Theorem signsvtn0 30647

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 4317	. . . . . . . . . . . 12 |- ( F e. (Word RR \ { (/) } ) <-> ( F e. Word RR /\ F =/= (/) ) )
2	1	biimpi 206	. . . . . . . . . . 11 |- ( F e. (Word RR \ { (/) } ) -> ( F e. Word RR /\ F =/= (/) ) )
3	2	adantr 481	. . . . . . . . . 10 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F e. Word RR /\ F =/= (/) ) )
4	3	simpld 475	. . . . . . . . 9 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> F e. Word RR )
5	4	adantr 481	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> F e. Word RR )
6		wrdf 13310	. . . . . . . 8 |- ( F e. Word RR -> F : ( 0..^ ( # ` F ) ) --> RR )
7	5, 6	syl 17	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> F : ( 0..^ ( # ` F ) ) --> RR )
8		lennncl 13325	. . . . . . . . . 10 |- ( ( F e. Word RR /\ F =/= (/) ) -> ( # ` F ) e. NN )
9	3, 8	syl 17	. . . . . . . . 9 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( # ` F ) e. NN )
10	9	adantr 481	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> ( # ` F ) e. NN )
11		lbfzo0 12507	. . . . . . . 8 |- ( 0 e. ( 0..^ ( # ` F ) ) <-> ( # ` F ) e. NN )
12	10, 11	sylibr 224	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> 0 e. ( 0..^ ( # ` F ) ) )
13	7, 12	ffvelrnd 6360	. . . . . 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 30645	. . . . . 6 |- ( ( F ` 0 ) e. RR -> ( T ` <" ( F ` 0 ) "> ) = <" (sgn ` ( F ` 0 ) ) "> )
19	13, 18	syl 17	. . . . 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 477	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> N = 1 )
22	20, 21	syl5eqr 2670	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> ( # ` F ) = 1 )
23		eqs1 13392	. . . . . . 7 |- ( ( F e. Word RR /\ ( # ` F ) = 1 ) -> F = <" ( F ` 0 ) "> )
24	5, 22, 23	syl2anc 693	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> F = <" ( F ` 0 ) "> )
25	24	fveq2d 6195	. . . . 5 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> ( T ` F ) = ( T ` <" ( F ` 0 ) "> ) )
26		oveq1 6657	. . . . . . . . . 10 |- ( N = 1 -> ( N - 1 ) = ( 1 - 1 ) )
27		1m1e0 11089	. . . . . . . . . 10 |- ( 1 - 1 ) = 0
28	26, 27	syl6eq 2672	. . . . . . . . 9 |- ( N = 1 -> ( N - 1 ) = 0 )
29	28	fveq2d 6195	. . . . . . . 8 |- ( N = 1 -> ( F ` ( N - 1 ) ) = ( F ` 0 ) )
30	29	fveq2d 6195	. . . . . . 7 |- ( N = 1 -> (sgn ` ( F ` ( N - 1 ) ) ) = (sgn ` ( F ` 0 ) ) )
31	30	s1eqd 13381	. . . . . 6 |- ( N = 1 -> <" (sgn ` ( F ` ( N - 1 ) ) ) "> = <" (sgn ` ( F ` 0 ) ) "> )
32	21, 31	syl 17	. . . . 5 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> <" (sgn ` ( F ` ( N - 1 ) ) ) "> = <" (sgn ` ( F ` 0 ) ) "> )
33	19, 25, 32	3eqtr4d 2666	. . . 4 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> ( T ` F ) = <" (sgn ` ( F ` ( N - 1 ) ) ) "> )
34	21, 28	syl 17	. . . 4 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> ( N - 1 ) = 0 )
35	33, 34	fveq12d 6197	. . 3 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> ( ( T ` F ) ` ( N - 1 ) ) = ( <" (sgn ` ( F ` ( N - 1 ) ) ) "> ` 0 ) )
36	4, 6	syl 17	. . . . . . . 8 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> F : ( 0..^ ( # ` F ) ) --> RR )
37	20	oveq1i 6660	. . . . . . . . 9 |- ( N - 1 ) = ( ( # ` F ) - 1 )
38		fzo0end 12560	. . . . . . . . . 10 |- ( ( # ` F ) e. NN -> ( ( # ` F ) - 1 ) e. ( 0..^ ( # ` F ) ) )
39	3, 8, 38	3syl 18	. . . . . . . . 9 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( ( # ` F ) - 1 ) e. ( 0..^ ( # ` F ) ) )
40	37, 39	syl5eqel 2705	. . . . . . . 8 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( N - 1 ) e. ( 0..^ ( # ` F ) ) )
41	36, 40	ffvelrnd 6360	. . . . . . 7 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F ` ( N - 1 ) ) e. RR )
42	41	rexrd 10089	. . . . . 6 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F ` ( N - 1 ) ) e. RR* )
43		sgncl 30600	. . . . . 6 |- ( ( F ` ( N - 1 ) ) e. RR* -> (sgn ` ( F ` ( N - 1 ) ) ) e. { -u 1 , 0 , 1 } )
44	42, 43	syl 17	. . . . 5 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> (sgn ` ( F ` ( N - 1 ) ) ) e. { -u 1 , 0 , 1 } )
45	44	adantr 481	. . . 4 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> (sgn ` ( F ` ( N - 1 ) ) ) e. { -u 1 , 0 , 1 } )
46		s1fv 13390	. . . 4 |- ( (sgn ` ( F ` ( N - 1 ) ) ) e. { -u 1 , 0 , 1 } -> ( <" (sgn ` ( F ` ( N - 1 ) ) ) "> ` 0 ) = (sgn ` ( F ` ( N - 1 ) ) ) )
47	45, 46	syl 17	. . 3 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> ( <" (sgn ` ( F ` ( N - 1 ) ) ) "> ` 0 ) = (sgn ` ( F ` ( N - 1 ) ) ) )
48	35, 47	eqtrd 2656	. 2 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N = 1 ) -> ( ( T ` F ) ` ( N - 1 ) ) = (sgn ` ( F ` ( N - 1 ) ) ) )
49		fzossfz 12488	. . . . . . . . . 10 |- ( 0..^ ( # ` F ) ) C_ ( 0 ... ( # ` F ) )
50	49, 39	sseldi 3601	. . . . . . . . 9 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( ( # ` F ) - 1 ) e. ( 0 ... ( # ` F ) ) )
51		swrd0val 13421	. . . . . . . . 9 |- ( ( F e. Word RR /\ ( ( # ` F ) - 1 ) e. ( 0 ... ( # ` F ) ) ) -> ( F substr <. 0 , ( ( # ` F ) - 1 ) >. ) = ( F |` ( 0..^ ( ( # ` F ) - 1 ) ) ) )
52	4, 50, 51	syl2anc 693	. . . . . . . 8 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F substr <. 0 , ( ( # ` F ) - 1 ) >. ) = ( F |` ( 0..^ ( ( # ` F ) - 1 ) ) ) )
53	52	oveq1d 6665	. . . . . . 7 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( ( F substr <. 0 , ( ( # ` F ) - 1 ) >. ) ++ <" ( lastS ` F ) "> ) = ( ( F |` ( 0..^ ( ( # ` F ) - 1 ) ) ) ++ <" ( lastS ` F ) "> ) )
54		swrdccatwrd 13468	. . . . . . . . 9 |- ( ( F e. Word RR /\ F =/= (/) ) -> ( ( F substr <. 0 , ( ( # ` F ) - 1 ) >. ) ++ <" ( lastS ` F ) "> ) = F )
55	54	eqcomd 2628	. . . . . . . 8 |- ( ( F e. Word RR /\ F =/= (/) ) -> F = ( ( F substr <. 0 , ( ( # ` F ) - 1 ) >. ) ++ <" ( lastS ` F ) "> ) )
56	3, 55	syl 17	. . . . . . 7 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> F = ( ( F substr <. 0 , ( ( # ` F ) - 1 ) >. ) ++ <" ( lastS ` F ) "> ) )
57	37	oveq2i 6661	. . . . . . . . . 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 5396	. . . . . . . 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 6195	. . . . . . . . . 10 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F ` ( N - 1 ) ) = ( F ` ( ( # ` F ) - 1 ) ) )
62		lsw 13351	. . . . . . . . . . 11 |- ( F e. (Word RR \ { (/) } ) -> ( lastS ` F ) = ( F ` ( ( # ` F ) - 1 ) ) )
63	62	adantr 481	. . . . . . . . . 10 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( lastS ` F ) = ( F ` ( ( # ` F ) - 1 ) ) )
64	61, 63	eqtr4d 2659	. . . . . . . . 9 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F ` ( N - 1 ) ) = ( lastS ` F ) )
65	64	s1eqd 13381	. . . . . . . 8 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> <" ( F ` ( N - 1 ) ) "> = <" ( lastS ` F ) "> )
66	59, 65	oveq12d 6668	. . . . . . 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 2666	. . . . . 6 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> F = ( ( F |` ( 0..^ ( N - 1 ) ) ) ++ <" ( F ` ( N - 1 ) ) "> ) )
68	67	fveq2d 6195	. . . . 5 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( T ` F ) = ( T ` ( ( F |` ( 0..^ ( N - 1 ) ) ) ++ <" ( F ` ( N - 1 ) ) "> ) ) )
69		ffn 6045	. . . . . . . . . . 11 |- ( F : ( 0..^ ( # ` F ) ) --> RR -> F Fn ( 0..^ ( # ` F ) ) )
70	4, 6, 69	3syl 18	. . . . . . . . . 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 6666	. . . . . . . . . . 11 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( 0..^ N ) = ( 0..^ ( # ` F ) ) )
73	72	fneq2d 5982	. . . . . . . . . 10 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F Fn ( 0..^ N ) <-> F Fn ( 0..^ ( # ` F ) ) ) )
74	70, 73	mpbird 247	. . . . . . . . 9 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> F Fn ( 0..^ N ) )
75	20, 9	syl5eqel 2705	. . . . . . . . . . 11 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> N e. NN )
76	75	nnnn0d 11351	. . . . . . . . . 10 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> N e. NN0 )
77		nn0z 11400	. . . . . . . . . 10 |- ( N e. NN0 -> N e. ZZ )
78		fzossrbm1 12497	. . . . . . . . . 10 |- ( N e. ZZ -> ( 0..^ ( N - 1 ) ) C_ ( 0..^ N ) )
79	76, 77, 78	3syl 18	. . . . . . . . 9 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( 0..^ ( N - 1 ) ) C_ ( 0..^ N ) )
80		fnssres 6004	. . . . . . . . 9 |- ( ( F Fn ( 0..^ N ) /\ ( 0..^ ( N - 1 ) ) C_ ( 0..^ N ) ) -> ( F |` ( 0..^ ( N - 1 ) ) ) Fn ( 0..^ ( N - 1 ) ) )
81	74, 79, 80	syl2anc 693	. . . . . . . 8 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F |` ( 0..^ ( N - 1 ) ) ) Fn ( 0..^ ( N - 1 ) ) )
82		hashfn 13164	. . . . . . . 8 |- ( ( F |` ( 0..^ ( N - 1 ) ) ) Fn ( 0..^ ( N - 1 ) ) -> ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) = ( # ` ( 0..^ ( N - 1 ) ) ) )
83	81, 82	syl 17	. . . . . . 7 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) = ( # ` ( 0..^ ( N - 1 ) ) ) )
84		nnm1nn0 11334	. . . . . . . 8 |- ( N e. NN -> ( N - 1 ) e. NN0 )
85		hashfzo0 13217	. . . . . . . 8 |- ( ( N - 1 ) e. NN0 -> ( # ` ( 0..^ ( N - 1 ) ) ) = ( N - 1 ) )
86	75, 84, 85	3syl 18	. . . . . . 7 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( # ` ( 0..^ ( N - 1 ) ) ) = ( N - 1 ) )
87	83, 86	eqtrd 2656	. . . . . 6 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) = ( N - 1 ) )
88	87	eqcomd 2628	. . . . 5 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( N - 1 ) = ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) )
89	68, 88	fveq12d 6197	. . . 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 481	. . 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 2659	. . . . . . . 8 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F substr <. 0 , ( ( # ` F ) - 1 ) >. ) = ( F |` ( 0..^ ( N - 1 ) ) ) )
92		swrdcl 13419	. . . . . . . . 9 |- ( F e. Word RR -> ( F substr <. 0 , ( ( # ` F ) - 1 ) >. ) e. Word RR )
93	4, 92	syl 17	. . . . . . . 8 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F substr <. 0 , ( ( # ` F ) - 1 ) >. ) e. Word RR )
94	91, 93	eqeltrrd 2702	. . . . . . 7 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F |` ( 0..^ ( N - 1 ) ) ) e. Word RR )
95	94	adantr 481	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( F |` ( 0..^ ( N - 1 ) ) ) e. Word RR )
96	87	adantr 481	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) = ( N - 1 ) )
97	75	adantr 481	. . . . . . . . . 10 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> N e. NN )
98	97	nncnd 11036	. . . . . . . . 9 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> N e. CC )
99		1cnd 10056	. . . . . . . . 9 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> 1 e. CC )
100		simpr 477	. . . . . . . . 9 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> N =/= 1 )
101	98, 99, 100	subne0d 10401	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( N - 1 ) =/= 0 )
102	96, 101	eqnetrd 2861	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) =/= 0 )
103		fveq2 6191	. . . . . . . . 9 |- ( ( F |` ( 0..^ ( N - 1 ) ) ) = (/) -> ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) = ( # ` (/) ) )
104		hash0 13158	. . . . . . . . 9 |- ( # ` (/) ) = 0
105	103, 104	syl6eq 2672	. . . . . . . 8 |- ( ( F |` ( 0..^ ( N - 1 ) ) ) = (/) -> ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) = 0 )
106	105	necon3i 2826	. . . . . . 7 |- ( ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) =/= 0 -> ( F |` ( 0..^ ( N - 1 ) ) ) =/= (/) )
107	102, 106	syl 17	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( F |` ( 0..^ ( N - 1 ) ) ) =/= (/) )
108	95, 107	jca 554	. . . . 5 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( ( F |` ( 0..^ ( N - 1 ) ) ) e. Word RR /\ ( F |` ( 0..^ ( N - 1 ) ) ) =/= (/) ) )
109		eldifsn 4317	. . . . 5 |- ( ( F |` ( 0..^ ( N - 1 ) ) ) e. (Word RR \ { (/) } ) <-> ( ( F |` ( 0..^ ( N - 1 ) ) ) e. Word RR /\ ( F |` ( 0..^ ( N - 1 ) ) ) =/= (/) ) )
110	108, 109	sylibr 224	. . . 4 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( F |` ( 0..^ ( N - 1 ) ) ) e. (Word RR \ { (/) } ) )
111	41	adantr 481	. . . 4 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( F ` ( N - 1 ) ) e. RR )
112	14, 15, 16, 17	signstfvn 30646	. . . 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 693	. . 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 13325	. . . . . 6 |- ( ( ( F |` ( 0..^ ( N - 1 ) ) ) e. Word RR /\ ( F |` ( 0..^ ( N - 1 ) ) ) =/= (/) ) -> ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) e. NN )
115		fzo0end 12560	. . . . . 6 |- ( ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) e. NN -> ( ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) - 1 ) e. ( 0..^ ( # ` ( F |` ( 0..^ ( N - 1 ) ) ) ) ) )
116	108, 114, 115	3syl 18	. . . . 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 30642	. . . . 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 693	. . . 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 481	. . . 4 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> (sgn ` ( F ` ( N - 1 ) ) ) e. { -u 1 , 0 , 1 } )
120		simpr 477	. . . . . 6 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( F ` ( N - 1 ) ) =/= 0 )
121		sgn0bi 30609	. . . . . . . 8 |- ( ( F ` ( N - 1 ) ) e. RR* -> ( (sgn ` ( F ` ( N - 1 ) ) ) = 0 <-> ( F ` ( N - 1 ) ) = 0 ) )
122	121	necon3bid 2838	. . . . . . 7 |- ( ( F ` ( N - 1 ) ) e. RR* -> ( (sgn ` ( F ` ( N - 1 ) ) ) =/= 0 <-> ( F ` ( N - 1 ) ) =/= 0 ) )
123	122	biimpar 502	. . . . . 6 |- ( ( ( F ` ( N - 1 ) ) e. RR* /\ ( F ` ( N - 1 ) ) =/= 0 ) -> (sgn ` ( F ` ( N - 1 ) ) ) =/= 0 )
124	42, 120, 123	syl2anc 693	. . . . 5 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> (sgn ` ( F ` ( N - 1 ) ) ) =/= 0 )
125	124	adantr 481	. . . 4 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> (sgn ` ( F ` ( N - 1 ) ) ) =/= 0 )
126	14, 15	signswlid 30636	. . . 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 1325	. . 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 2660	. 2 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) /\ N =/= 1 ) -> ( ( T ` F ) ` ( N - 1 ) ) = (sgn ` ( F ` ( N - 1 ) ) ) )
129	48, 128	pm2.61dane 2881	1 |- ( ( F e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) =/= 0 ) -> ( ( T ` F ) ` ( N - 1 ) ) = (sgn ` ( F ` ( N - 1 ) ) ) )

Syntax hints:   -> wi 4   /\ 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   |-> cmpt 4729   |` cres 5116   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   RRcr 9935   0cc0 9936   1c1 9937   RR*cxr 10073   - cmin 10266   -ucneg 10267   NNcn 11020   NN0cn0 11292   ZZcz 11377   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291   lastS clsw 13292   ++ cconcat 13293   <"cs1 13294   substr csubstr 13295  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

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-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-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-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-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  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  df-mulg 17541  df-cntz 17750

This theorem is referenced by:  signsvfpn  30662  signsvfnn  30663