Theorem signsvfn 30659

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 473	. . . . . . 7 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> F e. (Word RR \ { (/) } ) )
2	1	eldifad 3586	. . . . . 6 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> F e. Word RR )
3		simpr 477	. . . . . . 7 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> K e. RR )
4	3	s1cld 13383	. . . . . 6 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> <" K "> e. Word RR )
5		ccatcl 13359	. . . . . 6 |- ( ( F e. Word RR /\ <" K "> e. Word RR ) -> ( F ++ <" K "> ) e. Word RR )
6	2, 4, 5	syl2anc 693	. . . . 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 30655	. . . . 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 13360	. . . . . . . 8 |- ( ( F e. Word RR /\ <" K "> e. Word RR ) -> ( # ` ( F ++ <" K "> ) ) = ( ( # ` F ) + ( # ` <" K "> ) ) )
14	2, 4, 13	syl2anc 693	. . . . . . 7 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( # ` ( F ++ <" K "> ) ) = ( ( # ` F ) + ( # ` <" K "> ) ) )
15		s1len 13385	. . . . . . . 8 |- ( # ` <" K "> ) = 1
16	15	oveq2i 6661	. . . . . . 7 |- ( ( # ` F ) + ( # ` <" K "> ) ) = ( ( # ` F ) + 1 )
17	14, 16	syl6eq 2672	. . . . . 6 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( # ` ( F ++ <" K "> ) ) = ( ( # ` F ) + 1 ) )
18	17	oveq2d 6666	. . . . 5 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( 1..^ ( # ` ( F ++ <" K "> ) ) ) = ( 1..^ ( ( # ` F ) + 1 ) ) )
19	18	sumeq1d 14431	. . . 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 4317	. . . . . . . 8 |- ( F e. (Word RR \ { (/) } ) <-> ( F e. Word RR /\ F =/= (/) ) )
21		lennncl 13325	. . . . . . . 8 |- ( ( F e. Word RR /\ F =/= (/) ) -> ( # ` F ) e. NN )
22	20, 21	sylbi 207	. . . . . . 7 |- ( F e. (Word RR \ { (/) } ) -> ( # ` F ) e. NN )
23		nnuz 11723	. . . . . . 7 |- NN = ( ZZ>= ` 1 )
24	22, 23	syl6eleq 2711	. . . . . 6 |- ( F e. (Word RR \ { (/) } ) -> ( # ` F ) e. ( ZZ>= ` 1 ) )
25	24	adantr 481	. . . . 5 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( # ` F ) e. ( ZZ>= ` 1 ) )
26		1cnd 10056	. . . . . 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 10033	. . . . . 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 4120	. . . . 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 6191	. . . . . . 7 |- ( j = ( # ` F ) -> ( ( T ` ( F ++ <" K "> ) ) ` j ) = ( ( T ` ( F ++ <" K "> ) ) ` ( # ` F ) ) )
30		oveq1 6657	. . . . . . . 8 |- ( j = ( # ` F ) -> ( j - 1 ) = ( ( # ` F ) - 1 ) )
31	30	fveq2d 6195	. . . . . . 7 |- ( j = ( # ` F ) -> ( ( T ` ( F ++ <" K "> ) ) ` ( j - 1 ) ) = ( ( T ` ( F ++ <" K "> ) ) ` ( ( # ` F ) - 1 ) ) )
32	29, 31	neeq12d 2855	. . . . . 6 |- ( j = ( # ` F ) -> ( ( ( T ` ( F ++ <" K "> ) ) ` j ) =/= ( ( T ` ( F ++ <" K "> ) ) ` ( j - 1 ) ) <-> ( ( T ` ( F ++ <" K "> ) ) ` ( # ` F ) ) =/= ( ( T ` ( F ++ <" K "> ) ) ` ( ( # ` F ) - 1 ) ) ) )
33	32	ifbid 4108	. . . . 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 14481	. . . 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 2660	. . 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 751	. 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 481	. . . . . . . . 9 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> F e. Word RR )
38	3	adantr 481	. . . . . . . . 9 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> K e. RR )
39		fzo0ss1 12498	. . . . . . . . . . 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 3603	. . . . . . . . 9 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> j e. ( 0..^ ( # ` F ) ) )
42	7, 8, 9, 10	signstfvp 30648	. . . . . . . . 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 1326	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> ( ( T ` ( F ++ <" K "> ) ) ` j ) = ( ( T ` F ) ` j ) )
44		elfzoel2 12469	. . . . . . . . . . . . 13 |- ( j e. ( 1..^ ( # ` F ) ) -> ( # ` F ) e. ZZ )
45	44	adantl 482	. . . . . . . . . . . 12 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> ( # ` F ) e. ZZ )
46		1nn0 11308	. . . . . . . . . . . 12 |- 1 e. NN0
47		eluzmn 11694	. . . . . . . . . . . 12 |- ( ( ( # ` F ) e. ZZ /\ 1 e. NN0 ) -> ( # ` F ) e. ( ZZ>= ` ( ( # ` F ) - 1 ) ) )
48	45, 46, 47	sylancl 694	. . . . . . . . . . 11 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> ( # ` F ) e. ( ZZ>= ` ( ( # ` F ) - 1 ) ) )
49		fzoss2 12496	. . . . . . . . . . 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 477	. . . . . . . . . . 11 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> j e. ( 1..^ ( # ` F ) ) )
52		elfzoelz 12470	. . . . . . . . . . . . 13 |- ( j e. ( 1..^ ( # ` F ) ) -> j e. ZZ )
53	52	adantl 482	. . . . . . . . . . . 12 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> j e. ZZ )
54		elfzom1b 12567	. . . . . . . . . . . 12 |- ( ( j e. ZZ /\ ( # ` F ) e. ZZ ) -> ( j e. ( 1..^ ( # ` F ) ) <-> ( j - 1 ) e. ( 0..^ ( ( # ` F ) - 1 ) ) ) )
55	53, 45, 54	syl2anc 693	. . . . . . . . . . 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 222	. . . . . . . . . 10 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> ( j - 1 ) e. ( 0..^ ( ( # ` F ) - 1 ) ) )
57	50, 56	sseldd 3604	. . . . . . . . 9 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ j e. ( 1..^ ( # ` F ) ) ) -> ( j - 1 ) e. ( 0..^ ( # ` F ) ) )
58	7, 8, 9, 10	signstfvp 30648	. . . . . . . . 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 1326	. . . . . . . 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 2855	. . . . . . 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 4108	. . . . . 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 14433	. . . . 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 30655	. . . . . 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 2659	. . . 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 751	. . 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 30646	. . . . . . 7 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( ( T ` ( F ++ <" K "> ) ) ` ( # ` F ) ) = ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) .+^ (sgn ` K ) ) )
68	67	adantlr 751	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( ( T ` ( F ++ <" K "> ) ) ` ( # ` F ) ) = ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) .+^ (sgn ` K ) ) )
69	2	adantlr 751	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> F e. Word RR )
70		simpr 477	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> K e. RR )
71	22	ad2antrr 762	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( # ` F ) e. NN )
72		fzo0end 12560	. . . . . . . 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 30648	. . . . . . 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 1326	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( ( T ` ( F ++ <" K "> ) ) ` ( ( # ` F ) - 1 ) ) = ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) )
76	68, 75	neeq12d 2855	. . . . 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 30650	. . . . . . 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 487	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) e. { -u 1 , 1 } )
79	70	rexrd 10089	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> K e. RR* )
80		sgncl 30600	. . . . . . 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 30638	. . . . . 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 693	. . . . 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 30610	. . . . . . . . 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 6666	. . . . . . 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 4663	. . . . . 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 11125	. . . . . . . . 9 |- -u 1 e. RR
89		1re 10039	. . . . . . . . 9 |- 1 e. RR
90		prssi 4353	. . . . . . . . 9 |- ( ( -u 1 e. RR /\ 1 e. RR ) -> { -u 1 , 1 } C_ RR )
91	88, 89, 90	mp2an 708	. . . . . . . 8 |- { -u 1 , 1 } C_ RR
92	91, 78	sseldi 3601	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) e. RR )
93		sgnclre 30601	. . . . . . . 8 |- ( K e. RR -> (sgn ` K ) e. RR )
94	93	adantl 482	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ K e. RR ) -> (sgn ` K ) e. RR )
95		sgnmulsgn 30611	. . . . . . 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 693	. . . . . 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 30611	. . . . . . 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 693	. . . . . 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 300	. . . . 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 294	. . . 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 4108	. . 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 6668	. 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 2656	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 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   C_ wss 3574   (/)c0 3915   ifcif 4086   {csn 4177   {cpr 4179   {ctp 4181   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   RR*cxr 10073   < clt 10074   - cmin 10266   -ucneg 10267   NNcn 11020   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291   ++ cconcat 13293   <"cs1 13294  sgncsgn 13826   sum_csu 14416   ndxcnx 15854   Basecbs 15857   +g cplusg 15941   gsum_g cgsu 16101

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-sgn 13827  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mulg 17541  df-cntz 17750

This theorem is referenced by:  signsvtp  30660  signsvtn  30661  signlem0  30664