Theorem signsvtn0 27135

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 4111	. . . . . . . . . . . 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 465	. . . . . . . . . 10 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F e. Word RR /\ F =/= (/) ) )
4	3	simpld 459	. . . . . . . . 9 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> F e. Word RR )
5	4	adantr 465	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> F e. Word RR )
6		wrdf 12361	. . . . . . . 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 12371	. . . . . . . . . 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 465	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> ( # ` F ) e. NN )
11		lbfzo0 11706	. . . . . . . 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 5956	. . . . . 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 27133	. . . . . 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	. . . . . . . . 9 |- N = ( # ` F )
21	20	a1i 11	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> N = ( # ` F ) )
22		simpr 461	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> N = 1 )
23	21, 22	eqtr3d 2497	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> ( # ` F ) = 1 )
24		eqs1 12421	. . . . . . 7 |- ( ( F e. Word RR /\ ( # ` F ) = 1 ) -> F = <" ( F ` 0 ) "> )
25	5, 23, 24	syl2anc 661	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> F = <" ( F ` 0 ) "> )
26	25	fveq2d 5806	. . . . 5 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> ( T ` F ) = ( T ` <" ( F ` 0 ) "> ) )
27		oveq1 6210	. . . . . . . . . 10 |- ( N = 1 -> ( N - 1 ) = ( 1 - 1 ) )
28		1m1e0 10504	. . . . . . . . . 10 |- ( 1 - 1 ) = 0
29	27, 28	syl6eq 2511	. . . . . . . . 9 |- ( N = 1 -> ( N - 1 ) = 0 )
30	29	fveq2d 5806	. . . . . . . 8 |- ( N = 1 -> ( F ` ( N - 1 ) ) = ( F ` 0 ) )
31	30	fveq2d 5806	. . . . . . 7 |- ( N = 1 -> (sgn ` ( F ` ( N - 1 ) ) ) = (sgn ` ( F ` 0 ) ) )
32	31	s1eqd 12413	. . . . . 6 |- ( N = 1 -> <" (sgn ` ( F ` ( N - 1 ) ) ) "> = <" (sgn ` ( F ` 0 ) ) "> )
33	22, 32	syl 16	. . . . 5 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> <" (sgn ` ( F ` ( N - 1 ) ) ) "> = <" (sgn ` ( F ` 0 ) ) "> )
34	19, 26, 33	3eqtr4d 2505	. . . 4 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> ( T ` F ) = <" (sgn ` ( F ` ( N - 1 ) ) ) "> )
35	22, 29	syl 16	. . . 4 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> ( N - 1 ) = 0 )
36	34, 35	fveq12d 5808	. . 3 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> ( ( T ` F ) ` ( N - 1 ) ) = ( <" (sgn ` ( F ` ( N - 1 ) ) ) "> ` 0 ) )
37	4, 6	syl 16	. . . . . . . 8 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> F : ( 0..^ ( # ` F ) ) --> RR )
38	20	oveq1i 6213	. . . . . . . . 9 |- ( N - 1 ) = ( ( # ` F ) - 1 )
39		fzo0end 11739	. . . . . . . . . 10 |- ( ( # ` F ) e. NN -> ( ( # ` F ) - 1 ) e. ( 0..^ ( # ` F ) ) )
40	3, 8, 39	3syl 20	. . . . . . . . 9 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( ( # ` F ) - 1 ) e. ( 0..^ ( # ` F ) ) )
41	38, 40	syl5eqel 2546	. . . . . . . 8 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( N - 1 ) e. ( 0..^ ( # ` F ) ) )
42	37, 41	ffvelrnd 5956	. . . . . . 7 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F ` ( N - 1 ) ) e. RR )
43	42	rexrd 9547	. . . . . 6 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F ` ( N - 1 ) ) e. RR* )
44		sgncl 27085	. . . . . 6 |- ( ( F ` ( N - 1 ) ) e. RR* -> (sgn ` ( F ` ( N - 1 ) ) ) e. { -u 1 , 0 , 1 } )
45	43, 44	syl 16	. . . . 5 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> (sgn ` ( F ` ( N - 1 ) ) ) e. { -u 1 , 0 , 1 } )
46	45	adantr 465	. . . 4 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> (sgn ` ( F ` ( N - 1 ) ) ) e. { -u 1 , 0 , 1 } )
47		s1fv 12419	. . . 4 |- ( (sgn ` ( F ` ( N - 1 ) ) ) e. { -u 1 , 0 , 1 } -> ( <" (sgn ` ( F ` ( N - 1 ) ) ) "> ` 0 ) = (sgn ` ( F ` ( N - 1 ) ) ) )
48	46, 47	syl 16	. . 3 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> ( <" (sgn ` ( F ` ( N - 1 ) ) ) "> ` 0 ) = (sgn ` ( F ` ( N - 1 ) ) ) )
49	36, 48	eqtrd 2495	. 2 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> ( ( T ` F ) ` ( N - 1 ) ) = (sgn ` ( F ` ( N - 1 ) ) ) )
50		fzossfz 11690	. . . . . . . . . 10 |- ( 0..^ ( # ` F ) ) C_ ( 0 ... ( # ` F ) )
51	50, 40	sseldi 3465	. . . . . . . . 9 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( ( # ` F ) - 1 ) e. ( 0 ... ( # ` F ) ) )
52		swrd0val 12438	. . . . . . . . 9 |- ( ( F e. Word RR /\ ( ( # ` F ) - 1 ) e. ( 0 ... ( # ` F ) ) ) -> ( F substr <. 0 , ( ( # ` F ) - 1 ) >. ) = ( F |` ( 0..^ ( ( # ` F ) - 1 ) ) ) )
53	4, 51, 52	syl2anc 661	. . . . . . . 8 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F substr <. 0 , ( ( # ` F ) - 1 ) >. ) = ( F |` ( 0..^ ( ( # ` F ) - 1 ) ) ) )
54	53	oveq1d 6218	. . . . . . 7 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( ( F substr <. 0 , ( ( # ` F ) - 1 ) >. ) concat <" ( F ` ( ( # ` F ) - 1 ) ) "> ) = ( ( F |` ( 0..^ ( ( # ` F ) - 1 ) ) ) concat <" ( F ` ( ( # ` F ) - 1 ) ) "> ) )
55		wrdeqcats1 12489	. . . . . . . 8 |- ( ( F e. Word RR /\ F =/= (/) ) -> F = ( ( F substr <. 0 , ( ( # ` F ) - 1 ) >. ) concat <" ( F ` ( ( # ` F ) - 1 ) ) "> ) )
56	3, 55	syl 16	. . . . . . 7 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> F = ( ( F substr <. 0 , ( ( # ` F ) - 1 ) >. ) concat <" ( F ` ( ( # ` F ) - 1 ) ) "> ) )
57	38	oveq2i 6214	. . . . . . . . . 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 5221	. . . . . . . 8 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F |` ( 0..^ ( N - 1 ) ) ) = ( F |` ( 0..^ ( ( # ` F ) - 1 ) ) ) )
60	38	a1i 11	. . . . . . . . . 10 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( N - 1 ) = ( ( # ` F ) - 1 ) )
61	60	fveq2d 5806	. . . . . . . . 9 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F ` ( N - 1 ) ) = ( F ` ( ( # ` F ) - 1 ) ) )
62	61	s1eqd 12413	. . . . . . . 8 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> <" ( F ` ( N - 1 ) ) "> = <" ( F ` ( ( # ` F ) - 1 ) ) "> )
63	59, 62	oveq12d 6221	. . . . . . 7 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( ( F |` ( 0..^ ( N - 1 ) ) ) concat <" ( F ` ( N - 1 ) ) "> ) = ( ( F |` ( 0..^ ( ( # ` F ) - 1 ) ) ) concat <" ( F ` ( ( # ` F ) - 1 ) ) "> ) )
64	54, 56, 63	3eqtr4d 2505	. . . . . 6 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> F = ( ( F |` ( 0..^ ( N - 1 ) ) ) concat <" ( F ` ( N - 1 ) ) "> ) )
65	64	fveq2d 5806	. . . . 5 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( T ` F ) = ( T ` ( ( F |` ( 0..^ ( N - 1 ) ) ) concat <" ( F ` ( N - 1 ) ) "> ) ) )
66		ffn 5670	. . . . . . . . . . 11 |- ( F : ( 0..^ ( # ` F ) ) --> RR -> F Fn ( 0..^ ( # ` F ) ) )
67	37, 66	syl 16	. . . . . . . . . 10 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> F Fn ( 0..^ ( # ` F ) ) )
68	20	a1i 11	. . . . . . . . . . . 12 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> N = ( # ` F ) )
69	68	oveq2d 6219	. . . . . . . . . . 11 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( 0..^ N ) = ( 0..^ ( # ` F ) ) )
70	69	fneq2d 5613	. . . . . . . . . 10 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F Fn ( 0..^ N ) <-> F Fn ( 0..^ ( # ` F ) ) ) )
71	67, 70	mpbird 232	. . . . . . . . 9 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> F Fn ( 0..^ N ) )
72	20, 9	syl5eqel 2546	. . . . . . . . . . 11 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> N e. NN )
73	72	nnnn0d 10750	. . . . . . . . . 10 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> N e. NN0 )
74		fzossrbm1 11698	. . . . . . . . . 10 |- ( N e. NN0 -> ( 0..^ ( N - 1 ) ) C_ ( 0..^ N ) )
75	73, 74	syl 16	. . . . . . . . 9 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( 0..^ ( N - 1 ) ) C_ ( 0..^ N ) )
76		fnssres 5635	. . . . . . . . 9 |- ( ( F Fn ( 0..^ N ) /\ ( 0..^ ( N - 1 ) ) C_ ( 0..^ N ) ) -> ( F |` ( 0..^ ( N - 1 ) ) ) Fn ( 0..^ ( N - 1 ) ) )
77	71, 75, 76	syl2anc 661	. . . . . . . 8 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F |` ( 0..^ ( N - 1 ) ) ) Fn ( 0..^ ( N - 1 ) ) )
78		hashfn 12259	. . . . . . . 8 |- ( ( F |` ( 0..^ ( N - 1 ) ) ) Fn ( 0..^ ( N - 1 ) ) -> ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) = ( # ` ( 0..^ ( N - 1 ) ) ) )
79	77, 78	syl 16	. . . . . . 7 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) = ( # ` ( 0..^ ( N - 1 ) ) ) )
80		nnm1nn0 10735	. . . . . . . . 9 |- ( N e. NN -> ( N - 1 ) e. NN0 )
81	72, 80	syl 16	. . . . . . . 8 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( N - 1 ) e. NN0 )
82		hashfzo0 12312	. . . . . . . 8 |- ( ( N - 1 ) e. NN0 -> ( # ` ( 0..^ ( N - 1 ) ) ) = ( N - 1 ) )
83	81, 82	syl 16	. . . . . . 7 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( # ` ( 0..^ ( N - 1 ) ) ) = ( N - 1 ) )
84	79, 83	eqtrd 2495	. . . . . 6 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) = ( N - 1 ) )
85	84	eqcomd 2462	. . . . 5 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( N - 1 ) = ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) )
86	65, 85	fveq12d 5808	. . . 4 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( ( T ` F ) ` ( N - 1 ) ) = ( ( T ` ( ( F |` ( 0..^ ( N - 1 ) ) ) concat <" ( F ` ( N - 1 ) ) "> ) ) ` ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) ) )
87	86	adantr 465	. . 3 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( ( T ` F ) ` ( N - 1 ) ) = ( ( T ` ( ( F |` ( 0..^ ( N - 1 ) ) ) concat <" ( F ` ( N - 1 ) ) "> ) ) ` ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) ) )
88	53, 59	eqtr4d 2498	. . . . . . . . 9 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F substr <. 0 , ( ( # ` F ) - 1 ) >. ) = ( F |` ( 0..^ ( N - 1 ) ) ) )
89		swrdcl 12436	. . . . . . . . . 10 |- ( F e. Word RR -> ( F substr <. 0 , ( ( # ` F ) - 1 ) >. ) e. Word RR )
90	4, 89	syl 16	. . . . . . . . 9 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F substr <. 0 , ( ( # ` F ) - 1 ) >. ) e. Word RR )
91	88, 90	eqeltrrd 2543	. . . . . . . 8 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F |` ( 0..^ ( N - 1 ) ) ) e. Word RR )
92	91	adantr 465	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( F |` ( 0..^ ( N - 1 ) ) ) e. Word RR )
93	84	adantr 465	. . . . . . . . 9 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) = ( N - 1 ) )
94	72	adantr 465	. . . . . . . . . . 11 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> N e. NN )
95	94	nncnd 10452	. . . . . . . . . 10 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> N e. CC )
96		ax-1cn 9454	. . . . . . . . . . 11 |- 1 e. CC
97	96	a1i 11	. . . . . . . . . 10 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> 1 e. CC )
98		simpr 461	. . . . . . . . . 10 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> N =/= 1 )
99	95, 97, 98	subne0d 9842	. . . . . . . . 9 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( N - 1 ) =/= 0 )
100	93, 99	eqnetrd 2745	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) =/= 0 )
101		fveq2 5802	. . . . . . . . . 10 |- ( ( F |` ( 0..^ ( N - 1 ) ) ) = (/) -> ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) = ( # ` (/) ) )
102		hash0 12255	. . . . . . . . . 10 |- ( # ` (/) ) = 0
103	101, 102	syl6eq 2511	. . . . . . . . 9 |- ( ( F |` ( 0..^ ( N - 1 ) ) ) = (/) -> ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) = 0 )
104	103	necon3i 2692	. . . . . . . 8 |- ( ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) =/= 0 -> ( F |` ( 0..^ ( N - 1 ) ) ) =/= (/) )
105	100, 104	syl 16	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( F |` ( 0..^ ( N - 1 ) ) ) =/= (/) )
106	92, 105	jca 532	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( ( F |` ( 0..^ ( N - 1 ) ) ) e. Word RR /\ ( F |` ( 0..^ ( N - 1 ) ) ) =/= (/) ) )
107		eldifsn 4111	. . . . . 6 |- ( ( F |` ( 0..^ ( N - 1 ) ) ) e. (Word RR \ { (/) } ) <-> ( ( F |` ( 0..^ ( N - 1 ) ) ) e. Word RR /\ ( F |` ( 0..^ ( N - 1 ) ) ) =/= (/) ) )
108	106, 107	sylibr 212	. . . . 5 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( F |` ( 0..^ ( N - 1 ) ) ) e. (Word RR \ { (/) } ) )
109	42	adantr 465	. . . . 5 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( F ` ( N - 1 ) ) e. RR )
110	14, 15, 16, 17	signstfvn 27134	. . . . 5 |- ( ( ( F |` ( 0..^ ( N - 1 ) ) ) e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) e. RR ) -> ( ( T ` ( ( F |` ( 0..^ ( N - 1 ) ) ) concat <" ( F ` ( N - 1 ) ) "> ) ) ` ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) ) = ( ( ( T ` ( F |` ( 0..^ ( N - 1 ) ) ) ) ` ( ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) - 1 ) ) .+^ (sgn ` ( F ` ( N - 1 ) ) ) ) )
111	108, 109, 110	syl2anc 661	. . . 4 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( ( T ` ( ( F |` ( 0..^ ( N - 1 ) ) ) concat <" ( F ` ( N - 1 ) ) "> ) ) ` ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) ) = ( ( ( T ` ( F |` ( 0..^ ( N - 1 ) ) ) ) ` ( ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) - 1 ) ) .+^ (sgn ` ( F ` ( N - 1 ) ) ) ) )
112	106	simpld 459	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( F |` ( 0..^ ( N - 1 ) ) ) e. Word RR )
113		lennncl 12371	. . . . . . 7 |- ( ( ( F |` ( 0..^ ( N - 1 ) ) ) e. Word RR /\ ( F |` ( 0..^ ( N - 1 ) ) ) =/= (/) ) -> ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) e. NN )
114		fzo0end 11739	. . . . . . 7 |- ( ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) e. NN -> ( ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) - 1 ) e. ( 0..^ ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) ) )
115	106, 113, 114	3syl 20	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) - 1 ) e. ( 0..^ ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) ) )
116	14, 15, 16, 17	signstcl 27130	. . . . . 6 |- ( ( ( 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 } )
117	112, 115, 116	syl2anc 661	. . . . 5 |- ( ( ( 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 } )
118	45	adantr 465	. . . . 5 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> (sgn ` ( F ` ( N - 1 ) ) ) e. { -u 1 , 0 , 1 } )
119		simpr 461	. . . . . . 7 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F ` ( N - 1 ) ) =/= 0 )
120		sgn0bi 27094	. . . . . . . . 9 |- ( ( F ` ( N - 1 ) ) e. RR* -> ( (sgn ` ( F ` ( N - 1 ) ) ) = 0 <-> ( F ` ( N - 1 ) ) = 0 ) )
121	120	necon3bid 2710	. . . . . . . 8 |- ( ( F ` ( N - 1 ) ) e. RR* -> ( (sgn ` ( F ` ( N - 1 ) ) ) =/= 0 <-> ( F ` ( N - 1 ) ) =/= 0 ) )
122	121	biimpar 485	. . . . . . 7 |- ( ( ( F ` ( N - 1 ) ) e. RR* /\ ( F ` ( N - 1 ) ) =/= 0 ) -> (sgn ` ( F ` ( N - 1 ) ) ) =/= 0 )
123	43, 119, 122	syl2anc 661	. . . . . 6 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> (sgn ` ( F ` ( N - 1 ) ) ) =/= 0 )
124	123	adantr 465	. . . . 5 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> (sgn ` ( F ` ( N - 1 ) ) ) =/= 0 )
125	14, 15	signswlid 27124	. . . . 5 |- ( ( ( ( ( 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 ) ) ) )
126	117, 118, 124, 125	syl21anc 1218	. . . 4 |- ( ( ( 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 ) ) ) )
127	111, 126	eqtrd 2495	. . 3 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( ( T ` ( ( F |` ( 0..^ ( N - 1 ) ) ) concat <" ( F ` ( N - 1 ) ) "> ) ) ` ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) ) = (sgn ` ( F ` ( N - 1 ) ) ) )
128	87, 127	eqtrd 2495	. 2 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( ( T ` F ) ` ( N - 1 ) ) = (sgn ` ( F ` ( N - 1 ) ) ) )
129		exmidne 2658	. . 3 |- ( N = 1 \/ N =/= 1 )
130	129	a1i 11	. 2 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( N = 1 \/ N =/= 1 ) )
131	49, 128, 130	mpjaodan 784	1 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( ( T ` F ) ` ( N - 1 ) ) = (sgn ` ( F ` ( N - 1 ) ) ) )

Syntax hints:   -> wi 4   \/ wo 368   /\ wa 369   = wceq 1370   e. wcel 1758   =/= wne 2648   \ cdif 3436   C_ wss 3439   (/)c0 3748   ifcif 3902   {csn 3988   {cpr 3990   {ctp 3992   <.cop 3994   |-> cmpt 4461   |` cres 4953   Fn wfn 5524   -->wf 5525   ` cfv 5529  (class class class)co 6203   |-> cmpt2 6205   CCcc 9394   RRcr 9395   0cc0 9396   1c1 9397   RR*cxr 9531   - cmin 9709   -ucneg 9710   NNcn 10436   NN0cn0 10693   ...cfz 11557  ..^cfzo 11668   #chash 12223  Word cword 12342   concat cconcat 12344   <"cs1 12345   substr csubstr 12346  sgncsgn 12696   sum_csu 13284   ndxcnx 14292   Basecbs 14295   +g cplusg 14360   gsum_g cgsu 14501

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7961  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-supp 6804  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-oi 7838  df-card 8223  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-n0 10694  df-z 10761  df-uz 10976  df-fz 11558  df-fzo 11669  df-seq 11927  df-hash 12224  df-word 12350  df-concat 12352  df-s1 12353  df-substr 12354  df-sgn 12697  df-struct 14297  df-ndx 14298  df-slot 14299  df-base 14300  df-plusg 14373  df-0g 14502  df-gsum 14503  df-mnd 15537  df-mulg 15670  df-cntz 15957

This theorem is referenced by:  signsvfpn  27150  signsvfnn  27151