28166	. . . . . . 7 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( ( T ` ( F concat <" K "> ) ) ` ( # ` F ) ) = ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) .+^ (sgn ` K ) ) )
72	71	adantlr 714	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( ( T ` ( F concat <" K "> ) ) ` ( # ` F ) ) = ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) .+^ (sgn ` K ) ) )
73	2	adantlr 714	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> F e. Word RR )
74		simpr 461	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> K e. RR )
75	22	ad2antrr 725	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( # ` F ) e. NN )
76		fzo0end 11868	. . . . . . . 8 |- ( ( # ` F ) e. NN -> ( ( # ` F ) - 1 ) e. ( 0..^ ( # ` F ) ) )
77	75, 76	syl 16	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( ( # ` F ) - 1 ) e. ( 0..^ ( # ` F ) ) )
78	7, 8, 9, 10	signstfvp 28168	. . . . . . 7 |- ( ( F e. Word RR /\ K e. RR /\ ( ( # ` F ) - 1 ) e. ( 0..^ ( # ` F ) ) ) -> ( ( T ` ( F concat <" K "> ) ) ` ( ( # ` F ) - 1 ) ) = ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) )
79	73, 74, 77, 78	syl3anc 1228	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( ( T ` ( F concat <" K "> ) ) ` ( ( # ` F ) - 1 ) ) = ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) )
80	72, 79	neeq12d 2746	. . . . 5 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( ( ( T ` ( F concat <" K "> ) ) ` ( # ` F ) ) =/= ( ( T ` ( F concat <" K "> ) ) ` ( ( # ` F ) - 1 ) ) <-> ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) .+^ (sgn ` K ) ) =/= ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) ) )
81	7, 8, 9, 10	signstfvcl 28170	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( ( # ` F ) - 1 ) e. ( 0..^ ( # ` F ) ) ) -> ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) e. { -u 1 , 1 } )
82	77, 81	syldan 470	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) e. { -u 1 , 1 } )
83	74	rexrd 9639	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> K e. RR* )
84		sgncl 28117	. . . . . . 7 |- ( K e. RR* -> (sgn ` K ) e. { -u 1 , 0 , 1 } )
85	83, 84	syl 16	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> (sgn ` K ) e. { -u 1 , 0 , 1 } )
86	7, 8	signswch 28158	. . . . . 6 |- ( ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) e. { -u 1 , 1 } /\ (sgn ` K ) e. { -u 1 , 0 , 1 } ) -> ( ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) .+^ (sgn ` K ) ) =/= ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) <-> ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) x. (sgn ` K ) ) < 0 ) )
87	82, 85, 86	syl2anc 661	. . . . 5 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) .+^ (sgn ` K ) ) =/= ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) <-> ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) x. (sgn ` K ) ) < 0 ) )
88		sgnsgn 28127	. . . . . . . . 9 |- ( K e. RR* -> (sgn ` (sgn ` K ) ) = (sgn ` K ) )
89	83, 88	syl 16	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> (sgn ` (sgn ` K ) ) = (sgn ` K ) )
90	89	oveq2d 6298	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( (sgn ` ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) ) x. (sgn ` (sgn ` K ) ) ) = ( (sgn ` ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) ) x. (sgn ` K ) ) )
91	90	breq1d 4457	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( ( (sgn ` ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) ) x. (sgn ` (sgn ` K ) ) ) < 0 <-> ( (sgn ` ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) ) x. (sgn ` K ) ) < 0 ) )
92		neg1rr 10636	. . . . . . . . 9 |- -u 1 e. RR
93		1re 9591	. . . . . . . . 9 |- 1 e. RR
94		prssi 4183	. . . . . . . . 9 |- ( ( -u 1 e. RR /\ 1 e. RR ) -> { -u 1 , 1 } C_ RR )
95	92, 93, 94	mp2an 672	. . . . . . . 8 |- { -u 1 , 1 } C_ RR
96	95, 82	sseldi 3502	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) e. RR )
97		sgnclre 28118	. . . . . . . 8 |- ( K e. RR -> (sgn ` K ) e. RR )
98	74, 97	syl 16	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> (sgn ` K ) e. RR )
99		sgnmulsgn 28128	. . . . . . 7 |- ( ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) e. RR /\ (sgn ` K ) e. RR ) -> ( ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) x. (sgn ` K ) ) < 0 <-> ( (sgn ` ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) ) x. (sgn ` (sgn ` K ) ) ) < 0 ) )
100	96, 98, 99	syl2anc 661	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) x. (sgn ` K ) ) < 0 <-> ( (sgn ` ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) ) x. (sgn ` (sgn ` K ) ) ) < 0 ) )
101		sgnmulsgn 28128	. . . . . . 7 |- ( ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) e. RR /\ K e. RR ) -> ( ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) x. K ) < 0 <-> ( (sgn ` ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) ) x. (sgn ` K ) ) < 0 ) )
102	96, 74, 101	syl2anc 661	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) x. K ) < 0 <-> ( (sgn ` ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) ) x. (sgn ` K ) ) < 0 ) )
103	91, 100, 102	3bitr4d 285	. . . . 5 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) x. (sgn ` K ) ) < 0 <-> ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) x. K ) < 0 ) )
104	80, 87, 103	3bitrd 279	. . . 4 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( ( ( T ` ( F concat <" K "> ) ) ` ( # ` F ) ) =/= ( ( T ` ( F concat <" K "> ) ) ` ( ( # ` F ) - 1 ) ) <-> ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) x. K ) < 0 ) )
105	104	ifbid 3961	. . 3 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> if ( ( ( T ` ( F concat <" K "> ) ) ` ( # ` F ) ) =/= ( ( T ` ( F concat <" K "> ) ) ` ( ( # ` F ) - 1 ) ) , 1 , 0 ) = if ( ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) x. K ) < 0 , 1 , 0 ) )
106	70, 105	oveq12d 6300	. 2 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( sum_ j e. ( 1..^ ( # ` F ) ) if ( ( ( T ` ( F concat <" K "> ) ) ` j ) =/= ( ( T ` ( F concat <" K "> ) ) ` ( j - 1 ) ) , 1 , 0 ) + if ( ( ( T ` ( F concat <" K "> ) ) ` ( # ` F ) ) =/= ( ( T ` ( F concat <" K "> ) ) ` ( ( # ` F ) - 1 ) ) , 1 , 0 ) ) = ( ( V ` F ) + if ( ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) x. K ) < 0 , 1 , 0 ) ) )
107	37, 106	eqtrd 2508	1 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( V ` ( F concat <" K "> ) ) = ( ( V ` F ) + if ( ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) x. K ) < 0 , 1 , 0 ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   e. wcel 1767   =/= wne 2662   \ cdif 3473   C_ wss 3476   (/)c0 3785   ifcif 3939   {csn 4027   {cpr 4029   {ctp 4031   <.cop 4033   class class class wbr 4447   |-> cmpt 4505   ` cfv 5586  (class class class)co 6282   |-> cmpt2 6284   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489   + caddc 9491   x. cmul 9493   RR*cxr 9623   < clt 9624   - cmin 9801   -ucneg 9802   NNcn 10532   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11078   ...cfz 11668  ..^cfzo 11788   #chash 12369  Word cword 12496   concat cconcat 12498   <"cs1 12499  sgncsgn 12878   sum_csu 13467   ndxcnx 14483   Basecbs 14486   +g cplusg 14551   gsum_g cgsu 14692

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-fz 11669  df-fzo 11789  df-seq 12072  df-exp 12131  df-hash 12370  df-word 12504  df-concat 12506  df-s1 12507  df-substr 12508  df-sgn 12879  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270  df-sum 13468  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-plusg 14564  df-0g 14693  df-gsum 14694  df-mnd 15728  df-mulg 15861  df-cntz 16150

This theorem is referenced by:  signsvtp  28180  signsvtn  28181  signlem0  28184