Theorem signsvfn 29543

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 464	. . . . . . 7 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> F e. (Word RR \ { (/) } ) )
2	1	eldifad 3402	. . . . . 6 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> F e. Word RR )
3		simpr 468	. . . . . . 7 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> K e. RR )
4	3	s1cld 12795	. . . . . 6 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> <" K "> e. Word RR )
5		ccatcl 12771	. . . . . 6 |- ( ( F e. Word RR /\ <" K "> e. Word RR ) -> ( F ++ <" K "> ) e. Word RR )
6	2, 4, 5	syl2anc 673	. . . . 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 29539	. . . . 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 17	. . . 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 12772	. . . . . . . 8 |- ( ( F e. Word RR /\ <" K "> e. Word RR ) -> ( # ` ( F ++ <" K "> ) ) = ( ( # ` F ) + ( # ` <" K "> ) ) )
14	2, 4, 13	syl2anc 673	. . . . . . 7 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( # ` ( F ++ <" K "> ) ) = ( ( # ` F ) + ( # ` <" K "> ) ) )
15		s1len 12797	. . . . . . . 8 |- ( # ` <" K "> ) = 1
16	15	oveq2i 6319	. . . . . . 7 |- ( ( # ` F ) + ( # ` <" K "> ) ) = ( ( # ` F ) + 1 )
17	14, 16	syl6eq 2521	. . . . . 6 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( # ` ( F ++ <" K "> ) ) = ( ( # ` F ) + 1 ) )
18	17	oveq2d 6324	. . . . 5 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( 1..^ ( # ` ( F ++ <" K "> ) ) ) = ( 1..^ ( ( # ` F ) + 1 ) ) )
19	18	sumeq1d 13844	. . . 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 4088	. . . . . . . 8 |- ( F e. (Word RR \ { (/) } ) <-> ( F e. Word RR /\ F =/= (/) ) )
21		lennncl 12738	. . . . . . . 8 |- ( ( F e. Word RR /\ F =/= (/) ) -> ( # ` F ) e. NN )
22	20, 21	sylbi 200	. . . . . . 7 |- ( F e. (Word RR \ { (/) } ) -> ( # ` F ) e. NN )
23		nnuz 11218	. . . . . . 7 |- NN = ( ZZ>= ` 1 )
24	22, 23	syl6eleq 2559	. . . . . 6 |- ( F e. (Word RR \ { (/) } ) -> ( # ` F ) e. ( ZZ>= ` 1 ) )
25	24	adantr 472	. . . . 5 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( # ` F ) e. ( ZZ>= ` 1 ) )
26		1cnd 9677	. . . . . 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 9654	. . . . . 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 3904	. . . . 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 5879	. . . . . . 7 |- ( j = ( # ` F ) -> ( ( T ` ( F ++ <" K "> ) ) ` j ) = ( ( T ` ( F ++ <" K "> ) ) ` ( # ` F ) ) )
30		oveq1 6315	. . . . . . . 8 |- ( j = ( # ` F ) -> ( j - 1 ) = ( ( # ` F ) - 1 ) )
31	30	fveq2d 5883	. . . . . . 7 |- ( j = ( # ` F ) -> ( ( T ` ( F ++ <" K "> ) ) ` ( j - 1 ) ) = ( ( T ` ( F ++ <" K "> ) ) ` ( ( # ` F ) - 1 ) ) )
32	29, 31	neeq12d 2704	. . . . . 6 |- ( j = ( # ` F ) -> ( ( ( T ` ( F ++ <" K "> ) ) ` j ) =/= ( ( T ` ( F ++ <" K "> ) ) ` ( j - 1 ) ) <-> ( ( T ` ( F ++ <" K "> ) ) ` ( # ` F ) ) =/= ( ( T ` ( F ++ <" K "> ) ) ` ( ( # ` F ) - 1 ) ) ) )
33	32	ifbid 3894	. . . . 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 13890	. . . 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 2509	. . 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 729	. 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 472	. . . . . . . . 9 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> F e. Word RR )
38	3	adantr 472	. . . . . . . . 9 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> K e. RR )
39		fzo0ss1 11975	. . . . . . . . . . 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 3418	. . . . . . . . 9 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> j e. ( 0..^ ( # ` F ) ) )
42	7, 8, 9, 10	signstfvp 29532	. . . . . . . . 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 1292	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> ( ( T ` ( F ++ <" K "> ) ) ` j ) = ( ( T ` F ) ` j ) )
44		elfzoel2 11946	. . . . . . . . . . . . 13 |- ( j e. ( 1..^ ( # ` F ) ) -> ( # ` F ) e. ZZ )
45	44	adantl 473	. . . . . . . . . . . 12 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> ( # ` F ) e. ZZ )
46		1nn0 10909	. . . . . . . . . . . 12 |- 1 e. NN0
47		eluzmn 11189	. . . . . . . . . . . 12 |- ( ( ( # ` F ) e. ZZ /\ 1 e. NN0 ) -> ( # ` F ) e. ( ZZ>= ` ( ( # ` F ) - 1 ) ) )
48	45, 46, 47	sylancl 675	. . . . . . . . . . 11 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> ( # ` F ) e. ( ZZ>= ` ( ( # ` F ) - 1 ) ) )
49		fzoss2 11973	. . . . . . . . . . 11 |- ( ( # ` F ) e. ( ZZ>= ` ( ( # ` F ) - 1 ) ) -> ( 0..^ ( ( # ` F ) - 1 ) ) C_ ( 0..^ ( # ` F ) ) )
50	48, 49	syl 17	. . . . . . . . . 10 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> ( 0..^ ( ( # ` F ) - 1 ) ) C_ ( 0..^ ( # ` F ) ) )
51		simpr 468	. . . . . . . . . . 11 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> j e. ( 1..^ ( # ` F ) ) )
52		elfzoelz 11947	. . . . . . . . . . . . 13 |- ( j e. ( 1..^ ( # ` F ) ) -> j e. ZZ )
53	52	adantl 473	. . . . . . . . . . . 12 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> j e. ZZ )
54		elfzom1b 12039	. . . . . . . . . . . 12 |- ( ( j e. ZZ /\ ( # ` F ) e. ZZ ) -> ( j e. ( 1..^ ( # ` F ) ) <-> ( j - 1 ) e. ( 0..^ ( ( # ` F ) - 1 ) ) ) )
55	53, 45, 54	syl2anc 673	. . . . . . . . . . 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 215	. . . . . . . . . 10 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> ( j - 1 ) e. ( 0..^ ( ( # ` F ) - 1 ) ) )
57	50, 56	sseldd 3419	. . . . . . . . 9 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> ( j - 1 ) e. ( 0..^ ( # ` F ) ) )
58	7, 8, 9, 10	signstfvp 29532	. . . . . . . . 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 1292	. . . . . . . 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 2704	. . . . . . 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 3894	. . . . . 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 13846	. . . . 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 29539	. . . . . 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 17	. . . . 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 2508	. . . 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 729	. . 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 29530	. . . . . . 7 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( ( T ` ( F ++ <" K "> ) ) ` ( # ` F ) ) = ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) .+^ (sgn ` K ) ) )
68	67	adantlr 729	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( ( T ` ( F ++ <" K "> ) ) ` ( # ` F ) ) = ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) .+^ (sgn ` K ) ) )
69	2	adantlr 729	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> F e. Word RR )
70		simpr 468	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> K e. RR )
71	22	ad2antrr 740	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( # ` F ) e. NN )
72		fzo0end 12032	. . . . . . . 8 |- ( ( # ` F ) e. NN -> ( ( # ` F ) - 1 ) e. ( 0..^ ( # ` F ) ) )
73	71, 72	syl 17	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( ( # ` F ) - 1 ) e. ( 0..^ ( # ` F ) ) )
74	7, 8, 9, 10	signstfvp 29532	. . . . . . 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 1292	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( ( T ` ( F ++ <" K "> ) ) ` ( ( # ` F ) - 1 ) ) = ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) )
76	68, 75	neeq12d 2704	. . . . 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 29534	. . . . . . 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 478	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) e. { -u 1 , 1 } )
79	70	rexrd 9708	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> K e. RR* )
80		sgncl 29482	. . . . . . 7 |- ( K e. RR* -> (sgn ` K ) e. { -u 1 , 0 , 1 } )
81	79, 80	syl 17	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> (sgn ` K ) e. { -u 1 , 0 , 1 } )
82	7, 8	signswch 29522	. . . . . 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 673	. . . . 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 29492	. . . . . . . . 9 |- ( K e. RR* -> (sgn ` (sgn ` K ) ) = (sgn ` K ) )
85	79, 84	syl 17	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> (sgn ` (sgn ` K ) ) = (sgn ` K ) )
86	85	oveq2d 6324	. . . . . . 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 4405	. . . . . 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 10736	. . . . . . . . 9 |- -u 1 e. RR
89		1re 9660	. . . . . . . . 9 |- 1 e. RR
90		prssi 4119	. . . . . . . . 9 |- ( ( -u 1 e. RR /\ 1 e. RR ) -> { -u 1 , 1 } C_ RR )
91	88, 89, 90	mp2an 686	. . . . . . . 8 |- { -u 1 , 1 } C_ RR
92	91, 78	sseldi 3416	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) e. RR )
93		sgnclre 29483	. . . . . . . 8 |- ( K e. RR -> (sgn ` K ) e. RR )
94	93	adantl 473	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> (sgn ` K ) e. RR )
95		sgnmulsgn 29493	. . . . . . 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 673	. . . . . 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 29493	. . . . . . 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 673	. . . . . 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 293	. . . . 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 287	. . . 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 3894	. . 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 6326	. 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 2505	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 189   /\ wa 376   = wceq 1452   e. wcel 1904   =/= wne 2641   \ cdif 3387   C_ wss 3390   (/)c0 3722   ifcif 3872   {csn 3959   {cpr 3961   {ctp 3963   <.cop 3965   class class class wbr 4395   |-> cmpt 4454   ` cfv 5589  (class class class)co 6308   |-> cmpt2 6310   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558   + caddc 9560   x. cmul 9562   RR*cxr 9692   < clt 9693   - cmin 9880   -ucneg 9881   NNcn 10631   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   ...cfz 11810  ..^cfzo 11942   #chash 12553  Word cword 12703   ++ cconcat 12705   <"cs1 12706  sgncsgn 13226   sum_csu 13829   ndxcnx 15196   Basecbs 15199   +g cplusg 15268   gsum_g cgsu 15417

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

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-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-oi 8043  df-card 8391  df-cda 8616  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-seq 12252  df-exp 12311  df-hash 12554  df-word 12711  df-lsw 12712  df-concat 12713  df-s1 12714  df-substr 12715  df-sgn 13227  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-sum 13830  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-plusg 15281  df-0g 15418  df-gsum 15419  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-mulg 16754  df-cntz 17049

This theorem is referenced by:  signsvtp  29544  signsvtn  29545  signlem0  29548