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Theorem wrdind 12665
Description: Perform induction over the structure of a word. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
Hypotheses
Ref Expression
wrdind.1  |-  ( x  =  (/)  ->  ( ph  <->  ps ) )
wrdind.2  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
wrdind.3  |-  ( x  =  ( y concat  <" z "> )  ->  ( ph  <->  th )
)
wrdind.4  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
wrdind.5  |-  ps
wrdind.6  |-  ( ( y  e. Word  B  /\  z  e.  B )  ->  ( ch  ->  th )
)
Assertion
Ref Expression
wrdind  |-  ( A  e. Word  B  ->  ta )
Distinct variable groups:    x, A    x, y, z, B    ch, x    ph, y, z    ta, x    th, x
Allowed substitution hints:    ph( x)    ps( x, y, z)    ch( y,
z)    th( y, z)    ta( y, z)    A( y, z)

Proof of Theorem wrdind
Dummy variables  n  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lencl 12528 . . 3  |-  ( A  e. Word  B  ->  ( # `
 A )  e. 
NN0 )
2 eqeq2 2482 . . . . . 6  |-  ( n  =  0  ->  (
( # `  x )  =  n  <->  ( # `  x
)  =  0 ) )
32imbi1d 317 . . . . 5  |-  ( n  =  0  ->  (
( ( # `  x
)  =  n  ->  ph )  <->  ( ( # `  x )  =  0  ->  ph ) ) )
43ralbidv 2903 . . . 4  |-  ( n  =  0  ->  ( A. x  e. Word  B ( ( # `  x
)  =  n  ->  ph )  <->  A. x  e. Word  B
( ( # `  x
)  =  0  ->  ph ) ) )
5 eqeq2 2482 . . . . . 6  |-  ( n  =  m  ->  (
( # `  x )  =  n  <->  ( # `  x
)  =  m ) )
65imbi1d 317 . . . . 5  |-  ( n  =  m  ->  (
( ( # `  x
)  =  n  ->  ph )  <->  ( ( # `  x )  =  m  ->  ph ) ) )
76ralbidv 2903 . . . 4  |-  ( n  =  m  ->  ( A. x  e. Word  B ( ( # `  x
)  =  n  ->  ph )  <->  A. x  e. Word  B
( ( # `  x
)  =  m  ->  ph ) ) )
8 eqeq2 2482 . . . . . 6  |-  ( n  =  ( m  + 
1 )  ->  (
( # `  x )  =  n  <->  ( # `  x
)  =  ( m  +  1 ) ) )
98imbi1d 317 . . . . 5  |-  ( n  =  ( m  + 
1 )  ->  (
( ( # `  x
)  =  n  ->  ph )  <->  ( ( # `  x )  =  ( m  +  1 )  ->  ph ) ) )
109ralbidv 2903 . . . 4  |-  ( n  =  ( m  + 
1 )  ->  ( A. x  e. Word  B ( ( # `  x
)  =  n  ->  ph )  <->  A. x  e. Word  B
( ( # `  x
)  =  ( m  +  1 )  ->  ph ) ) )
11 eqeq2 2482 . . . . . 6  |-  ( n  =  ( # `  A
)  ->  ( ( # `
 x )  =  n  <->  ( # `  x
)  =  ( # `  A ) ) )
1211imbi1d 317 . . . . 5  |-  ( n  =  ( # `  A
)  ->  ( (
( # `  x )  =  n  ->  ph )  <->  ( ( # `  x
)  =  ( # `  A )  ->  ph )
) )
1312ralbidv 2903 . . . 4  |-  ( n  =  ( # `  A
)  ->  ( A. x  e. Word  B (
( # `  x )  =  n  ->  ph )  <->  A. x  e. Word  B ( ( # `  x
)  =  ( # `  A )  ->  ph )
) )
14 hasheq0 12401 . . . . . 6  |-  ( x  e. Word  B  ->  (
( # `  x )  =  0  <->  x  =  (/) ) )
15 wrdind.5 . . . . . . 7  |-  ps
16 wrdind.1 . . . . . . 7  |-  ( x  =  (/)  ->  ( ph  <->  ps ) )
1715, 16mpbiri 233 . . . . . 6  |-  ( x  =  (/)  ->  ph )
1814, 17syl6bi 228 . . . . 5  |-  ( x  e. Word  B  ->  (
( # `  x )  =  0  ->  ph )
)
1918rgen 2824 . . . 4  |-  A. x  e. Word  B ( ( # `  x )  =  0  ->  ph )
20 fveq2 5866 . . . . . . . 8  |-  ( x  =  y  ->  ( # `
 x )  =  ( # `  y
) )
2120eqeq1d 2469 . . . . . . 7  |-  ( x  =  y  ->  (
( # `  x )  =  m  <->  ( # `  y
)  =  m ) )
22 wrdind.2 . . . . . . 7  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
2321, 22imbi12d 320 . . . . . 6  |-  ( x  =  y  ->  (
( ( # `  x
)  =  m  ->  ph )  <->  ( ( # `  y )  =  m  ->  ch ) ) )
2423cbvralv 3088 . . . . 5  |-  ( A. x  e. Word  B (
( # `  x )  =  m  ->  ph )  <->  A. y  e. Word  B ( ( # `  y
)  =  m  ->  ch ) )
25 swrdcl 12609 . . . . . . . . . . . 12  |-  ( x  e. Word  B  ->  (
x substr  <. 0 ,  ( ( # `  x
)  -  1 )
>. )  e. Word  B )
2625ad2antrl 727 . . . . . . . . . . 11  |-  ( ( ( m  e.  NN0  /\ 
A. y  e. Word  B
( ( # `  y
)  =  m  ->  ch ) )  /\  (
x  e. Word  B  /\  ( # `  x )  =  ( m  + 
1 ) ) )  ->  ( x substr  <. 0 ,  ( ( # `  x )  -  1 ) >. )  e. Word  B
)
27 simplr 754 . . . . . . . . . . 11  |-  ( ( ( m  e.  NN0  /\ 
A. y  e. Word  B
( ( # `  y
)  =  m  ->  ch ) )  /\  (
x  e. Word  B  /\  ( # `  x )  =  ( m  + 
1 ) ) )  ->  A. y  e. Word  B
( ( # `  y
)  =  m  ->  ch ) )
28 simprl 755 . . . . . . . . . . . . 13  |-  ( ( ( m  e.  NN0  /\ 
A. y  e. Word  B
( ( # `  y
)  =  m  ->  ch ) )  /\  (
x  e. Word  B  /\  ( # `  x )  =  ( m  + 
1 ) ) )  ->  x  e. Word  B
)
29 fzossfz 11814 . . . . . . . . . . . . . 14  |-  ( 0..^ ( # `  x
) )  C_  (
0 ... ( # `  x
) )
30 simprr 756 . . . . . . . . . . . . . . . 16  |-  ( ( ( m  e.  NN0  /\ 
A. y  e. Word  B
( ( # `  y
)  =  m  ->  ch ) )  /\  (
x  e. Word  B  /\  ( # `  x )  =  ( m  + 
1 ) ) )  ->  ( # `  x
)  =  ( m  +  1 ) )
31 nn0p1nn 10835 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  NN0  ->  ( m  +  1 )  e.  NN )
3231ad2antrr 725 . . . . . . . . . . . . . . . 16  |-  ( ( ( m  e.  NN0  /\ 
A. y  e. Word  B
( ( # `  y
)  =  m  ->  ch ) )  /\  (
x  e. Word  B  /\  ( # `  x )  =  ( m  + 
1 ) ) )  ->  ( m  + 
1 )  e.  NN )
3330, 32eqeltrd 2555 . . . . . . . . . . . . . . 15  |-  ( ( ( m  e.  NN0  /\ 
A. y  e. Word  B
( ( # `  y
)  =  m  ->  ch ) )  /\  (
x  e. Word  B  /\  ( # `  x )  =  ( m  + 
1 ) ) )  ->  ( # `  x
)  e.  NN )
34 fzo0end 11872 . . . . . . . . . . . . . . 15  |-  ( (
# `  x )  e.  NN  ->  ( ( # `
 x )  - 
1 )  e.  ( 0..^ ( # `  x
) ) )
3533, 34syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( m  e.  NN0  /\ 
A. y  e. Word  B
( ( # `  y
)  =  m  ->  ch ) )  /\  (
x  e. Word  B  /\  ( # `  x )  =  ( m  + 
1 ) ) )  ->  ( ( # `  x )  -  1 )  e.  ( 0..^ ( # `  x
) ) )
3629, 35sseldi 3502 . . . . . . . . . . . . 13  |-  ( ( ( m  e.  NN0  /\ 
A. y  e. Word  B
( ( # `  y
)  =  m  ->  ch ) )  /\  (
x  e. Word  B  /\  ( # `  x )  =  ( m  + 
1 ) ) )  ->  ( ( # `  x )  -  1 )  e.  ( 0 ... ( # `  x
) ) )
37 swrd0len 12612 . . . . . . . . . . . . 13  |-  ( ( x  e. Word  B  /\  ( ( # `  x
)  -  1 )  e.  ( 0 ... ( # `  x
) ) )  -> 
( # `  ( x substr  <. 0 ,  ( (
# `  x )  -  1 ) >.
) )  =  ( ( # `  x
)  -  1 ) )
3828, 36, 37syl2anc 661 . . . . . . . . . . . 12  |-  ( ( ( m  e.  NN0  /\ 
A. y  e. Word  B
( ( # `  y
)  =  m  ->  ch ) )  /\  (
x  e. Word  B  /\  ( # `  x )  =  ( m  + 
1 ) ) )  ->  ( # `  (
x substr  <. 0 ,  ( ( # `  x
)  -  1 )
>. ) )  =  ( ( # `  x
)  -  1 ) )
3930oveq1d 6299 . . . . . . . . . . . 12  |-  ( ( ( m  e.  NN0  /\ 
A. y  e. Word  B
( ( # `  y
)  =  m  ->  ch ) )  /\  (
x  e. Word  B  /\  ( # `  x )  =  ( m  + 
1 ) ) )  ->  ( ( # `  x )  -  1 )  =  ( ( m  +  1 )  -  1 ) )
40 nn0cn 10805 . . . . . . . . . . . . . 14  |-  ( m  e.  NN0  ->  m  e.  CC )
4140ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( m  e.  NN0  /\ 
A. y  e. Word  B
( ( # `  y
)  =  m  ->  ch ) )  /\  (
x  e. Word  B  /\  ( # `  x )  =  ( m  + 
1 ) ) )  ->  m  e.  CC )
42 ax-1cn 9550 . . . . . . . . . . . . 13  |-  1  e.  CC
43 pncan 9826 . . . . . . . . . . . . 13  |-  ( ( m  e.  CC  /\  1  e.  CC )  ->  ( ( m  + 
1 )  -  1 )  =  m )
4441, 42, 43sylancl 662 . . . . . . . . . . . 12  |-  ( ( ( m  e.  NN0  /\ 
A. y  e. Word  B
( ( # `  y
)  =  m  ->  ch ) )  /\  (
x  e. Word  B  /\  ( # `  x )  =  ( m  + 
1 ) ) )  ->  ( ( m  +  1 )  - 
1 )  =  m )
4538, 39, 443eqtrd 2512 . . . . . . . . . . 11  |-  ( ( ( m  e.  NN0  /\ 
A. y  e. Word  B
( ( # `  y
)  =  m  ->  ch ) )  /\  (
x  e. Word  B  /\  ( # `  x )  =  ( m  + 
1 ) ) )  ->  ( # `  (
x substr  <. 0 ,  ( ( # `  x
)  -  1 )
>. ) )  =  m )
46 fveq2 5866 . . . . . . . . . . . . . 14  |-  ( y  =  ( x substr  <. 0 ,  ( ( # `  x )  -  1 ) >. )  ->  ( # `
 y )  =  ( # `  (
x substr  <. 0 ,  ( ( # `  x
)  -  1 )
>. ) ) )
4746eqeq1d 2469 . . . . . . . . . . . . 13  |-  ( y  =  ( x substr  <. 0 ,  ( ( # `  x )  -  1 ) >. )  ->  (
( # `  y )  =  m  <->  ( # `  (
x substr  <. 0 ,  ( ( # `  x
)  -  1 )
>. ) )  =  m ) )
48 vex 3116 . . . . . . . . . . . . . . 15  |-  y  e. 
_V
4948, 22sbcie 3366 . . . . . . . . . . . . . 14  |-  ( [. y  /  x ]. ph  <->  ch )
50 dfsbcq 3333 . . . . . . . . . . . . . 14  |-  ( y  =  ( x substr  <. 0 ,  ( ( # `  x )  -  1 ) >. )  ->  ( [. y  /  x ]. ph  <->  [. ( x substr  <. 0 ,  ( ( # `  x )  -  1 ) >. )  /  x ]. ph ) )
5149, 50syl5bbr 259 . . . . . . . . . . . . 13  |-  ( y  =  ( x substr  <. 0 ,  ( ( # `  x )  -  1 ) >. )  ->  ( ch 
<-> 
[. ( x substr  <. 0 ,  ( ( # `  x )  -  1 ) >. )  /  x ]. ph ) )
5247, 51imbi12d 320 . . . . . . . . . . . 12  |-  ( y  =  ( x substr  <. 0 ,  ( ( # `  x )  -  1 ) >. )  ->  (
( ( # `  y
)  =  m  ->  ch )  <->  ( ( # `  ( x substr  <. 0 ,  ( ( # `  x )  -  1 ) >. ) )  =  m  ->  [. ( x substr  <. 0 ,  ( (
# `  x )  -  1 ) >.
)  /  x ]. ph ) ) )
5352rspcv 3210 . . . . . . . . . . 11  |-  ( ( x substr  <. 0 ,  ( ( # `  x
)  -  1 )
>. )  e. Word  B  -> 
( A. y  e. Word  B ( ( # `  y )  =  m  ->  ch )  -> 
( ( # `  (
x substr  <. 0 ,  ( ( # `  x
)  -  1 )
>. ) )  =  m  ->  [. ( x substr  <. 0 ,  ( ( # `  x )  -  1 ) >. )  /  x ]. ph ) ) )
5426, 27, 45, 53syl3c 61 . . . . . . . . . 10  |-  ( ( ( m  e.  NN0  /\ 
A. y  e. Word  B
( ( # `  y
)  =  m  ->  ch ) )  /\  (
x  e. Word  B  /\  ( # `  x )  =  ( m  + 
1 ) ) )  ->  [. ( x substr  <. 0 ,  ( ( # `  x )  -  1 ) >. )  /  x ]. ph )
55 wrdf 12519 . . . . . . . . . . . . 13  |-  ( x  e. Word  B  ->  x : ( 0..^ (
# `  x )
) --> B )
5655ad2antrl 727 . . . . . . . . . . . 12  |-  ( ( ( m  e.  NN0  /\ 
A. y  e. Word  B
( ( # `  y
)  =  m  ->  ch ) )  /\  (
x  e. Word  B  /\  ( # `  x )  =  ( m  + 
1 ) ) )  ->  x : ( 0..^ ( # `  x
) ) --> B )
5756, 35ffvelrnd 6022 . . . . . . . . . . 11  |-  ( ( ( m  e.  NN0  /\ 
A. y  e. Word  B
( ( # `  y
)  =  m  ->  ch ) )  /\  (
x  e. Word  B  /\  ( # `  x )  =  ( m  + 
1 ) ) )  ->  ( x `  ( ( # `  x
)  -  1 ) )  e.  B )
58 oveq1 6291 . . . . . . . . . . . . . 14  |-  ( y  =  ( x substr  <. 0 ,  ( ( # `  x )  -  1 ) >. )  ->  (
y concat  <" z "> )  =  ( ( x substr  <. 0 ,  ( ( # `  x )  -  1 ) >. ) concat  <" z "> ) )
59 dfsbcq 3333 . . . . . . . . . . . . . 14  |-  ( ( y concat  <" z "> )  =  ( ( x substr  <. 0 ,  ( ( # `  x )  -  1 ) >. ) concat  <" z "> )  ->  ( [. ( y concat  <" z "> )  /  x ]. ph  <->  [. ( ( x substr  <. 0 ,  ( (
# `  x )  -  1 ) >.
) concat  <" z "> )  /  x ]. ph ) )
6058, 59syl 16 . . . . . . . . . . . . 13  |-  ( y  =  ( x substr  <. 0 ,  ( ( # `  x )  -  1 ) >. )  ->  ( [. ( y concat  <" z "> )  /  x ]. ph  <->  [. ( ( x substr  <. 0 ,  ( (
# `  x )  -  1 ) >.
) concat  <" z "> )  /  x ]. ph ) )
6150, 60imbi12d 320 . . . . . . . . . . . 12  |-  ( y  =  ( x substr  <. 0 ,  ( ( # `  x )  -  1 ) >. )  ->  (
( [. y  /  x ]. ph  ->  [. ( y concat  <" z "> )  /  x ]. ph )  <->  (
[. ( x substr  <. 0 ,  ( ( # `  x )  -  1 ) >. )  /  x ]. ph  ->  [. ( ( x substr  <. 0 ,  ( ( # `  x
)  -  1 )
>. ) concat  <" z "> )  /  x ]. ph ) ) )
62 s1eq 12575 . . . . . . . . . . . . . . 15  |-  ( z  =  ( x `  ( ( # `  x
)  -  1 ) )  ->  <" z ">  =  <" (
x `  ( ( # `
 x )  - 
1 ) ) "> )
6362oveq2d 6300 . . . . . . . . . . . . . 14  |-  ( z  =  ( x `  ( ( # `  x
)  -  1 ) )  ->  ( (
x substr  <. 0 ,  ( ( # `  x
)  -  1 )
>. ) concat  <" z "> )  =  ( ( x substr  <. 0 ,  ( ( # `  x )  -  1 ) >. ) concat  <" (
x `  ( ( # `
 x )  - 
1 ) ) "> ) )
64 dfsbcq 3333 . . . . . . . . . . . . . 14  |-  ( ( ( x substr  <. 0 ,  ( ( # `  x )  -  1 ) >. ) concat  <" z "> )  =  ( ( x substr  <. 0 ,  ( ( # `  x )  -  1 ) >. ) concat  <" (
x `  ( ( # `
 x )  - 
1 ) ) "> )  ->  ( [. ( ( x substr  <. 0 ,  ( ( # `  x )  -  1 ) >. ) concat  <" z "> )  /  x ]. ph  <->  [. ( ( x substr  <. 0 ,  ( (
# `  x )  -  1 ) >.
) concat  <" ( x `
 ( ( # `  x )  -  1 ) ) "> )  /  x ]. ph )
)
6563, 64syl 16 . . . . . . . . . . . . 13  |-  ( z  =  ( x `  ( ( # `  x
)  -  1 ) )  ->  ( [. ( ( x substr  <. 0 ,  ( ( # `  x )  -  1 ) >. ) concat  <" z "> )  /  x ]. ph  <->  [. ( ( x substr  <. 0 ,  ( (
# `  x )  -  1 ) >.
) concat  <" ( x `
 ( ( # `  x )  -  1 ) ) "> )  /  x ]. ph )
)
6665imbi2d 316 . . . . . . . . . . . 12  |-  ( z  =  ( x `  ( ( # `  x
)  -  1 ) )  ->  ( ( [. ( x substr  <. 0 ,  ( ( # `  x )  -  1 ) >. )  /  x ]. ph  ->  [. ( ( x substr  <. 0 ,  ( ( # `  x
)  -  1 )
>. ) concat  <" z "> )  /  x ]. ph )  <->  ( [. ( x substr  <. 0 ,  ( ( # `  x
)  -  1 )
>. )  /  x ]. ph  ->  [. ( ( x substr  <. 0 ,  ( ( # `  x
)  -  1 )
>. ) concat  <" (
x `  ( ( # `
 x )  - 
1 ) ) "> )  /  x ]. ph ) ) )
67 wrdind.6 . . . . . . . . . . . . 13  |-  ( ( y  e. Word  B  /\  z  e.  B )  ->  ( ch  ->  th )
)
68 ovex 6309 . . . . . . . . . . . . . 14  |-  ( y concat  <" z "> )  e.  _V
69 wrdind.3 . . . . . . . . . . . . . 14  |-  ( x  =  ( y concat  <" z "> )  ->  ( ph  <->  th )
)
7068, 69sbcie 3366 . . . . . . . . . . . . 13  |-  ( [. ( y concat  <" z "> )  /  x ]. ph  <->  th )
7167, 49, 703imtr4g 270 . . . . . . . . . . . 12  |-  ( ( y  e. Word  B  /\  z  e.  B )  ->  ( [. y  /  x ]. ph  ->  [. (
y concat  <" z "> )  /  x ]. ph ) )
7261, 66, 71vtocl2ga 3179 . . . . . . . . . . 11  |-  ( ( ( x substr  <. 0 ,  ( ( # `  x )  -  1 ) >. )  e. Word  B  /\  ( x `  (
( # `  x )  -  1 ) )  e.  B )  -> 
( [. ( x substr  <. 0 ,  ( ( # `  x )  -  1 ) >. )  /  x ]. ph  ->  [. ( ( x substr  <. 0 ,  ( ( # `  x
)  -  1 )
>. ) concat  <" (
x `  ( ( # `
 x )  - 
1 ) ) "> )  /  x ]. ph ) )
7326, 57, 72syl2anc 661 . . . . . . . . . 10  |-  ( ( ( m  e.  NN0  /\ 
A. y  e. Word  B
( ( # `  y
)  =  m  ->  ch ) )  /\  (
x  e. Word  B  /\  ( # `  x )  =  ( m  + 
1 ) ) )  ->  ( [. (
x substr  <. 0 ,  ( ( # `  x
)  -  1 )
>. )  /  x ]. ph  ->  [. ( ( x substr  <. 0 ,  ( ( # `  x
)  -  1 )
>. ) concat  <" (
x `  ( ( # `
 x )  - 
1 ) ) "> )  /  x ]. ph ) )
7454, 73mpd 15 . . . . . . . . 9  |-  ( ( ( m  e.  NN0  /\ 
A. y  e. Word  B
( ( # `  y
)  =  m  ->  ch ) )  /\  (
x  e. Word  B  /\  ( # `  x )  =  ( m  + 
1 ) ) )  ->  [. ( ( x substr  <. 0 ,  ( (
# `  x )  -  1 ) >.
) concat  <" ( x `
 ( ( # `  x )  -  1 ) ) "> )  /  x ]. ph )
75 wrdfin 12527 . . . . . . . . . . . . . 14  |-  ( x  e. Word  B  ->  x  e.  Fin )
7675ad2antrl 727 . . . . . . . . . . . . 13  |-  ( ( ( m  e.  NN0  /\ 
A. y  e. Word  B
( ( # `  y
)  =  m  ->  ch ) )  /\  (
x  e. Word  B  /\  ( # `  x )  =  ( m  + 
1 ) ) )  ->  x  e.  Fin )
77 hashnncl 12404 . . . . . . . . . . . . 13  |-  ( x  e.  Fin  ->  (
( # `  x )  e.  NN  <->  x  =/=  (/) ) )
7876, 77syl 16 . . . . . . . . . . . 12  |-  ( ( ( m  e.  NN0  /\ 
A. y  e. Word  B
( ( # `  y
)  =  m  ->  ch ) )  /\  (
x  e. Word  B  /\  ( # `  x )  =  ( m  + 
1 ) ) )  ->  ( ( # `  x )  e.  NN  <->  x  =/=  (/) ) )
7933, 78mpbid 210 . . . . . . . . . . 11  |-  ( ( ( m  e.  NN0  /\ 
A. y  e. Word  B
( ( # `  y
)  =  m  ->  ch ) )  /\  (
x  e. Word  B  /\  ( # `  x )  =  ( m  + 
1 ) ) )  ->  x  =/=  (/) )
80 wrdeqcats1 12662 . . . . . . . . . . 11  |-  ( ( x  e. Word  B  /\  x  =/=  (/) )  ->  x  =  ( ( x substr  <. 0 ,  ( (
# `  x )  -  1 ) >.
) concat  <" ( x `
 ( ( # `  x )  -  1 ) ) "> ) )
8128, 79, 80syl2anc 661 . . . . . . . . . 10  |-  ( ( ( m  e.  NN0  /\ 
A. y  e. Word  B
( ( # `  y
)  =  m  ->  ch ) )  /\  (
x  e. Word  B  /\  ( # `  x )  =  ( m  + 
1 ) ) )  ->  x  =  ( ( x substr  <. 0 ,  ( ( # `  x )  -  1 ) >. ) concat  <" (
x `  ( ( # `
 x )  - 
1 ) ) "> ) )
82 sbceq1a 3342 . . . . . . . . . 10  |-  ( x  =  ( ( x substr  <. 0 ,  ( (
# `  x )  -  1 ) >.
) concat  <" ( x `
 ( ( # `  x )  -  1 ) ) "> )  ->  ( ph  <->  [. ( ( x substr  <. 0 ,  ( ( # `  x
)  -  1 )
>. ) concat  <" (
x `  ( ( # `
 x )  - 
1 ) ) "> )  /  x ]. ph ) )
8381, 82syl 16 . . . . . . . . 9  |-  ( ( ( m  e.  NN0  /\ 
A. y  e. Word  B
( ( # `  y
)  =  m  ->  ch ) )  /\  (
x  e. Word  B  /\  ( # `  x )  =  ( m  + 
1 ) ) )  ->  ( ph  <->  [. ( ( x substr  <. 0 ,  ( ( # `  x
)  -  1 )
>. ) concat  <" (
x `  ( ( # `
 x )  - 
1 ) ) "> )  /  x ]. ph ) )
8474, 83mpbird 232 . . . . . . . 8  |-  ( ( ( m  e.  NN0  /\ 
A. y  e. Word  B
( ( # `  y
)  =  m  ->  ch ) )  /\  (
x  e. Word  B  /\  ( # `  x )  =  ( m  + 
1 ) ) )  ->  ph )
8584expr 615 . . . . . . 7  |-  ( ( ( m  e.  NN0  /\ 
A. y  e. Word  B
( ( # `  y
)  =  m  ->  ch ) )  /\  x  e. Word  B )  ->  (
( # `  x )  =  ( m  + 
1 )  ->  ph )
)
8685ralrimiva 2878 . . . . . 6  |-  ( ( m  e.  NN0  /\  A. y  e. Word  B ( ( # `  y
)  =  m  ->  ch ) )  ->  A. x  e. Word  B ( ( # `  x )  =  ( m  +  1 )  ->  ph ) )
8786ex 434 . . . . 5  |-  ( m  e.  NN0  ->  ( A. y  e. Word  B (
( # `  y )  =  m  ->  ch )  ->  A. x  e. Word  B
( ( # `  x
)  =  ( m  +  1 )  ->  ph ) ) )
8824, 87syl5bi 217 . . . 4  |-  ( m  e.  NN0  ->  ( A. x  e. Word  B (
( # `  x )  =  m  ->  ph )  ->  A. x  e. Word  B
( ( # `  x
)  =  ( m  +  1 )  ->  ph ) ) )
894, 7, 10, 13, 19, 88nn0ind 10957 . . 3  |-  ( (
# `  A )  e.  NN0  ->  A. x  e. Word  B ( ( # `  x )  =  (
# `  A )  ->  ph ) )
901, 89syl 16 . 2  |-  ( A  e. Word  B  ->  A. x  e. Word  B ( ( # `  x )  =  (
# `  A )  ->  ph ) )
91 eqidd 2468 . 2  |-  ( A  e. Word  B  ->  ( # `
 A )  =  ( # `  A
) )
92 fveq2 5866 . . . . 5  |-  ( x  =  A  ->  ( # `
 x )  =  ( # `  A
) )
9392eqeq1d 2469 . . . 4  |-  ( x  =  A  ->  (
( # `  x )  =  ( # `  A
)  <->  ( # `  A
)  =  ( # `  A ) ) )
94 wrdind.4 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
9593, 94imbi12d 320 . . 3  |-  ( x  =  A  ->  (
( ( # `  x
)  =  ( # `  A )  ->  ph )  <->  ( ( # `  A
)  =  ( # `  A )  ->  ta ) ) )
9695rspcv 3210 . 2  |-  ( A  e. Word  B  ->  ( A. x  e. Word  B ( ( # `  x
)  =  ( # `  A )  ->  ph )  ->  ( ( # `  A
)  =  ( # `  A )  ->  ta ) ) )
9790, 91, 96mp2d 45 1  |-  ( A  e. Word  B  ->  ta )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   [.wsbc 3331   (/)c0 3785   <.cop 4033   -->wf 5584   ` cfv 5588  (class class class)co 6284   Fincfn 7516   CCcc 9490   0cc0 9492   1c1 9493    + caddc 9495    - cmin 9805   NNcn 10536   NN0cn0 10795   ...cfz 11672  ..^cfzo 11792   #chash 12373  Word cword 12500   concat cconcat 12502   <"cs1 12503   substr csubstr 12504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-fzo 11793  df-hash 12374  df-word 12508  df-concat 12510  df-s1 12511  df-substr 12512
This theorem is referenced by:  frmdgsum  15862  gsumwrev  16206  gsmsymgrfix  16258  efginvrel2  16551  signstfvneq0  28197  signstfvc  28199
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