Theorem signsvfn 28714

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 ++ <" 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 3483	. . . . . 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 12623	. . . . . 6 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> <" K "> e. Word RR )
5		ccatcl 12601	. . . . . 6 |- ( ( F e. Word RR /\ <" K "> e. Word RR ) -> ( F ++ <" K "> ) e. Word RR )
6	2, 4, 5	syl2anc 661	. . . . 5 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( F ++ <" 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 28710	. . . . 5 |- ( ( F ++ <" K "> ) e. Word RR -> ( V ` ( F ++ <" K "> ) ) = sum_ j e. ( 1..^ ( # ` ( F ++ <" K "> ) ) ) if ( ( ( T ` ( F ++ <" K "> ) ) ` j ) =/= ( ( T ` ( F ++ <" K "> ) ) ` ( j - 1 ) ) , 1 , 0 ) )
12	6, 11	syl 16	. . . 4 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( V ` ( F ++ <" K "> ) ) = sum_ j e. ( 1..^ ( # ` ( F ++ <" K "> ) ) ) if ( ( ( T ` ( F ++ <" K "> ) ) ` j ) =/= ( ( T ` ( F ++ <" K "> ) ) ` ( j - 1 ) ) , 1 , 0 ) )
13		ccatlen 12602	. . . . . . . 8 |- ( ( F e. Word RR /\ <" K "> e. Word RR ) -> ( # ` ( F ++ <" K "> ) ) = ( ( # ` F ) + ( # ` <" K "> ) ) )
14	2, 4, 13	syl2anc 661	. . . . . . 7 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( # ` ( F ++ <" K "> ) ) = ( ( # ` F ) + ( # ` <" K "> ) ) )
15		s1len 12625	. . . . . . . 8 |- ( # ` <" K "> ) = 1
16	15	oveq2i 6307	. . . . . . 7 |- ( ( # ` F ) + ( # ` <" K "> ) ) = ( ( # ` F ) + 1 )
17	14, 16	syl6eq 2514	. . . . . 6 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( # ` ( F ++ <" K "> ) ) = ( ( # ` F ) + 1 ) )
18	17	oveq2d 6312	. . . . 5 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( 1..^ ( # ` ( F ++ <" K "> ) ) ) = ( 1..^ ( ( # ` F ) + 1 ) ) )
19	18	sumeq1d 13534	. . . 4 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> sum_ j e. ( 1..^ ( # ` ( F ++ <" K "> ) ) ) if ( ( ( T ` ( F ++ <" K "> ) ) ` j ) =/= ( ( T ` ( F ++ <" K "> ) ) ` ( j - 1 ) ) , 1 , 0 ) = sum_ j e. ( 1..^ ( ( # ` F ) + 1 ) ) if ( ( ( T ` ( F ++ <" K "> ) ) ` j ) =/= ( ( T ` ( F ++ <" K "> ) ) ` ( j - 1 ) ) , 1 , 0 ) )
20		eldifsn 4157	. . . . . . . 8 |- ( F e. (Word RR \ { (/) } ) <-> ( F e. Word RR /\ F =/= (/) ) )
21		lennncl 12569	. . . . . . . 8 |- ( ( F e. Word RR /\ F =/= (/) ) -> ( # ` F ) e. NN )
22	20, 21	sylbi 195	. . . . . . 7 |- ( F e. (Word RR \ { (/) } ) -> ( # ` F ) e. NN )
23		nnuz 11141	. . . . . . 7 |- NN = ( ZZ>= ` 1 )
24	22, 23	syl6eleq 2555	. . . . . 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		1cnd 9629	. . . . . 6 |- ( ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1 ... ( # ` F ) ) ) /\ ( ( T ` ( F ++ <" K "> ) ) ` j ) =/= ( ( T ` ( F ++ <" K "> ) ) ` ( j - 1 ) ) ) -> 1 e. CC )
27		0cnd 9606	. . . . . 6 |- ( ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1 ... ( # ` F ) ) ) /\ -. ( ( T ` ( F ++ <" K "> ) ) ` j ) =/= ( ( T ` ( F ++ <" K "> ) ) ` ( j - 1 ) ) ) -> 0 e. CC )
28	26, 27	ifclda 3976	. . . . 5 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1 ... ( # ` F ) ) ) -> if ( ( ( T ` ( F ++ <" K "> ) ) ` j ) =/= ( ( T ` ( F ++ <" K "> ) ) ` ( j - 1 ) ) , 1 , 0 ) e. CC )
29		fveq2 5872	. . . . . . 7 |- ( j = ( # ` F ) -> ( ( T ` ( F ++ <" K "> ) ) ` j ) = ( ( T ` ( F ++ <" K "> ) ) ` ( # ` F ) ) )
30		oveq1 6303	. . . . . . . 8 |- ( j = ( # ` F ) -> ( j - 1 ) = ( ( # ` F ) - 1 ) )
31	30	fveq2d 5876	. . . . . . 7 |- ( j = ( # ` F ) -> ( ( T ` ( F ++ <" K "> ) ) ` ( j - 1 ) ) = ( ( T ` ( F ++ <" K "> ) ) ` ( ( # ` F ) - 1 ) ) )
32	29, 31	neeq12d 2736	. . . . . 6 |- ( j = ( # ` F ) -> ( ( ( T ` ( F ++ <" K "> ) ) ` j ) =/= ( ( T ` ( F ++ <" K "> ) ) ` ( j - 1 ) ) <-> ( ( T ` ( F ++ <" K "> ) ) ` ( # ` F ) ) =/= ( ( T ` ( F ++ <" K "> ) ) ` ( ( # ` F ) - 1 ) ) ) )
33	32	ifbid 3966	. . . . 5 |- ( j = ( # ` F ) -> if ( ( ( T ` ( F ++ <" K "> ) ) ` j ) =/= ( ( T ` ( F ++ <" K "> ) ) ` ( j - 1 ) ) , 1 , 0 ) = if ( ( ( T ` ( F ++ <" K "> ) ) ` ( # ` F ) ) =/= ( ( T ` ( F ++ <" K "> ) ) ` ( ( # ` F ) - 1 ) ) , 1 , 0 ) )
34	25, 28, 33	fzosump1 13578	. . . 4 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> sum_ j e. ( 1..^ ( ( # ` F ) + 1 ) ) if ( ( ( T ` ( F ++ <" K "> ) ) ` j ) =/= ( ( T ` ( F ++ <" K "> ) ) ` ( j - 1 ) ) , 1 , 0 ) = ( sum_ j e. ( 1..^ ( # ` F ) ) if ( ( ( T ` ( F ++ <" K "> ) ) ` j ) =/= ( ( T ` ( F ++ <" K "> ) ) ` ( j - 1 ) ) , 1 , 0 ) + if ( ( ( T ` ( F ++ <" K "> ) ) ` ( # ` F ) ) =/= ( ( T ` ( F ++ <" K "> ) ) ` ( ( # ` F ) - 1 ) ) , 1 , 0 ) ) )
35	12, 19, 34	3eqtrd 2502	. . 3 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( V ` ( F ++ <" K "> ) ) = ( sum_ j e. ( 1..^ ( # ` F ) ) if ( ( ( T ` ( F ++ <" K "> ) ) ` j ) =/= ( ( T ` ( F ++ <" K "> ) ) ` ( j - 1 ) ) , 1 , 0 ) + if ( ( ( T ` ( F ++ <" K "> ) ) ` ( # ` F ) ) =/= ( ( T ` ( F ++ <" K "> ) ) ` ( ( # ` F ) - 1 ) ) , 1 , 0 ) ) )
36	35	adantlr 714	. 2 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( V ` ( F ++ <" K "> ) ) = ( sum_ j e. ( 1..^ ( # ` F ) ) if ( ( ( T ` ( F ++ <" K "> ) ) ` j ) =/= ( ( T ` ( F ++ <" K "> ) ) ` ( j - 1 ) ) , 1 , 0 ) + if ( ( ( T ` ( F ++ <" K "> ) ) ` ( # ` F ) ) =/= ( ( T ` ( F ++ <" K "> ) ) ` ( ( # ` F ) - 1 ) ) , 1 , 0 ) ) )
37	2	adantr 465	. . . . . . . . 9 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> F e. Word RR )
38	3	adantr 465	. . . . . . . . 9 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> K e. RR )
39		fzo0ss1 11853	. . . . . . . . . . 11 |- ( 1..^ ( # ` F ) ) C_ ( 0..^ ( # ` F ) )
40	39	a1i 11	. . . . . . . . . 10 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( 1..^ ( # ` F ) ) C_ ( 0..^ ( # ` F ) ) )
41	40	sselda 3499	. . . . . . . . 9 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> j e. ( 0..^ ( # ` F ) ) )
42	7, 8, 9, 10	signstfvp 28703	. . . . . . . . 9 |- ( ( F e. Word RR /\ K e. RR /\ j e. ( 0..^ ( # ` F ) ) ) -> ( ( T ` ( F ++ <" K "> ) ) ` j ) = ( ( T ` F ) ` j ) )
43	37, 38, 41, 42	syl3anc 1228	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> ( ( T ` ( F ++ <" K "> ) ) ` j ) = ( ( T ` F ) ` j ) )
44		elfzoel2 11824	. . . . . . . . . . . . 13 |- ( j e. ( 1..^ ( # ` F ) ) -> ( # ` F ) e. ZZ )
45	44	adantl 466	. . . . . . . . . . . 12 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> ( # ` F ) e. ZZ )
46		1nn0 10832	. . . . . . . . . . . 12 |- 1 e. NN0
47		eluzmn 28666	. . . . . . . . . . . 12 |- ( ( ( # ` F ) e. ZZ /\ 1 e. NN0 ) -> ( # ` F ) e. ( ZZ>= ` ( ( # ` F ) - 1 ) ) )
48	45, 46, 47	sylancl 662	. . . . . . . . . . 11 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> ( # ` F ) e. ( ZZ>= ` ( ( # ` F ) - 1 ) ) )
49		fzoss2 11851	. . . . . . . . . . 11 |- ( ( # ` F ) e. ( ZZ>= ` ( ( # ` F ) - 1 ) ) -> ( 0..^ ( ( # ` F ) - 1 ) ) C_ ( 0..^ ( # ` F ) ) )
50	48, 49	syl 16	. . . . . . . . . 10 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> ( 0..^ ( ( # ` F ) - 1 ) ) C_ ( 0..^ ( # ` F ) ) )
51		simpr 461	. . . . . . . . . . 11 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> j e. ( 1..^ ( # ` F ) ) )
52		elfzoelz 11825	. . . . . . . . . . . . 13 |- ( j e. ( 1..^ ( # ` F ) ) -> j e. ZZ )
53	52	adantl 466	. . . . . . . . . . . 12 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> j e. ZZ )
54		elfzom1b 11913	. . . . . . . . . . . 12 |- ( ( j e. ZZ /\ ( # ` F ) e. ZZ ) -> ( j e. ( 1..^ ( # ` F ) ) <-> ( j - 1 ) e. ( 0..^ ( ( # ` F ) - 1 ) ) ) )
55	53, 45, 54	syl2anc 661	. . . . . . . . . . 11 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> ( j e. ( 1..^ ( # ` F ) ) <-> ( j - 1 ) e. ( 0..^ ( ( # ` F ) - 1 ) ) ) )
56	51, 55	mpbid 210	. . . . . . . . . 10 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> ( j - 1 ) e. ( 0..^ ( ( # ` F ) - 1 ) ) )
57	50, 56	sseldd 3500	. . . . . . . . 9 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> ( j - 1 ) e. ( 0..^ ( # ` F ) ) )
58	7, 8, 9, 10	signstfvp 28703	. . . . . . . . 9 |- ( ( F e. Word RR /\ K e. RR /\ ( j - 1 ) e. ( 0..^ ( # ` F ) ) ) -> ( ( T ` ( F ++ <" K "> ) ) ` ( j - 1 ) ) = ( ( T ` F ) ` ( j - 1 ) ) )
59	37, 38, 57, 58	syl3anc 1228	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> ( ( T ` ( F ++ <" K "> ) ) ` ( j - 1 ) ) = ( ( T ` F ) ` ( j - 1 ) ) )
60	43, 59	neeq12d 2736	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> ( ( ( T ` ( F ++ <" K "> ) ) ` j ) =/= ( ( T ` ( F ++ <" K "> ) ) ` ( j - 1 ) ) <-> ( ( T ` F ) ` j ) =/= ( ( T ` F ) ` ( j - 1 ) ) ) )
61	60	ifbid 3966	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> if ( ( ( T ` ( F ++ <" K "> ) ) ` j ) =/= ( ( T ` ( F ++ <" K "> ) ) ` ( j - 1 ) ) , 1 , 0 ) = if ( ( ( T ` F ) ` j ) =/= ( ( T ` F ) ` ( j - 1 ) ) , 1 , 0 ) )
62	61	sumeq2dv 13536	. . . . 5 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> sum_ j e. ( 1..^ ( # ` F ) ) if ( ( ( T ` ( F ++ <" K "> ) ) ` j ) =/= ( ( T ` ( F ++ <" K "> ) ) ` ( j - 1 ) ) , 1 , 0 ) = sum_ j e. ( 1..^ ( # ` F ) ) if ( ( ( T ` F ) ` j ) =/= ( ( T ` F ) ` ( j - 1 ) ) , 1 , 0 ) )
63	7, 8, 9, 10	signsvvfval 28710	. . . . . 6 |- ( F e. Word RR -> ( V ` F ) = sum_ j e. ( 1..^ ( # ` F ) ) if ( ( ( T ` F ) ` j ) =/= ( ( T ` F ) ` ( j - 1 ) ) , 1 , 0 ) )
64	2, 63	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 ) )
65	62, 64	eqtr4d 2501	. . . 4 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> sum_ j e. ( 1..^ ( # ` F ) ) if ( ( ( T ` ( F ++ <" K "> ) ) ` j ) =/= ( ( T ` ( F ++ <" K "> ) ) ` ( j - 1 ) ) , 1 , 0 ) = ( V ` F ) )
66	65	adantlr 714	. . 3 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> sum_ j e. ( 1..^ ( # ` F ) ) if ( ( ( T ` ( F ++ <" K "> ) ) ` j ) =/= ( ( T ` ( F ++ <" K "> ) ) ` ( j - 1 ) ) , 1 , 0 ) = ( V ` F ) )
67	7, 8, 9, 10	signstfvn 28701	. . . . . . 7 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( ( T ` ( F ++ <" K "> ) ) ` ( # ` F ) ) = ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) .+^ (sgn ` K ) ) )
68	67	adantlr 714	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( ( T ` ( F ++ <" K "> ) ) ` ( # ` F ) ) = ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) .+^ (sgn ` K ) ) )
69	2	adantlr 714	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> F e. Word RR )
70		simpr 461	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> K e. RR )
71	22	ad2antrr 725	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( # ` F ) e. NN )
72		fzo0end 11906	. . . . . . . 8 |- ( ( # ` F ) e. NN -> ( ( # ` F ) - 1 ) e. ( 0..^ ( # ` F ) ) )
73	71, 72	syl 16	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( ( # ` F ) - 1 ) e. ( 0..^ ( # ` F ) ) )
74	7, 8, 9, 10	signstfvp 28703	. . . . . . 7 |- ( ( F e. Word RR /\ K e. RR /\ ( ( # ` F ) - 1 ) e. ( 0..^ ( # ` F ) ) ) -> ( ( T ` ( F ++ <" K "> ) ) ` ( ( # ` F ) - 1 ) ) = ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) )
75	69, 70, 73, 74	syl3anc 1228	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( ( T ` ( F ++ <" K "> ) ) ` ( ( # ` F ) - 1 ) ) = ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) )
76	68, 75	neeq12d 2736	. . . . 5 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( ( ( T ` ( F ++ <" K "> ) ) ` ( # ` F ) ) =/= ( ( T ` ( F ++ <" K "> ) ) ` ( ( # ` F ) - 1 ) ) <-> ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) .+^ (sgn ` K ) ) =/= ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) ) )
77	7, 8, 9, 10	signstfvcl 28705	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( ( # ` F ) - 1 ) e. ( 0..^ ( # ` F ) ) ) -> ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) e. { -u 1 , 1 } )
78	73, 77	syldan 470	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) e. { -u 1 , 1 } )
79	70	rexrd 9660	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> K e. RR* )
80		sgncl 28652	. . . . . . 7 |- ( K e. RR* -> (sgn ` K ) e. { -u 1 , 0 , 1 } )
81	79, 80	syl 16	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> (sgn ` K ) e. { -u 1 , 0 , 1 } )
82	7, 8	signswch 28693	. . . . . 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 ) )
83	78, 81, 82	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 ) )
84		sgnsgn 28662	. . . . . . . . 9 |- ( K e. RR* -> (sgn ` (sgn ` K ) ) = (sgn ` K ) )
85	79, 84	syl 16	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> (sgn ` (sgn ` K ) ) = (sgn ` K ) )
86	85	oveq2d 6312	. . . . . . 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 ) ) )
87	86	breq1d 4466	. . . . . 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 ) )
88		neg1rr 10661	. . . . . . . . 9 |- -u 1 e. RR
89		1re 9612	. . . . . . . . 9 |- 1 e. RR
90		prssi 4188	. . . . . . . . 9 |- ( ( -u 1 e. RR /\ 1 e. RR ) -> { -u 1 , 1 } C_ RR )
91	88, 89, 90	mp2an 672	. . . . . . . 8 |- { -u 1 , 1 } C_ RR
92	91, 78	sseldi 3497	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) e. RR )
93		sgnclre 28653	. . . . . . . 8 |- ( K e. RR -> (sgn ` K ) e. RR )
94	93	adantl 466	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> (sgn ` K ) e. RR )
95		sgnmulsgn 28663	. . . . . . 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 ) )
96	92, 94, 95	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 ) )
97		sgnmulsgn 28663	. . . . . . 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 ) )
98	92, 70, 97	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 ) )
99	87, 96, 98	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 ) )
100	76, 83, 99	3bitrd 279	. . . 4 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( ( ( T ` ( F ++ <" K "> ) ) ` ( # ` F ) ) =/= ( ( T ` ( F ++ <" K "> ) ) ` ( ( # ` F ) - 1 ) ) <-> ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) x. K ) < 0 ) )
101	100	ifbid 3966	. . 3 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> if ( ( ( T ` ( F ++ <" K "> ) ) ` ( # ` F ) ) =/= ( ( T ` ( F ++ <" K "> ) ) ` ( ( # ` F ) - 1 ) ) , 1 , 0 ) = if ( ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) x. K ) < 0 , 1 , 0 ) )
102	66, 101	oveq12d 6314	. 2 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( sum_ j e. ( 1..^ ( # ` F ) ) if ( ( ( T ` ( F ++ <" K "> ) ) ` j ) =/= ( ( T ` ( F ++ <" K "> ) ) ` ( j - 1 ) ) , 1 , 0 ) + if ( ( ( T ` ( F ++ <" K "> ) ) ` ( # ` F ) ) =/= ( ( T ` ( F ++ <" K "> ) ) ` ( ( # ` F ) - 1 ) ) , 1 , 0 ) ) = ( ( V ` F ) + if ( ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) x. K ) < 0 , 1 , 0 ) ) )
103	36, 102	eqtrd 2498	1 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( V ` ( F ++ <" K "> ) ) = ( ( V ` F ) + if ( ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) x. K ) < 0 , 1 , 0 ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1395   e. wcel 1819   =/= wne 2652   \ cdif 3468   C_ wss 3471   (/)c0 3793   ifcif 3944   {csn 4032   {cpr 4034   {ctp 4036   <.cop 4038   class class class wbr 4456   |-> cmpt 4515   ` cfv 5594  (class class class)co 6296   |-> cmpt2 6298   CCcc 9507   RRcr 9508   0cc0 9509   1c1 9510   + caddc 9512   x. cmul 9514   RR*cxr 9644   < clt 9645   - cmin 9824   -ucneg 9825   NNcn 10556   NN0cn0 10816   ZZcz 10885   ZZ>=cuz 11106   ...cfz 11697  ..^cfzo 11820   #chash 12407  Word cword 12537   ++ cconcat 12539   <"cs1 12540  sgncsgn 12930   sum_csu 13519   ndxcnx 14640   Basecbs 14643   +g cplusg 14711   gsum_g cgsu 14857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-fz 11698  df-fzo 11821  df-seq 12110  df-exp 12169  df-hash 12408  df-word 12545  df-lsw 12546  df-concat 12547  df-s1 12548  df-substr 12549  df-sgn 12931  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-clim 13322  df-sum 13520  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-plusg 14724  df-0g 14858  df-gsum 14859  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-mulg 16186  df-cntz 16481

This theorem is referenced by:  signsvtp  28715  signsvtn  28716  signlem0  28719