Theorem signstfvn 30646

Description: Zero-skipping sign in a word compared to a shorter word. (Contributed by Thierry Arnoux, 8-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
signstfvn	|- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( ( T ` ( F ++ <" K "> ) ) ` ( # ` F ) ) = ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) .+^ (sgn ` K ) ) )

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

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

Proof of Theorem signstfvn

Step	Hyp	Ref	Expression
1		signsv.p	. . . . 5 |- .+^ = ( a e. { -u 1 , 0 , 1 } , b e. { -u 1 , 0 , 1 } |-> if ( b = 0 , a , b ) )
2		signsv.w	. . . . 5 |- W = { <. ( Base ` ndx ) , { -u 1 , 0 , 1 } >. , <. ( +g ` ndx ) , .+^ >. }
3	1, 2	signswbase 30631	. . . 4 |- { -u 1 , 0 , 1 } = ( Base ` W )
4	1, 2	signswmnd 30634	. . . . 5 |- W e. Mnd
5	4	a1i 11	. . . 4 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> W e. Mnd )
6		eldifi 3732	. . . . . . . . 9 |- ( F e. (Word RR \ { (/) } ) -> F e. Word RR )
7		lencl 13324	. . . . . . . . 9 |- ( F e. Word RR -> ( # ` F ) e. NN0 )
8	6, 7	syl 17	. . . . . . . 8 |- ( F e. (Word RR \ { (/) } ) -> ( # ` F ) e. NN0 )
9		eldifsn 4317	. . . . . . . . 9 |- ( F e. (Word RR \ { (/) } ) <-> ( F e. Word RR /\ F =/= (/) ) )
10		hasheq0 13154	. . . . . . . . . . 11 |- ( F e. Word RR -> ( ( # ` F ) = 0 <-> F = (/) ) )
11	10	necon3bid 2838	. . . . . . . . . 10 |- ( F e. Word RR -> ( ( # ` F ) =/= 0 <-> F =/= (/) ) )
12	11	biimpar 502	. . . . . . . . 9 |- ( ( F e. Word RR /\ F =/= (/) ) -> ( # ` F ) =/= 0 )
13	9, 12	sylbi 207	. . . . . . . 8 |- ( F e. (Word RR \ { (/) } ) -> ( # ` F ) =/= 0 )
14		elnnne0 11306	. . . . . . . 8 |- ( ( # ` F ) e. NN <-> ( ( # ` F ) e. NN0 /\ ( # ` F ) =/= 0 ) )
15	8, 13, 14	sylanbrc 698	. . . . . . 7 |- ( F e. (Word RR \ { (/) } ) -> ( # ` F ) e. NN )
16	15	adantr 481	. . . . . 6 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( # ` F ) e. NN )
17		nnm1nn0 11334	. . . . . 6 |- ( ( # ` F ) e. NN -> ( ( # ` F ) - 1 ) e. NN0 )
18	16, 17	syl 17	. . . . 5 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( ( # ` F ) - 1 ) e. NN0 )
19		nn0uz 11722	. . . . 5 |- NN0 = ( ZZ>= ` 0 )
20	18, 19	syl6eleq 2711	. . . 4 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( ( # ` F ) - 1 ) e. ( ZZ>= ` 0 ) )
21		s1cl 13382	. . . . . . . . . 10 |- ( K e. RR -> <" K "> e. Word RR )
22		ccatcl 13359	. . . . . . . . . 10 |- ( ( F e. Word RR /\ <" K "> e. Word RR ) -> ( F ++ <" K "> ) e. Word RR )
23	6, 21, 22	syl2an 494	. . . . . . . . 9 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( F ++ <" K "> ) e. Word RR )
24	23	adantr 481	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ i e. ( 0 ... ( ( # ` F ) - 1 ) ) ) -> ( F ++ <" K "> ) e. Word RR )
25		wrdf 13310	. . . . . . . 8 |- ( ( F ++ <" K "> ) e. Word RR -> ( F ++ <" K "> ) : ( 0..^ ( # ` ( F ++ <" K "> ) ) ) --> RR )
26	24, 25	syl 17	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ i e. ( 0 ... ( ( # ` F ) - 1 ) ) ) -> ( F ++ <" K "> ) : ( 0..^ ( # ` ( F ++ <" K "> ) ) ) --> RR )
27	8	adantr 481	. . . . . . . . . . . 12 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( # ` F ) e. NN0 )
28	27	nn0zd 11480	. . . . . . . . . . 11 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( # ` F ) e. ZZ )
29		fzoval 12471	. . . . . . . . . . 11 |- ( ( # ` F ) e. ZZ -> ( 0..^ ( # ` F ) ) = ( 0 ... ( ( # ` F ) - 1 ) ) )
30	28, 29	syl 17	. . . . . . . . . 10 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( 0..^ ( # ` F ) ) = ( 0 ... ( ( # ` F ) - 1 ) ) )
31		fzossfz 12488	. . . . . . . . . 10 |- ( 0..^ ( # ` F ) ) C_ ( 0 ... ( # ` F ) )
32	30, 31	syl6eqssr 3656	. . . . . . . . 9 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( 0 ... ( ( # ` F ) - 1 ) ) C_ ( 0 ... ( # ` F ) ) )
33		ccatlen 13360	. . . . . . . . . . . . 13 |- ( ( F e. Word RR /\ <" K "> e. Word RR ) -> ( # ` ( F ++ <" K "> ) ) = ( ( # ` F ) + ( # ` <" K "> ) ) )
34	6, 21, 33	syl2an 494	. . . . . . . . . . . 12 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( # ` ( F ++ <" K "> ) ) = ( ( # ` F ) + ( # ` <" K "> ) ) )
35		s1len 13385	. . . . . . . . . . . . 13 |- ( # ` <" K "> ) = 1
36	35	oveq2i 6661	. . . . . . . . . . . 12 |- ( ( # ` F ) + ( # ` <" K "> ) ) = ( ( # ` F ) + 1 )
37	34, 36	syl6eq 2672	. . . . . . . . . . 11 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( # ` ( F ++ <" K "> ) ) = ( ( # ` F ) + 1 ) )
38	37	oveq2d 6666	. . . . . . . . . 10 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( 0..^ ( # ` ( F ++ <" K "> ) ) ) = ( 0..^ ( ( # ` F ) + 1 ) ) )
39	28	peano2zd 11485	. . . . . . . . . . 11 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( ( # ` F ) + 1 ) e. ZZ )
40		fzoval 12471	. . . . . . . . . . 11 |- ( ( ( # ` F ) + 1 ) e. ZZ -> ( 0..^ ( ( # ` F ) + 1 ) ) = ( 0 ... ( ( ( # ` F ) + 1 ) - 1 ) ) )
41	39, 40	syl 17	. . . . . . . . . 10 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( 0..^ ( ( # ` F ) + 1 ) ) = ( 0 ... ( ( ( # ` F ) + 1 ) - 1 ) ) )
42	27	nn0cnd 11353	. . . . . . . . . . . 12 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( # ` F ) e. CC )
43		1cnd 10056	. . . . . . . . . . . 12 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> 1 e. CC )
44	42, 43	pncand 10393	. . . . . . . . . . 11 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( ( ( # ` F ) + 1 ) - 1 ) = ( # ` F ) )
45	44	oveq2d 6666	. . . . . . . . . 10 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( 0 ... ( ( ( # ` F ) + 1 ) - 1 ) ) = ( 0 ... ( # ` F ) ) )
46	38, 41, 45	3eqtrd 2660	. . . . . . . . 9 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( 0..^ ( # ` ( F ++ <" K "> ) ) ) = ( 0 ... ( # ` F ) ) )
47	32, 46	sseqtr4d 3642	. . . . . . . 8 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( 0 ... ( ( # ` F ) - 1 ) ) C_ ( 0..^ ( # ` ( F ++ <" K "> ) ) ) )
48	47	sselda 3603	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ i e. ( 0 ... ( ( # ` F ) - 1 ) ) ) -> i e. ( 0..^ ( # ` ( F ++ <" K "> ) ) ) )
49	26, 48	ffvelrnd 6360	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ i e. ( 0 ... ( ( # ` F ) - 1 ) ) ) -> ( ( F ++ <" K "> ) ` i ) e. RR )
50	49	rexrd 10089	. . . . 5 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ i e. ( 0 ... ( ( # ` F ) - 1 ) ) ) -> ( ( F ++ <" K "> ) ` i ) e. RR* )
51		sgncl 30600	. . . . 5 |- ( ( ( F ++ <" K "> ) ` i ) e. RR* -> (sgn ` ( ( F ++ <" K "> ) ` i ) ) e. { -u 1 , 0 , 1 } )
52	50, 51	syl 17	. . . 4 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ i e. ( 0 ... ( ( # ` F ) - 1 ) ) ) -> (sgn ` ( ( F ++ <" K "> ) ` i ) ) e. { -u 1 , 0 , 1 } )
53	1, 2	signswplusg 30632	. . . 4 |- .+^ = ( +g ` W )
54		simpr 477	. . . . . 6 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> K e. RR )
55	54	rexrd 10089	. . . . 5 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> K e. RR* )
56		sgncl 30600	. . . . 5 |- ( K e. RR* -> (sgn ` K ) e. { -u 1 , 0 , 1 } )
57	55, 56	syl 17	. . . 4 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> (sgn ` K ) e. { -u 1 , 0 , 1 } )
58		simpr 477	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ i = ( ( ( # ` F ) - 1 ) + 1 ) ) -> i = ( ( ( # ` F ) - 1 ) + 1 ) )
59	42, 43	npcand 10396	. . . . . . . . 9 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( ( ( # ` F ) - 1 ) + 1 ) = ( # ` F ) )
60	59	adantr 481	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ i = ( ( ( # ` F ) - 1 ) + 1 ) ) -> ( ( ( # ` F ) - 1 ) + 1 ) = ( # ` F ) )
61	58, 60	eqtrd 2656	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ i = ( ( ( # ` F ) - 1 ) + 1 ) ) -> i = ( # ` F ) )
62	61	fveq2d 6195	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ i = ( ( ( # ` F ) - 1 ) + 1 ) ) -> ( ( F ++ <" K "> ) ` i ) = ( ( F ++ <" K "> ) ` ( # ` F ) ) )
63	6	adantr 481	. . . . . . . . 9 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> F e. Word RR )
64	54, 21	syl 17	. . . . . . . . 9 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> <" K "> e. Word RR )
65		c0ex 10034	. . . . . . . . . . . . 13 |- 0 e. _V
66	65	snid 4208	. . . . . . . . . . . 12 |- 0 e. { 0 }
67		fzo01 12550	. . . . . . . . . . . 12 |- ( 0..^ 1 ) = { 0 }
68	66, 67	eleqtrri 2700	. . . . . . . . . . 11 |- 0 e. ( 0..^ 1 )
69	35	oveq2i 6661	. . . . . . . . . . 11 |- ( 0..^ ( # ` <" K "> ) ) = ( 0..^ 1 )
70	68, 69	eleqtrri 2700	. . . . . . . . . 10 |- 0 e. ( 0..^ ( # ` <" K "> ) )
71	70	a1i 11	. . . . . . . . 9 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> 0 e. ( 0..^ ( # ` <" K "> ) ) )
72		ccatval3 13363	. . . . . . . . 9 |- ( ( F e. Word RR /\ <" K "> e. Word RR /\ 0 e. ( 0..^ ( # ` <" K "> ) ) ) -> ( ( F ++ <" K "> ) ` ( 0 + ( # ` F ) ) ) = ( <" K "> ` 0 ) )
73	63, 64, 71, 72	syl3anc 1326	. . . . . . . 8 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( ( F ++ <" K "> ) ` ( 0 + ( # ` F ) ) ) = ( <" K "> ` 0 ) )
74	42	addid2d 10237	. . . . . . . . 9 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( 0 + ( # ` F ) ) = ( # ` F ) )
75	74	fveq2d 6195	. . . . . . . 8 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( ( F ++ <" K "> ) ` ( 0 + ( # ` F ) ) ) = ( ( F ++ <" K "> ) ` ( # ` F ) ) )
76		s1fv 13390	. . . . . . . . 9 |- ( K e. RR -> ( <" K "> ` 0 ) = K )
77	54, 76	syl 17	. . . . . . . 8 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( <" K "> ` 0 ) = K )
78	73, 75, 77	3eqtr3d 2664	. . . . . . 7 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( ( F ++ <" K "> ) ` ( # ` F ) ) = K )
79	78	adantr 481	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ i = ( ( ( # ` F ) - 1 ) + 1 ) ) -> ( ( F ++ <" K "> ) ` ( # ` F ) ) = K )
80	62, 79	eqtrd 2656	. . . . 5 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ i = ( ( ( # ` F ) - 1 ) + 1 ) ) -> ( ( F ++ <" K "> ) ` i ) = K )
81	80	fveq2d 6195	. . . 4 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ i = ( ( ( # ` F ) - 1 ) + 1 ) ) -> (sgn ` ( ( F ++ <" K "> ) ` i ) ) = (sgn ` K ) )
82	3, 5, 20, 52, 53, 57, 81	gsumnunsn 30615	. . 3 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( W gsumg ( i e. ( 0 ... ( ( ( # ` F ) - 1 ) + 1 ) ) |-> (sgn ` ( ( F ++ <" K "> ) ` i ) ) ) ) = ( ( W gsumg ( i e. ( 0 ... ( ( # ` F ) - 1 ) ) |-> (sgn ` ( ( F ++ <" K "> ) ` i ) ) ) ) .+^ (sgn ` K ) ) )
83	59	oveq2d 6666	. . . . 5 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( 0 ... ( ( ( # ` F ) - 1 ) + 1 ) ) = ( 0 ... ( # ` F ) ) )
84	83	mpteq1d 4738	. . . 4 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( i e. ( 0 ... ( ( ( # ` F ) - 1 ) + 1 ) ) |-> (sgn ` ( ( F ++ <" K "> ) ` i ) ) ) = ( i e. ( 0 ... ( # ` F ) ) |-> (sgn ` ( ( F ++ <" K "> ) ` i ) ) ) )
85	84	oveq2d 6666	. . 3 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( W gsumg ( i e. ( 0 ... ( ( ( # ` F ) - 1 ) + 1 ) ) |-> (sgn ` ( ( F ++ <" K "> ) ` i ) ) ) ) = ( W gsumg ( i e. ( 0 ... ( # ` F ) ) |-> (sgn ` ( ( F ++ <" K "> ) ` i ) ) ) ) )
86	63	adantr 481	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ i e. ( 0 ... ( ( # ` F ) - 1 ) ) ) -> F e. Word RR )
87	64	adantr 481	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ i e. ( 0 ... ( ( # ` F ) - 1 ) ) ) -> <" K "> e. Word RR )
88	30	eleq2d 2687	. . . . . . . . 9 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( i e. ( 0..^ ( # ` F ) ) <-> i e. ( 0 ... ( ( # ` F ) - 1 ) ) ) )
89	88	biimpar 502	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ i e. ( 0 ... ( ( # ` F ) - 1 ) ) ) -> i e. ( 0..^ ( # ` F ) ) )
90		ccatval1 13361	. . . . . . . 8 |- ( ( F e. Word RR /\ <" K "> e. Word RR /\ i e. ( 0..^ ( # ` F ) ) ) -> ( ( F ++ <" K "> ) ` i ) = ( F ` i ) )
91	86, 87, 89, 90	syl3anc 1326	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ i e. ( 0 ... ( ( # ` F ) - 1 ) ) ) -> ( ( F ++ <" K "> ) ` i ) = ( F ` i ) )
92	91	fveq2d 6195	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) /\ i e. ( 0 ... ( ( # ` F ) - 1 ) ) ) -> (sgn ` ( ( F ++ <" K "> ) ` i ) ) = (sgn ` ( F ` i ) ) )
93	92	mpteq2dva 4744	. . . . 5 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( i e. ( 0 ... ( ( # ` F ) - 1 ) ) |-> (sgn ` ( ( F ++ <" K "> ) ` i ) ) ) = ( i e. ( 0 ... ( ( # ` F ) - 1 ) ) |-> (sgn ` ( F ` i ) ) ) )
94	93	oveq2d 6666	. . . 4 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( W gsumg ( i e. ( 0 ... ( ( # ` F ) - 1 ) ) |-> (sgn ` ( ( F ++ <" K "> ) ` i ) ) ) ) = ( W gsumg ( i e. ( 0 ... ( ( # ` F ) - 1 ) ) |-> (sgn ` ( F ` i ) ) ) ) )
95	94	oveq1d 6665	. . 3 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( ( W gsumg ( i e. ( 0 ... ( ( # ` F ) - 1 ) ) |-> (sgn ` ( ( F ++ <" K "> ) ` i ) ) ) ) .+^ (sgn ` K ) ) = ( ( W gsumg ( i e. ( 0 ... ( ( # ` F ) - 1 ) ) |-> (sgn ` ( F ` i ) ) ) ) .+^ (sgn ` K ) ) )
96	82, 85, 95	3eqtr3d 2664	. 2 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( W gsumg ( i e. ( 0 ... ( # ` F ) ) |-> (sgn ` ( ( F ++ <" K "> ) ` i ) ) ) ) = ( ( W gsumg ( i e. ( 0 ... ( ( # ` F ) - 1 ) ) |-> (sgn ` ( F ` i ) ) ) ) .+^ (sgn ` K ) ) )
97		eqidd 2623	. . . . . 6 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( # ` F ) = ( # ` F ) )
98	97	olcd 408	. . . . 5 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( ( # ` F ) e. ( 0..^ ( # ` F ) ) \/ ( # ` F ) = ( # ` F ) ) )
99	27, 19	syl6eleq 2711	. . . . . 6 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( # ` F ) e. ( ZZ>= ` 0 ) )
100		fzosplitsni 12579	. . . . . 6 |- ( ( # ` F ) e. ( ZZ>= ` 0 ) -> ( ( # ` F ) e. ( 0..^ ( ( # ` F ) + 1 ) ) <-> ( ( # ` F ) e. ( 0..^ ( # ` F ) ) \/ ( # ` F ) = ( # ` F ) ) ) )
101	99, 100	syl 17	. . . . 5 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( ( # ` F ) e. ( 0..^ ( ( # ` F ) + 1 ) ) <-> ( ( # ` F ) e. ( 0..^ ( # ` F ) ) \/ ( # ` F ) = ( # ` F ) ) ) )
102	98, 101	mpbird 247	. . . 4 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( # ` F ) e. ( 0..^ ( ( # ` F ) + 1 ) ) )
103	102, 38	eleqtrrd 2704	. . 3 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( # ` F ) e. ( 0..^ ( # ` ( F ++ <" K "> ) ) ) )
104		signsv.t	. . . 4 |- T = ( f e. Word RR |-> ( n e. ( 0..^ ( # ` f ) ) |-> ( W gsumg ( i e. ( 0 ... n ) |-> (sgn ` ( f ` i ) ) ) ) ) )
105		signsv.v	. . . 4 |- V = ( f e. Word RR |-> sum_ j e. ( 1..^ ( # ` f ) ) if ( ( ( T ` f ) ` j ) =/= ( ( T ` f ) ` ( j - 1 ) ) , 1 , 0 ) )
106	1, 2, 104, 105	signstfval 30641	. . 3 |- ( ( ( F ++ <" K "> ) e. Word RR /\ ( # ` F ) e. ( 0..^ ( # ` ( F ++ <" K "> ) ) ) ) -> ( ( T ` ( F ++ <" K "> ) ) ` ( # ` F ) ) = ( W gsumg ( i e. ( 0 ... ( # ` F ) ) |-> (sgn ` ( ( F ++ <" K "> ) ` i ) ) ) ) )
107	23, 103, 106	syl2anc 693	. 2 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( ( T ` ( F ++ <" K "> ) ) ` ( # ` F ) ) = ( W gsumg ( i e. ( 0 ... ( # ` F ) ) |-> (sgn ` ( ( F ++ <" K "> ) ` i ) ) ) ) )
108		fzo0end 12560	. . . . . 6 |- ( ( # ` F ) e. NN -> ( ( # ` F ) - 1 ) e. ( 0..^ ( # ` F ) ) )
109	15, 108	syl 17	. . . . 5 |- ( F e. (Word RR \ { (/) } ) -> ( ( # ` F ) - 1 ) e. ( 0..^ ( # ` F ) ) )
110	1, 2, 104, 105	signstfval 30641	. . . . 5 |- ( ( F e. Word RR /\ ( ( # ` F ) - 1 ) e. ( 0..^ ( # ` F ) ) ) -> ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) = ( W gsumg ( i e. ( 0 ... ( ( # ` F ) - 1 ) ) |-> (sgn ` ( F ` i ) ) ) ) )
111	6, 109, 110	syl2anc 693	. . . 4 |- ( F e. (Word RR \ { (/) } ) -> ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) = ( W gsumg ( i e. ( 0 ... ( ( # ` F ) - 1 ) ) |-> (sgn ` ( F ` i ) ) ) ) )
112	111	adantr 481	. . 3 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) = ( W gsumg ( i e. ( 0 ... ( ( # ` F ) - 1 ) ) |-> (sgn ` ( F ` i ) ) ) ) )
113	112	oveq1d 6665	. 2 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) .+^ (sgn ` K ) ) = ( ( W gsumg ( i e. ( 0 ... ( ( # ` F ) - 1 ) ) |-> (sgn ` ( F ` i ) ) ) ) .+^ (sgn ` K ) ) )
114	96, 107, 113	3eqtr4d 2666	1 |- ( ( F e. (Word RR \ { (/) } ) /\ K e. RR ) -> ( ( T ` ( F ++ <" K "> ) ) ` ( # ` F ) ) = ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) .+^ (sgn ` K ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   (/)c0 3915   ifcif 4086   {csn 4177   {cpr 4179   {ctp 4181   <.cop 4183   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   RR*cxr 10073   - 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   Mndcmnd 17294

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-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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  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-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-sgn 13827  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

This theorem is referenced by:  signsvtn0  30647  signstfvneq0  30649  signstfveq0  30654  signsvfn  30659