Theorem signsvtn0 28791

Description: If the last letter is non zero, then this is the zero-skipping sign. (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 ) )
signsvtn0.1	|- N = ( # ` F )

Assertion

Ref	Expression
signsvtn0	|- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( ( T ` F ) ` ( N - 1 ) ) = (sgn ` ( F ` ( N - 1 ) ) ) )

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

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

Proof of Theorem signsvtn0

Step	Hyp	Ref	Expression
1		eldifsn 4141	. . . . . . . . . . . 12 |- ( F e. (Word RR \ { (/) } ) <-> ( F e. Word RR /\ F =/= (/) ) )
2	1	biimpi 194	. . . . . . . . . . 11 |- ( F e. (Word RR \ { (/) } ) -> ( F e. Word RR /\ F =/= (/) ) )
3	2	adantr 463	. . . . . . . . . 10 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F e. Word RR /\ F =/= (/) ) )
4	3	simpld 457	. . . . . . . . 9 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> F e. Word RR )
5	4	adantr 463	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> F e. Word RR )
6		wrdf 12538	. . . . . . . 8 |- ( F e. Word RR -> F : ( 0..^ ( # ` F ) ) --> RR )
7	5, 6	syl 16	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> F : ( 0..^ ( # ` F ) ) --> RR )
8		lennncl 12550	. . . . . . . . . 10 |- ( ( F e. Word RR /\ F =/= (/) ) -> ( # ` F ) e. NN )
9	3, 8	syl 16	. . . . . . . . 9 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( # ` F ) e. NN )
10	9	adantr 463	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> ( # ` F ) e. NN )
11		lbfzo0 11839	. . . . . . . 8 |- ( 0 e. ( 0..^ ( # ` F ) ) <-> ( # ` F ) e. NN )
12	10, 11	sylibr 212	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> 0 e. ( 0..^ ( # ` F ) ) )
13	7, 12	ffvelrnd 6008	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> ( F ` 0 ) e. RR )
14		signsv.p	. . . . . . 7 |- .+^ = ( a e. { -u 1 , 0 , 1 } , b e. { -u 1 , 0 , 1 } |-> if ( b = 0 , a , b ) )
15		signsv.w	. . . . . . 7 |- W = { <. ( Base ` ndx ) , { -u 1 , 0 , 1 } >. , <. ( +g ` ndx ) , .+^ >. }
16		signsv.t	. . . . . . 7 |- T = ( f e. Word RR |-> ( n e. ( 0..^ ( # ` f ) ) |-> ( W gsumg ( i e. ( 0 ... n ) |-> (sgn ` ( f ` i ) ) ) ) ) )
17		signsv.v	. . . . . . 7 |- V = ( f e. Word RR |-> sum_ j e. ( 1..^ ( # ` f ) ) if ( ( ( T ` f ) ` j ) =/= ( ( T ` f ) ` ( j - 1 ) ) , 1 , 0 ) )
18	14, 15, 16, 17	signstf0 28789	. . . . . 6 |- ( ( F ` 0 ) e. RR -> ( T ` <" ( F ` 0 ) "> ) = <" (sgn ` ( F ` 0 ) ) "> )
19	13, 18	syl 16	. . . . 5 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> ( T ` <" ( F ` 0 ) "> ) = <" (sgn ` ( F ` 0 ) ) "> )
20		signsvtn0.1	. . . . . . . 8 |- N = ( # ` F )
21		simpr 459	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> N = 1 )
22	20, 21	syl5eqr 2509	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> ( # ` F ) = 1 )
23		eqs1 12610	. . . . . . 7 |- ( ( F e. Word RR /\ ( # ` F ) = 1 ) -> F = <" ( F ` 0 ) "> )
24	5, 22, 23	syl2anc 659	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> F = <" ( F ` 0 ) "> )
25	24	fveq2d 5852	. . . . 5 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> ( T ` F ) = ( T ` <" ( F ` 0 ) "> ) )
26		oveq1 6277	. . . . . . . . . 10 |- ( N = 1 -> ( N - 1 ) = ( 1 - 1 ) )
27		1m1e0 10600	. . . . . . . . . 10 |- ( 1 - 1 ) = 0
28	26, 27	syl6eq 2511	. . . . . . . . 9 |- ( N = 1 -> ( N - 1 ) = 0 )
29	28	fveq2d 5852	. . . . . . . 8 |- ( N = 1 -> ( F ` ( N - 1 ) ) = ( F ` 0 ) )
30	29	fveq2d 5852	. . . . . . 7 |- ( N = 1 -> (sgn ` ( F ` ( N - 1 ) ) ) = (sgn ` ( F ` 0 ) ) )
31	30	s1eqd 12602	. . . . . 6 |- ( N = 1 -> <" (sgn ` ( F ` ( N - 1 ) ) ) "> = <" (sgn ` ( F ` 0 ) ) "> )
32	21, 31	syl 16	. . . . 5 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> <" (sgn ` ( F ` ( N - 1 ) ) ) "> = <" (sgn ` ( F ` 0 ) ) "> )
33	19, 25, 32	3eqtr4d 2505	. . . 4 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> ( T ` F ) = <" (sgn ` ( F ` ( N - 1 ) ) ) "> )
34	21, 28	syl 16	. . . 4 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> ( N - 1 ) = 0 )
35	33, 34	fveq12d 5854	. . 3 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> ( ( T ` F ) ` ( N - 1 ) ) = ( <" (sgn ` ( F ` ( N - 1 ) ) ) "> ` 0 ) )
36	4, 6	syl 16	. . . . . . . 8 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> F : ( 0..^ ( # ` F ) ) --> RR )
37	20	oveq1i 6280	. . . . . . . . 9 |- ( N - 1 ) = ( ( # ` F ) - 1 )
38		fzo0end 11885	. . . . . . . . . 10 |- ( ( # ` F ) e. NN -> ( ( # ` F ) - 1 ) e. ( 0..^ ( # ` F ) ) )
39	3, 8, 38	3syl 20	. . . . . . . . 9 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( ( # ` F ) - 1 ) e. ( 0..^ ( # ` F ) ) )
40	37, 39	syl5eqel 2546	. . . . . . . 8 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( N - 1 ) e. ( 0..^ ( # ` F ) ) )
41	36, 40	ffvelrnd 6008	. . . . . . 7 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F ` ( N - 1 ) ) e. RR )
42	41	rexrd 9632	. . . . . 6 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F ` ( N - 1 ) ) e. RR* )
43		sgncl 28741	. . . . . 6 |- ( ( F ` ( N - 1 ) ) e. RR* -> (sgn ` ( F ` ( N - 1 ) ) ) e. { -u 1 , 0 , 1 } )
44	42, 43	syl 16	. . . . 5 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> (sgn ` ( F ` ( N - 1 ) ) ) e. { -u 1 , 0 , 1 } )
45	44	adantr 463	. . . 4 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> (sgn ` ( F ` ( N - 1 ) ) ) e. { -u 1 , 0 , 1 } )
46		s1fv 12608	. . . 4 |- ( (sgn ` ( F ` ( N - 1 ) ) ) e. { -u 1 , 0 , 1 } -> ( <" (sgn ` ( F ` ( N - 1 ) ) ) "> ` 0 ) = (sgn ` ( F ` ( N - 1 ) ) ) )
47	45, 46	syl 16	. . 3 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> ( <" (sgn ` ( F ` ( N - 1 ) ) ) "> ` 0 ) = (sgn ` ( F ` ( N - 1 ) ) ) )
48	35, 47	eqtrd 2495	. 2 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> ( ( T ` F ) ` ( N - 1 ) ) = (sgn ` ( F ` ( N - 1 ) ) ) )
49		fzossfz 11822	. . . . . . . . . 10 |- ( 0..^ ( # ` F ) ) C_ ( 0 ... ( # ` F ) )
50	49, 39	sseldi 3487	. . . . . . . . 9 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( ( # ` F ) - 1 ) e. ( 0 ... ( # ` F ) ) )
51		swrd0val 12637	. . . . . . . . 9 |- ( ( F e. Word RR /\ ( ( # ` F ) - 1 ) e. ( 0 ... ( # ` F ) ) ) -> ( F substr <. 0 , ( ( # ` F ) - 1 ) >. ) = ( F |` ( 0..^ ( ( # ` F ) - 1 ) ) ) )
52	4, 50, 51	syl2anc 659	. . . . . . . 8 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F substr <. 0 , ( ( # ` F ) - 1 ) >. ) = ( F |` ( 0..^ ( ( # ` F ) - 1 ) ) ) )
53	52	oveq1d 6285	. . . . . . 7 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( ( F substr <. 0 , ( ( # ` F ) - 1 ) >. ) ++ <" ( lastS ` F ) "> ) = ( ( F |` ( 0..^ ( ( # ` F ) - 1 ) ) ) ++ <" ( lastS ` F ) "> ) )
54		swrdccatwrd 12684	. . . . . . . . 9 |- ( ( F e. Word RR /\ F =/= (/) ) -> ( ( F substr <. 0 , ( ( # ` F ) - 1 ) >. ) ++ <" ( lastS ` F ) "> ) = F )
55	54	eqcomd 2462	. . . . . . . 8 |- ( ( F e. Word RR /\ F =/= (/) ) -> F = ( ( F substr <. 0 , ( ( # ` F ) - 1 ) >. ) ++ <" ( lastS ` F ) "> ) )
56	3, 55	syl 16	. . . . . . 7 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> F = ( ( F substr <. 0 , ( ( # ` F ) - 1 ) >. ) ++ <" ( lastS ` F ) "> ) )
57	37	oveq2i 6281	. . . . . . . . . 10 |- ( 0..^ ( N - 1 ) ) = ( 0..^ ( ( # ` F ) - 1 ) )
58	57	a1i 11	. . . . . . . . 9 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( 0..^ ( N - 1 ) ) = ( 0..^ ( ( # ` F ) - 1 ) ) )
59	58	reseq2d 5262	. . . . . . . 8 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F |` ( 0..^ ( N - 1 ) ) ) = ( F |` ( 0..^ ( ( # ` F ) - 1 ) ) ) )
60	37	a1i 11	. . . . . . . . . . 11 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( N - 1 ) = ( ( # ` F ) - 1 ) )
61	60	fveq2d 5852	. . . . . . . . . 10 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F ` ( N - 1 ) ) = ( F ` ( ( # ` F ) - 1 ) ) )
62		lsw 12573	. . . . . . . . . . 11 |- ( F e. (Word RR \ { (/) } ) -> ( lastS ` F ) = ( F ` ( ( # ` F ) - 1 ) ) )
63	62	adantr 463	. . . . . . . . . 10 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( lastS ` F ) = ( F ` ( ( # ` F ) - 1 ) ) )
64	61, 63	eqtr4d 2498	. . . . . . . . 9 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F ` ( N - 1 ) ) = ( lastS ` F ) )
65	64	s1eqd 12602	. . . . . . . 8 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> <" ( F ` ( N - 1 ) ) "> = <" ( lastS ` F ) "> )
66	59, 65	oveq12d 6288	. . . . . . 7 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( ( F |` ( 0..^ ( N - 1 ) ) ) ++ <" ( F ` ( N - 1 ) ) "> ) = ( ( F |` ( 0..^ ( ( # ` F ) - 1 ) ) ) ++ <" ( lastS ` F ) "> ) )
67	53, 56, 66	3eqtr4d 2505	. . . . . 6 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> F = ( ( F |` ( 0..^ ( N - 1 ) ) ) ++ <" ( F ` ( N - 1 ) ) "> ) )
68	67	fveq2d 5852	. . . . 5 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( T ` F ) = ( T ` ( ( F |` ( 0..^ ( N - 1 ) ) ) ++ <" ( F ` ( N - 1 ) ) "> ) ) )
69		ffn 5713	. . . . . . . . . . 11 |- ( F : ( 0..^ ( # ` F ) ) --> RR -> F Fn ( 0..^ ( # ` F ) ) )
70	4, 6, 69	3syl 20	. . . . . . . . . 10 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> F Fn ( 0..^ ( # ` F ) ) )
71	20	a1i 11	. . . . . . . . . . . 12 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> N = ( # ` F ) )
72	71	oveq2d 6286	. . . . . . . . . . 11 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( 0..^ N ) = ( 0..^ ( # ` F ) ) )
73	72	fneq2d 5654	. . . . . . . . . 10 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F Fn ( 0..^ N ) <-> F Fn ( 0..^ ( # ` F ) ) ) )
74	70, 73	mpbird 232	. . . . . . . . 9 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> F Fn ( 0..^ N ) )
75	20, 9	syl5eqel 2546	. . . . . . . . . . 11 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> N e. NN )
76	75	nnnn0d 10848	. . . . . . . . . 10 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> N e. NN0 )
77		nn0z 10883	. . . . . . . . . 10 |- ( N e. NN0 -> N e. ZZ )
78		fzossrbm1 11831	. . . . . . . . . 10 |- ( N e. ZZ -> ( 0..^ ( N - 1 ) ) C_ ( 0..^ N ) )
79	76, 77, 78	3syl 20	. . . . . . . . 9 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( 0..^ ( N - 1 ) ) C_ ( 0..^ N ) )
80		fnssres 5676	. . . . . . . . 9 |- ( ( F Fn ( 0..^ N ) /\ ( 0..^ ( N - 1 ) ) C_ ( 0..^ N ) ) -> ( F |` ( 0..^ ( N - 1 ) ) ) Fn ( 0..^ ( N - 1 ) ) )
81	74, 79, 80	syl2anc 659	. . . . . . . 8 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F |` ( 0..^ ( N - 1 ) ) ) Fn ( 0..^ ( N - 1 ) ) )
82		hashfn 12426	. . . . . . . 8 |- ( ( F |` ( 0..^ ( N - 1 ) ) ) Fn ( 0..^ ( N - 1 ) ) -> ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) = ( # ` ( 0..^ ( N - 1 ) ) ) )
83	81, 82	syl 16	. . . . . . 7 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) = ( # ` ( 0..^ ( N - 1 ) ) ) )
84		nnm1nn0 10833	. . . . . . . 8 |- ( N e. NN -> ( N - 1 ) e. NN0 )
85		hashfzo0 12472	. . . . . . . 8 |- ( ( N - 1 ) e. NN0 -> ( # ` ( 0..^ ( N - 1 ) ) ) = ( N - 1 ) )
86	75, 84, 85	3syl 20	. . . . . . 7 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( # ` ( 0..^ ( N - 1 ) ) ) = ( N - 1 ) )
87	83, 86	eqtrd 2495	. . . . . 6 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) = ( N - 1 ) )
88	87	eqcomd 2462	. . . . 5 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( N - 1 ) = ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) )
89	68, 88	fveq12d 5854	. . . 4 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( ( T ` F ) ` ( N - 1 ) ) = ( ( T ` ( ( F |` ( 0..^ ( N - 1 ) ) ) ++ <" ( F ` ( N - 1 ) ) "> ) ) ` ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) ) )
90	89	adantr 463	. . 3 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( ( T ` F ) ` ( N - 1 ) ) = ( ( T ` ( ( F |` ( 0..^ ( N - 1 ) ) ) ++ <" ( F ` ( N - 1 ) ) "> ) ) ` ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) ) )
91	52, 59	eqtr4d 2498	. . . . . . . 8 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F substr <. 0 , ( ( # ` F ) - 1 ) >. ) = ( F |` ( 0..^ ( N - 1 ) ) ) )
92		swrdcl 12635	. . . . . . . . 9 |- ( F e. Word RR -> ( F substr <. 0 , ( ( # ` F ) - 1 ) >. ) e. Word RR )
93	4, 92	syl 16	. . . . . . . 8 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F substr <. 0 , ( ( # ` F ) - 1 ) >. ) e. Word RR )
94	91, 93	eqeltrrd 2543	. . . . . . 7 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F |` ( 0..^ ( N - 1 ) ) ) e. Word RR )
95	94	adantr 463	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( F |` ( 0..^ ( N - 1 ) ) ) e. Word RR )
96	87	adantr 463	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) = ( N - 1 ) )
97	75	adantr 463	. . . . . . . . . 10 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> N e. NN )
98	97	nncnd 10547	. . . . . . . . 9 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> N e. CC )
99		1cnd 9601	. . . . . . . . 9 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> 1 e. CC )
100		simpr 459	. . . . . . . . 9 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> N =/= 1 )
101	98, 99, 100	subne0d 9931	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( N - 1 ) =/= 0 )
102	96, 101	eqnetrd 2747	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) =/= 0 )
103		fveq2 5848	. . . . . . . . 9 |- ( ( F |` ( 0..^ ( N - 1 ) ) ) = (/) -> ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) = ( # ` (/) ) )
104		hash0 12420	. . . . . . . . 9 |- ( # ` (/) ) = 0
105	103, 104	syl6eq 2511	. . . . . . . 8 |- ( ( F |` ( 0..^ ( N - 1 ) ) ) = (/) -> ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) = 0 )
106	105	necon3i 2694	. . . . . . 7 |- ( ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) =/= 0 -> ( F |` ( 0..^ ( N - 1 ) ) ) =/= (/) )
107	102, 106	syl 16	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( F |` ( 0..^ ( N - 1 ) ) ) =/= (/) )
108	95, 107	jca 530	. . . . 5 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( ( F |` ( 0..^ ( N - 1 ) ) ) e. Word RR /\ ( F |` ( 0..^ ( N - 1 ) ) ) =/= (/) ) )
109		eldifsn 4141	. . . . 5 |- ( ( F |` ( 0..^ ( N - 1 ) ) ) e. (Word RR \ { (/) } ) <-> ( ( F |` ( 0..^ ( N - 1 ) ) ) e. Word RR /\ ( F |` ( 0..^ ( N - 1 ) ) ) =/= (/) ) )
110	108, 109	sylibr 212	. . . 4 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( F |` ( 0..^ ( N - 1 ) ) ) e. (Word RR \ { (/) } ) )
111	41	adantr 463	. . . 4 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( F ` ( N - 1 ) ) e. RR )
112	14, 15, 16, 17	signstfvn 28790	. . . 4 |- ( ( ( F |` ( 0..^ ( N - 1 ) ) ) e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) e. RR ) -> ( ( T ` ( ( F |` ( 0..^ ( N - 1 ) ) ) ++ <" ( F ` ( N - 1 ) ) "> ) ) ` ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) ) = ( ( ( T ` ( F |` ( 0..^ ( N - 1 ) ) ) ) ` ( ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) - 1 ) ) .+^ (sgn ` ( F ` ( N - 1 ) ) ) ) )
113	110, 111, 112	syl2anc 659	. . 3 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( ( T ` ( ( F |` ( 0..^ ( N - 1 ) ) ) ++ <" ( F ` ( N - 1 ) ) "> ) ) ` ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) ) = ( ( ( T ` ( F |` ( 0..^ ( N - 1 ) ) ) ) ` ( ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) - 1 ) ) .+^ (sgn ` ( F ` ( N - 1 ) ) ) ) )
114		lennncl 12550	. . . . . 6 |- ( ( ( F |` ( 0..^ ( N - 1 ) ) ) e. Word RR /\ ( F |` ( 0..^ ( N - 1 ) ) ) =/= (/) ) -> ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) e. NN )
115		fzo0end 11885	. . . . . 6 |- ( ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) e. NN -> ( ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) - 1 ) e. ( 0..^ ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) ) )
116	108, 114, 115	3syl 20	. . . . 5 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) - 1 ) e. ( 0..^ ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) ) )
117	14, 15, 16, 17	signstcl 28786	. . . . 5 |- ( ( ( F |` ( 0..^ ( N - 1 ) ) ) e. Word RR /\ ( ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) - 1 ) e. ( 0..^ ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) ) ) -> ( ( T ` ( F |` ( 0..^ ( N - 1 ) ) ) ) ` ( ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) - 1 ) ) e. { -u 1 , 0 , 1 } )
118	95, 116, 117	syl2anc 659	. . . 4 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( ( T ` ( F |` ( 0..^ ( N - 1 ) ) ) ) ` ( ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) - 1 ) ) e. { -u 1 , 0 , 1 } )
119	44	adantr 463	. . . 4 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> (sgn ` ( F ` ( N - 1 ) ) ) e. { -u 1 , 0 , 1 } )
120		simpr 459	. . . . . 6 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F ` ( N - 1 ) ) =/= 0 )
121		sgn0bi 28750	. . . . . . . 8 |- ( ( F ` ( N - 1 ) ) e. RR* -> ( (sgn ` ( F ` ( N - 1 ) ) ) = 0 <-> ( F ` ( N - 1 ) ) = 0 ) )
122	121	necon3bid 2712	. . . . . . 7 |- ( ( F ` ( N - 1 ) ) e. RR* -> ( (sgn ` ( F ` ( N - 1 ) ) ) =/= 0 <-> ( F ` ( N - 1 ) ) =/= 0 ) )
123	122	biimpar 483	. . . . . 6 |- ( ( ( F ` ( N - 1 ) ) e. RR* /\ ( F ` ( N - 1 ) ) =/= 0 ) -> (sgn ` ( F ` ( N - 1 ) ) ) =/= 0 )
124	42, 120, 123	syl2anc 659	. . . . 5 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> (sgn ` ( F ` ( N - 1 ) ) ) =/= 0 )
125	124	adantr 463	. . . 4 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> (sgn ` ( F ` ( N - 1 ) ) ) =/= 0 )
126	14, 15	signswlid 28780	. . . 4 |- ( ( ( ( ( T ` ( F |` ( 0..^ ( N - 1 ) ) ) ) ` ( ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) - 1 ) ) e. { -u 1 , 0 , 1 } /\ (sgn ` ( F ` ( N - 1 ) ) ) e. { -u 1 , 0 , 1 } ) /\ (sgn ` ( F ` ( N - 1 ) ) ) =/= 0 ) -> ( ( ( T ` ( F |` ( 0..^ ( N - 1 ) ) ) ) ` ( ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) - 1 ) ) .+^ (sgn ` ( F ` ( N - 1 ) ) ) ) = (sgn ` ( F ` ( N - 1 ) ) ) )
127	118, 119, 125, 126	syl21anc 1225	. . 3 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( ( ( T ` ( F |` ( 0..^ ( N - 1 ) ) ) ) ` ( ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) - 1 ) ) .+^ (sgn ` ( F ` ( N - 1 ) ) ) ) = (sgn ` ( F ` ( N - 1 ) ) ) )
128	90, 113, 127	3eqtrd 2499	. 2 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( ( T ` F ) ` ( N - 1 ) ) = (sgn ` ( F ` ( N - 1 ) ) ) )
129	48, 128	pm2.61dane 2772	1 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( ( T ` F ) ` ( N - 1 ) ) = (sgn ` ( F ` ( N - 1 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 367   = wceq 1398   e. wcel 1823   =/= wne 2649   \ cdif 3458   C_ wss 3461   (/)c0 3783   ifcif 3929   {csn 4016   {cpr 4018   {ctp 4020   <.cop 4022   |-> cmpt 4497   |` cres 4990   Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270   |-> cmpt2 6272   RRcr 9480   0cc0 9481   1c1 9482   RR*cxr 9616   - cmin 9796   -ucneg 9797   NNcn 10531   NN0cn0 10791   ZZcz 10860   ...cfz 11675  ..^cfzo 11799   #chash 12387  Word cword 12518   lastS clsw 12519   ++ cconcat 12520   <"cs1 12521   substr csubstr 12522  sgncsgn 13001   sum_csu 13590   ndxcnx 14713   Basecbs 14716   +g cplusg 14784   gsum_g cgsu 14930

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-seq 12090  df-hash 12388  df-word 12526  df-lsw 12527  df-concat 12528  df-s1 12529  df-substr 12530  df-sgn 13002  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-plusg 14797  df-0g 14931  df-gsum 14932  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-mulg 16259  df-cntz 16554

This theorem is referenced by:  signsvfpn  28806  signsvfnn  28807