Theorem signstfveq0 30654

Description: In case the last letter is zero, the zero skipping sign is the same as the previous letter. (Contributed by Thierry Arnoux, 11-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 ) )
signstfveq0.1	|- N = ( # ` F )

Assertion

Ref	Expression
signstfveq0	|- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( ( T ` F ) ` ( N - 1 ) ) = ( ( T ` F ) ` ( N - 2 ) ) )

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 signstfveq0

Step	Hyp	Ref	Expression
1		simpll 790	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> F e. (Word RR \ { (/) } ) )
2	1	eldifad 3586	. . . . 5 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> F e. Word RR )
3		swrdcl 13419	. . . . 5 |- ( F e. Word RR -> ( F substr <. 0 , ( N - 1 ) >. ) e. Word RR )
4	2, 3	syl 17	. . . 4 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( F substr <. 0 , ( N - 1 ) >. ) e. Word RR )
5		1nn0 11308	. . . . . . . . . . 11 |- 1 e. NN0
6	5	a1i 11	. . . . . . . . . 10 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> 1 e. NN0 )
7	6	nn0red 11352	. . . . . . . . . . 11 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> 1 e. RR )
8		2re 11090	. . . . . . . . . . . 12 |- 2 e. RR
9	8	a1i 11	. . . . . . . . . . 11 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> 2 e. RR )
10		signstfveq0.1	. . . . . . . . . . . . 13 |- N = ( # ` F )
11		lencl 13324	. . . . . . . . . . . . . 14 |- ( F e. Word RR -> ( # ` F ) e. NN0 )
12	2, 11	syl 17	. . . . . . . . . . . . 13 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( # ` F ) e. NN0 )
13	10, 12	syl5eqel 2705	. . . . . . . . . . . 12 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> N e. NN0 )
14	13	nn0red 11352	. . . . . . . . . . 11 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> N e. RR )
15		1le2 11241	. . . . . . . . . . . 12 |- 1 <_ 2
16	15	a1i 11	. . . . . . . . . . 11 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> 1 <_ 2 )
17		signsv.p	. . . . . . . . . . . . . 14 |- .+^ = ( a e. { -u 1 , 0 , 1 } , b e. { -u 1 , 0 , 1 } |-> if ( b = 0 , a , b ) )
18		signsv.w	. . . . . . . . . . . . . 14 |- W = { <. ( Base ` ndx ) , { -u 1 , 0 , 1 } >. , <. ( +g ` ndx ) , .+^ >. }
19		signsv.t	. . . . . . . . . . . . . 14 |- T = ( f e. Word RR |-> ( n e. ( 0..^ ( # ` f ) ) |-> ( W gsumg ( i e. ( 0 ... n ) |-> (sgn ` ( f ` i ) ) ) ) ) )
20		signsv.v	. . . . . . . . . . . . . 14 |- V = ( f e. Word RR |-> sum_ j e. ( 1..^ ( # ` f ) ) if ( ( ( T ` f ) ` j ) =/= ( ( T ` f ) ` ( j - 1 ) ) , 1 , 0 ) )
21	17, 18, 19, 20, 10	signstfveq0a 30653	. . . . . . . . . . . . 13 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> N e. ( ZZ>= ` 2 ) )
22		eluz2 11693	. . . . . . . . . . . . 13 |- ( N e. ( ZZ>= ` 2 ) <-> ( 2 e. ZZ /\ N e. ZZ /\ 2 <_ N ) )
23	21, 22	sylib 208	. . . . . . . . . . . 12 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( 2 e. ZZ /\ N e. ZZ /\ 2 <_ N ) )
24	23	simp3d 1075	. . . . . . . . . . 11 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> 2 <_ N )
25	7, 9, 14, 16, 24	letrd 10194	. . . . . . . . . 10 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> 1 <_ N )
26		fznn0 12432	. . . . . . . . . . 11 |- ( N e. NN0 -> ( 1 e. ( 0 ... N ) <-> ( 1 e. NN0 /\ 1 <_ N ) ) )
27	13, 26	syl 17	. . . . . . . . . 10 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( 1 e. ( 0 ... N ) <-> ( 1 e. NN0 /\ 1 <_ N ) ) )
28	6, 25, 27	mpbir2and 957	. . . . . . . . 9 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> 1 e. ( 0 ... N ) )
29		fznn0sub2 12446	. . . . . . . . 9 |- ( 1 e. ( 0 ... N ) -> ( N - 1 ) e. ( 0 ... N ) )
30	28, 29	syl 17	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( N - 1 ) e. ( 0 ... N ) )
31	10	oveq2i 6661	. . . . . . . 8 |- ( 0 ... N ) = ( 0 ... ( # ` F ) )
32	30, 31	syl6eleq 2711	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( N - 1 ) e. ( 0 ... ( # ` F ) ) )
33		swrd0len 13422	. . . . . . 7 |- ( ( F e. Word RR /\ ( N - 1 ) e. ( 0 ... ( # ` F ) ) ) -> ( # ` ( F substr <. 0 , ( N - 1 ) >. ) ) = ( N - 1 ) )
34	2, 32, 33	syl2anc 693	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( # ` ( F substr <. 0 , ( N - 1 ) >. ) ) = ( N - 1 ) )
35		uz2m1nn 11763	. . . . . . 7 |- ( N e. ( ZZ>= ` 2 ) -> ( N - 1 ) e. NN )
36	21, 35	syl 17	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( N - 1 ) e. NN )
37	34, 36	eqeltrd 2701	. . . . 5 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( # ` ( F substr <. 0 , ( N - 1 ) >. ) ) e. NN )
38		nnne0 11053	. . . . . 6 |- ( ( # ` ( F substr <. 0 , ( N - 1 ) >. ) ) e. NN -> ( # ` ( F substr <. 0 , ( N - 1 ) >. ) ) =/= 0 )
39		fveq2 6191	. . . . . . . 8 |- ( ( F substr <. 0 , ( N - 1 ) >. ) = (/) -> ( # ` ( F substr <. 0 , ( N - 1 ) >. ) ) = ( # ` (/) ) )
40		hash0 13158	. . . . . . . 8 |- ( # ` (/) ) = 0
41	39, 40	syl6eq 2672	. . . . . . 7 |- ( ( F substr <. 0 , ( N - 1 ) >. ) = (/) -> ( # ` ( F substr <. 0 , ( N - 1 ) >. ) ) = 0 )
42	41	necon3i 2826	. . . . . 6 |- ( ( # ` ( F substr <. 0 , ( N - 1 ) >. ) ) =/= 0 -> ( F substr <. 0 , ( N - 1 ) >. ) =/= (/) )
43	38, 42	syl 17	. . . . 5 |- ( ( # ` ( F substr <. 0 , ( N - 1 ) >. ) ) e. NN -> ( F substr <. 0 , ( N - 1 ) >. ) =/= (/) )
44	37, 43	syl 17	. . . 4 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( F substr <. 0 , ( N - 1 ) >. ) =/= (/) )
45		eldifsn 4317	. . . 4 |- ( ( F substr <. 0 , ( N - 1 ) >. ) e. (Word RR \ { (/) } ) <-> ( ( F substr <. 0 , ( N - 1 ) >. ) e. Word RR /\ ( F substr <. 0 , ( N - 1 ) >. ) =/= (/) ) )
46	4, 44, 45	sylanbrc 698	. . 3 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( F substr <. 0 , ( N - 1 ) >. ) e. (Word RR \ { (/) } ) )
47		simpr 477	. . . 4 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( F ` ( N - 1 ) ) = 0 )
48		0re 10040	. . . 4 |- 0 e. RR
49	47, 48	syl6eqel 2709	. . 3 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( F ` ( N - 1 ) ) e. RR )
50	17, 18, 19, 20	signstfvn 30646	. . 3 |- ( ( ( F substr <. 0 , ( N - 1 ) >. ) e. (Word RR \ { (/) } ) /\ ( F ` ( N - 1 ) ) e. RR ) -> ( ( T ` ( ( F substr <. 0 , ( N - 1 ) >. ) ++ <" ( F ` ( N - 1 ) ) "> ) ) ` ( # ` ( F substr <. 0 , ( N - 1 ) >. ) ) ) = ( ( ( T ` ( F substr <. 0 , ( N - 1 ) >. ) ) ` ( ( # ` ( F substr <. 0 , ( N - 1 ) >. ) ) - 1 ) ) .+^ (sgn ` ( F ` ( N - 1 ) ) ) ) )
51	46, 49, 50	syl2anc 693	. 2 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( ( T ` ( ( F substr <. 0 , ( N - 1 ) >. ) ++ <" ( F ` ( N - 1 ) ) "> ) ) ` ( # ` ( F substr <. 0 , ( N - 1 ) >. ) ) ) = ( ( ( T ` ( F substr <. 0 , ( N - 1 ) >. ) ) ` ( ( # ` ( F substr <. 0 , ( N - 1 ) >. ) ) - 1 ) ) .+^ (sgn ` ( F ` ( N - 1 ) ) ) ) )
52	10	oveq1i 6660	. . . . . . . . 9 |- ( N - 1 ) = ( ( # ` F ) - 1 )
53	52	opeq2i 4406	. . . . . . . 8 |- <. 0 , ( N - 1 ) >. = <. 0 , ( ( # ` F ) - 1 ) >.
54	53	oveq2i 6661	. . . . . . 7 |- ( F substr <. 0 , ( N - 1 ) >. ) = ( F substr <. 0 , ( ( # ` F ) - 1 ) >. )
55	54	a1i 11	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( F substr <. 0 , ( N - 1 ) >. ) = ( F substr <. 0 , ( ( # ` F ) - 1 ) >. ) )
56		lsw 13351	. . . . . . . . . 10 |- ( F e. (Word RR \ { (/) } ) -> ( lastS ` F ) = ( F ` ( ( # ` F ) - 1 ) ) )
57	56	ad2antrr 762	. . . . . . . . 9 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( lastS ` F ) = ( F ` ( ( # ` F ) - 1 ) ) )
58	10	eqcomi 2631	. . . . . . . . . . 11 |- ( # ` F ) = N
59	58	oveq1i 6660	. . . . . . . . . 10 |- ( ( # ` F ) - 1 ) = ( N - 1 )
60	59	fveq2i 6194	. . . . . . . . 9 |- ( F ` ( ( # ` F ) - 1 ) ) = ( F ` ( N - 1 ) )
61	57, 60	syl6eq 2672	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( lastS ` F ) = ( F ` ( N - 1 ) ) )
62	61	s1eqd 13381	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> <" ( lastS ` F ) "> = <" ( F ` ( N - 1 ) ) "> )
63	62	eqcomd 2628	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> <" ( F ` ( N - 1 ) ) "> = <" ( lastS ` F ) "> )
64	55, 63	oveq12d 6668	. . . . 5 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( ( F substr <. 0 , ( N - 1 ) >. ) ++ <" ( F ` ( N - 1 ) ) "> ) = ( ( F substr <. 0 , ( ( # ` F ) - 1 ) >. ) ++ <" ( lastS ` F ) "> ) )
65		eldifsn 4317	. . . . . . 7 |- ( F e. (Word RR \ { (/) } ) <-> ( F e. Word RR /\ F =/= (/) ) )
66	1, 65	sylib 208	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( F e. Word RR /\ F =/= (/) ) )
67		swrdccatwrd 13468	. . . . . 6 |- ( ( F e. Word RR /\ F =/= (/) ) -> ( ( F substr <. 0 , ( ( # ` F ) - 1 ) >. ) ++ <" ( lastS ` F ) "> ) = F )
68	66, 67	syl 17	. . . . 5 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( ( F substr <. 0 , ( ( # ` F ) - 1 ) >. ) ++ <" ( lastS ` F ) "> ) = F )
69	64, 68	eqtrd 2656	. . . 4 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( ( F substr <. 0 , ( N - 1 ) >. ) ++ <" ( F ` ( N - 1 ) ) "> ) = F )
70	69	fveq2d 6195	. . 3 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( T ` ( ( F substr <. 0 , ( N - 1 ) >. ) ++ <" ( F ` ( N - 1 ) ) "> ) ) = ( T ` F ) )
71	70, 34	fveq12d 6197	. 2 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( ( T ` ( ( F substr <. 0 , ( N - 1 ) >. ) ++ <" ( F ` ( N - 1 ) ) "> ) ) ` ( # ` ( F substr <. 0 , ( N - 1 ) >. ) ) ) = ( ( T ` F ) ` ( N - 1 ) ) )
72	13	nn0cnd 11353	. . . . . . . . . 10 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> N e. CC )
73		1cnd 10056	. . . . . . . . . 10 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> 1 e. CC )
74	72, 73, 73	subsub4d 10423	. . . . . . . . 9 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( ( N - 1 ) - 1 ) = ( N - ( 1 + 1 ) ) )
75		1p1e2 11134	. . . . . . . . . 10 |- ( 1 + 1 ) = 2
76	75	oveq2i 6661	. . . . . . . . 9 |- ( N - ( 1 + 1 ) ) = ( N - 2 )
77	74, 76	syl6eq 2672	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( ( N - 1 ) - 1 ) = ( N - 2 ) )
78		fzo0end 12560	. . . . . . . . 9 |- ( ( N - 1 ) e. NN -> ( ( N - 1 ) - 1 ) e. ( 0..^ ( N - 1 ) ) )
79	36, 78	syl 17	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( ( N - 1 ) - 1 ) e. ( 0..^ ( N - 1 ) ) )
80	77, 79	eqeltrrd 2702	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( N - 2 ) e. ( 0..^ ( N - 1 ) ) )
81	34	oveq2d 6666	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( 0..^ ( # ` ( F substr <. 0 , ( N - 1 ) >. ) ) ) = ( 0..^ ( N - 1 ) ) )
82	80, 81	eleqtrrd 2704	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( N - 2 ) e. ( 0..^ ( # ` ( F substr <. 0 , ( N - 1 ) >. ) ) ) )
83	17, 18, 19, 20	signstfvp 30648	. . . . . 6 |- ( ( ( F substr <. 0 , ( N - 1 ) >. ) e. Word RR /\ ( F ` ( N - 1 ) ) e. RR /\ ( N - 2 ) e. ( 0..^ ( # ` ( F substr <. 0 , ( N - 1 ) >. ) ) ) ) -> ( ( T ` ( ( F substr <. 0 , ( N - 1 ) >. ) ++ <" ( F ` ( N - 1 ) ) "> ) ) ` ( N - 2 ) ) = ( ( T ` ( F substr <. 0 , ( N - 1 ) >. ) ) ` ( N - 2 ) ) )
84	4, 49, 82, 83	syl3anc 1326	. . . . 5 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( ( T ` ( ( F substr <. 0 , ( N - 1 ) >. ) ++ <" ( F ` ( N - 1 ) ) "> ) ) ` ( N - 2 ) ) = ( ( T ` ( F substr <. 0 , ( N - 1 ) >. ) ) ` ( N - 2 ) ) )
85	69	eqcomd 2628	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> F = ( ( F substr <. 0 , ( N - 1 ) >. ) ++ <" ( F ` ( N - 1 ) ) "> ) )
86	85	fveq2d 6195	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( T ` F ) = ( T ` ( ( F substr <. 0 , ( N - 1 ) >. ) ++ <" ( F ` ( N - 1 ) ) "> ) ) )
87	86	fveq1d 6193	. . . . 5 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( ( T ` F ) ` ( N - 2 ) ) = ( ( T ` ( ( F substr <. 0 , ( N - 1 ) >. ) ++ <" ( F ` ( N - 1 ) ) "> ) ) ` ( N - 2 ) ) )
88	34	oveq1d 6665	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( ( # ` ( F substr <. 0 , ( N - 1 ) >. ) ) - 1 ) = ( ( N - 1 ) - 1 ) )
89	88, 74	eqtrd 2656	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( ( # ` ( F substr <. 0 , ( N - 1 ) >. ) ) - 1 ) = ( N - ( 1 + 1 ) ) )
90	89, 76	syl6eq 2672	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( ( # ` ( F substr <. 0 , ( N - 1 ) >. ) ) - 1 ) = ( N - 2 ) )
91	90	fveq2d 6195	. . . . 5 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( ( T ` ( F substr <. 0 , ( N - 1 ) >. ) ) ` ( ( # ` ( F substr <. 0 , ( N - 1 ) >. ) ) - 1 ) ) = ( ( T ` ( F substr <. 0 , ( N - 1 ) >. ) ) ` ( N - 2 ) ) )
92	84, 87, 91	3eqtr4rd 2667	. . . 4 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( ( T ` ( F substr <. 0 , ( N - 1 ) >. ) ) ` ( ( # ` ( F substr <. 0 , ( N - 1 ) >. ) ) - 1 ) ) = ( ( T ` F ) ` ( N - 2 ) ) )
93		fveq2 6191	. . . . . 6 |- ( ( F ` ( N - 1 ) ) = 0 -> (sgn ` ( F ` ( N - 1 ) ) ) = (sgn ` 0 ) )
94		sgn0 13829	. . . . . 6 |- (sgn ` 0 ) = 0
95	93, 94	syl6eq 2672	. . . . 5 |- ( ( F ` ( N - 1 ) ) = 0 -> (sgn ` ( F ` ( N - 1 ) ) ) = 0 )
96	95	adantl 482	. . . 4 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> (sgn ` ( F ` ( N - 1 ) ) ) = 0 )
97	92, 96	oveq12d 6668	. . 3 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( ( ( T ` ( F substr <. 0 , ( N - 1 ) >. ) ) ` ( ( # ` ( F substr <. 0 , ( N - 1 ) >. ) ) - 1 ) ) .+^ (sgn ` ( F ` ( N - 1 ) ) ) ) = ( ( ( T ` F ) ` ( N - 2 ) ) .+^ 0 ) )
98		uznn0sub 11719	. . . . . . . 8 |- ( N e. ( ZZ>= ` 2 ) -> ( N - 2 ) e. NN0 )
99	21, 98	syl 17	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( N - 2 ) e. NN0 )
100		eluz2nn 11726	. . . . . . . 8 |- ( N e. ( ZZ>= ` 2 ) -> N e. NN )
101	21, 100	syl 17	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> N e. NN )
102		2rp 11837	. . . . . . . . 9 |- 2 e. RR+
103	102	a1i 11	. . . . . . . 8 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> 2 e. RR+ )
104	14, 103	ltsubrpd 11904	. . . . . . 7 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( N - 2 ) < N )
105		elfzo0 12508	. . . . . . 7 |- ( ( N - 2 ) e. ( 0..^ N ) <-> ( ( N - 2 ) e. NN0 /\ N e. NN /\ ( N - 2 ) < N ) )
106	99, 101, 104, 105	syl3anbrc 1246	. . . . . 6 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( N - 2 ) e. ( 0..^ N ) )
107	10	oveq2i 6661	. . . . . 6 |- ( 0..^ N ) = ( 0..^ ( # ` F ) )
108	106, 107	syl6eleq 2711	. . . . 5 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( N - 2 ) e. ( 0..^ ( # ` F ) ) )
109	17, 18, 19, 20	signstcl 30642	. . . . 5 |- ( ( F e. Word RR /\ ( N - 2 ) e. ( 0..^ ( # ` F ) ) ) -> ( ( T ` F ) ` ( N - 2 ) ) e. { -u 1 , 0 , 1 } )
110	2, 108, 109	syl2anc 693	. . . 4 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( ( T ` F ) ` ( N - 2 ) ) e. { -u 1 , 0 , 1 } )
111	17, 18	signswrid 30635	. . . 4 |- ( ( ( T ` F ) ` ( N - 2 ) ) e. { -u 1 , 0 , 1 } -> ( ( ( T ` F ) ` ( N - 2 ) ) .+^ 0 ) = ( ( T ` F ) ` ( N - 2 ) ) )
112	110, 111	syl 17	. . 3 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( ( ( T ` F ) ` ( N - 2 ) ) .+^ 0 ) = ( ( T ` F ) ` ( N - 2 ) ) )
113	97, 112	eqtrd 2656	. 2 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( ( ( T ` ( F substr <. 0 , ( N - 1 ) >. ) ) ` ( ( # ` ( F substr <. 0 , ( N - 1 ) >. ) ) - 1 ) ) .+^ (sgn ` ( F ` ( N - 1 ) ) ) ) = ( ( T ` F ) ` ( N - 2 ) ) )
114	51, 71, 113	3eqtr3d 2664	1 |- ( ( ( F e. (Word RR \ { (/) } ) /\ ( F ` 0 ) =/= 0 ) /\ ( F ` ( N - 1 ) ) = 0 ) -> ( ( T ` F ) ` ( N - 1 ) ) = ( ( T ` F ) ` ( N - 2 ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   (/)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   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   - cmin 10266   -ucneg 10267   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832   ...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-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-rp 11833  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

This theorem is referenced by: (None)