Step |
Hyp |
Ref |
Expression |
1 |
|
signsv.p |
|- .+^ = ( a e. { -u 1 , 0 , 1 } , b e. { -u 1 , 0 , 1 } |-> if ( b = 0 , a , b ) ) |
2 |
|
signsv.w |
|- W = { <. ( Base ` ndx ) , { -u 1 , 0 , 1 } >. , <. ( +g ` ndx ) , .+^ >. } |
3 |
|
signsv.t |
|- T = ( f e. Word RR |-> ( n e. ( 0 ..^ ( # ` f ) ) |-> ( W gsum ( i e. ( 0 ... n ) |-> ( sgn ` ( f ` i ) ) ) ) ) ) |
4 |
|
signsv.v |
|- V = ( f e. Word RR |-> sum_ j e. ( 1 ..^ ( # ` f ) ) if ( ( ( T ` f ) ` j ) =/= ( ( T ` f ) ` ( j - 1 ) ) , 1 , 0 ) ) |
5 |
|
signsvf.e |
|- ( ph -> E e. ( Word RR \ { (/) } ) ) |
6 |
|
signsvf.0 |
|- ( ph -> ( E ` 0 ) =/= 0 ) |
7 |
|
signsvf.f |
|- ( ph -> F = ( E ++ <" A "> ) ) |
8 |
|
signsvf.a |
|- ( ph -> A e. RR ) |
9 |
|
signsvf.n |
|- N = ( # ` E ) |
10 |
|
signsvt.b |
|- B = ( ( T ` E ) ` ( N - 1 ) ) |
11 |
7
|
fveq2d |
|- ( ph -> ( V ` F ) = ( V ` ( E ++ <" A "> ) ) ) |
12 |
1 2 3 4
|
signsvfn |
|- ( ( ( E e. ( Word RR \ { (/) } ) /\ ( E ` 0 ) =/= 0 ) /\ A e. RR ) -> ( V ` ( E ++ <" A "> ) ) = ( ( V ` E ) + if ( ( ( ( T ` E ) ` ( ( # ` E ) - 1 ) ) x. A ) < 0 , 1 , 0 ) ) ) |
13 |
5 6 8 12
|
syl21anc |
|- ( ph -> ( V ` ( E ++ <" A "> ) ) = ( ( V ` E ) + if ( ( ( ( T ` E ) ` ( ( # ` E ) - 1 ) ) x. A ) < 0 , 1 , 0 ) ) ) |
14 |
11 13
|
eqtrd |
|- ( ph -> ( V ` F ) = ( ( V ` E ) + if ( ( ( ( T ` E ) ` ( ( # ` E ) - 1 ) ) x. A ) < 0 , 1 , 0 ) ) ) |
15 |
14
|
adantr |
|- ( ( ph /\ 0 < ( A x. B ) ) -> ( V ` F ) = ( ( V ` E ) + if ( ( ( ( T ` E ) ` ( ( # ` E ) - 1 ) ) x. A ) < 0 , 1 , 0 ) ) ) |
16 |
|
0red |
|- ( ( ph /\ 0 < ( A x. B ) ) -> 0 e. RR ) |
17 |
5
|
adantr |
|- ( ( ph /\ 0 < ( A x. B ) ) -> E e. ( Word RR \ { (/) } ) ) |
18 |
17
|
eldifad |
|- ( ( ph /\ 0 < ( A x. B ) ) -> E e. Word RR ) |
19 |
1 2 3 4
|
signstf |
|- ( E e. Word RR -> ( T ` E ) e. Word RR ) |
20 |
|
wrdf |
|- ( ( T ` E ) e. Word RR -> ( T ` E ) : ( 0 ..^ ( # ` ( T ` E ) ) ) --> RR ) |
21 |
18 19 20
|
3syl |
|- ( ( ph /\ 0 < ( A x. B ) ) -> ( T ` E ) : ( 0 ..^ ( # ` ( T ` E ) ) ) --> RR ) |
22 |
|
eldifsn |
|- ( E e. ( Word RR \ { (/) } ) <-> ( E e. Word RR /\ E =/= (/) ) ) |
23 |
5 22
|
sylib |
|- ( ph -> ( E e. Word RR /\ E =/= (/) ) ) |
24 |
23
|
adantr |
|- ( ( ph /\ 0 < ( A x. B ) ) -> ( E e. Word RR /\ E =/= (/) ) ) |
25 |
|
lennncl |
|- ( ( E e. Word RR /\ E =/= (/) ) -> ( # ` E ) e. NN ) |
26 |
|
fzo0end |
|- ( ( # ` E ) e. NN -> ( ( # ` E ) - 1 ) e. ( 0 ..^ ( # ` E ) ) ) |
27 |
24 25 26
|
3syl |
|- ( ( ph /\ 0 < ( A x. B ) ) -> ( ( # ` E ) - 1 ) e. ( 0 ..^ ( # ` E ) ) ) |
28 |
1 2 3 4
|
signstlen |
|- ( E e. Word RR -> ( # ` ( T ` E ) ) = ( # ` E ) ) |
29 |
18 28
|
syl |
|- ( ( ph /\ 0 < ( A x. B ) ) -> ( # ` ( T ` E ) ) = ( # ` E ) ) |
30 |
29
|
oveq2d |
|- ( ( ph /\ 0 < ( A x. B ) ) -> ( 0 ..^ ( # ` ( T ` E ) ) ) = ( 0 ..^ ( # ` E ) ) ) |
31 |
27 30
|
eleqtrrd |
|- ( ( ph /\ 0 < ( A x. B ) ) -> ( ( # ` E ) - 1 ) e. ( 0 ..^ ( # ` ( T ` E ) ) ) ) |
32 |
21 31
|
ffvelrnd |
|- ( ( ph /\ 0 < ( A x. B ) ) -> ( ( T ` E ) ` ( ( # ` E ) - 1 ) ) e. RR ) |
33 |
8
|
adantr |
|- ( ( ph /\ 0 < ( A x. B ) ) -> A e. RR ) |
34 |
32 33
|
remulcld |
|- ( ( ph /\ 0 < ( A x. B ) ) -> ( ( ( T ` E ) ` ( ( # ` E ) - 1 ) ) x. A ) e. RR ) |
35 |
|
simpr |
|- ( ( ph /\ 0 < ( A x. B ) ) -> 0 < ( A x. B ) ) |
36 |
9
|
oveq1i |
|- ( N - 1 ) = ( ( # ` E ) - 1 ) |
37 |
36
|
fveq2i |
|- ( ( T ` E ) ` ( N - 1 ) ) = ( ( T ` E ) ` ( ( # ` E ) - 1 ) ) |
38 |
10 37
|
eqtri |
|- B = ( ( T ` E ) ` ( ( # ` E ) - 1 ) ) |
39 |
38 32
|
eqeltrid |
|- ( ( ph /\ 0 < ( A x. B ) ) -> B e. RR ) |
40 |
39
|
recnd |
|- ( ( ph /\ 0 < ( A x. B ) ) -> B e. CC ) |
41 |
33
|
recnd |
|- ( ( ph /\ 0 < ( A x. B ) ) -> A e. CC ) |
42 |
40 41
|
mulcomd |
|- ( ( ph /\ 0 < ( A x. B ) ) -> ( B x. A ) = ( A x. B ) ) |
43 |
35 42
|
breqtrrd |
|- ( ( ph /\ 0 < ( A x. B ) ) -> 0 < ( B x. A ) ) |
44 |
38
|
oveq1i |
|- ( B x. A ) = ( ( ( T ` E ) ` ( ( # ` E ) - 1 ) ) x. A ) |
45 |
43 44
|
breqtrdi |
|- ( ( ph /\ 0 < ( A x. B ) ) -> 0 < ( ( ( T ` E ) ` ( ( # ` E ) - 1 ) ) x. A ) ) |
46 |
16 34 45
|
ltnsymd |
|- ( ( ph /\ 0 < ( A x. B ) ) -> -. ( ( ( T ` E ) ` ( ( # ` E ) - 1 ) ) x. A ) < 0 ) |
47 |
46
|
iffalsed |
|- ( ( ph /\ 0 < ( A x. B ) ) -> if ( ( ( ( T ` E ) ` ( ( # ` E ) - 1 ) ) x. A ) < 0 , 1 , 0 ) = 0 ) |
48 |
47
|
oveq2d |
|- ( ( ph /\ 0 < ( A x. B ) ) -> ( ( V ` E ) + if ( ( ( ( T ` E ) ` ( ( # ` E ) - 1 ) ) x. A ) < 0 , 1 , 0 ) ) = ( ( V ` E ) + 0 ) ) |
49 |
1 2 3 4
|
signsvvf |
|- V : Word RR --> NN0 |
50 |
49
|
a1i |
|- ( ( ph /\ 0 < ( A x. B ) ) -> V : Word RR --> NN0 ) |
51 |
50 18
|
ffvelrnd |
|- ( ( ph /\ 0 < ( A x. B ) ) -> ( V ` E ) e. NN0 ) |
52 |
51
|
nn0cnd |
|- ( ( ph /\ 0 < ( A x. B ) ) -> ( V ` E ) e. CC ) |
53 |
52
|
addid1d |
|- ( ( ph /\ 0 < ( A x. B ) ) -> ( ( V ` E ) + 0 ) = ( V ` E ) ) |
54 |
15 48 53
|
3eqtrd |
|- ( ( ph /\ 0 < ( A x. B ) ) -> ( V ` F ) = ( V ` E ) ) |