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Theorem efgred 15335
Description: The reduced word that forms the base of the sequence in efgsval 15318 is uniquely determined, given the terminal point. (Contributed by Mario Carneiro, 28-Sep-2015.)
Hypotheses
Ref Expression
efgval.w  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
efgval.r  |-  .~  =  ( ~FG  `  I )
efgval2.m  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
efgval2.t  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
efgred.d  |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `  x ) )
efgred.s  |-  S  =  ( m  e.  {
t  e.  (Word  W  \  { (/) } )  |  ( ( t ` 
0 )  e.  D  /\  A. k  e.  ( 1..^ ( # `  t
) ) ( t `
 k )  e. 
ran  ( T `  ( t `  (
k  -  1 ) ) ) ) } 
|->  ( m `  (
( # `  m )  -  1 ) ) )
Assertion
Ref Expression
efgred  |-  ( ( A  e.  dom  S  /\  B  e.  dom  S  /\  ( S `  A )  =  ( S `  B ) )  ->  ( A `  0 )  =  ( B `  0
) )
Distinct variable groups:    y, z    t, n, v, w, y, z, m, x    m, M    x, n, M, t, v, w    k, m, t, x, T    k, n, v, w, y, z, W, m, t, x    .~ , m, t, x, y, z    m, I, n, t, v, w, x, y, z    D, m, t
Allowed substitution hints:    A( x, y, z, w, v, t, k, m, n)    B( x, y, z, w, v, t, k, m, n)    D( x, y, z, w, v, k, n)    .~ ( w, v, k, n)    S( x, y, z, w, v, t, k, m, n)    T( y, z, w, v, n)    I( k)    M( y, z, k)

Proof of Theorem efgred
Dummy variables  a 
b  c  d  i are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 efgval.w . . . . . . . 8  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
2 fviss 5743 . . . . . . . 8  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
31, 2eqsstri 3338 . . . . . . 7  |-  W  C_ Word  ( I  X.  2o )
4 efgval.r . . . . . . . . . . 11  |-  .~  =  ( ~FG  `  I )
5 efgval2.m . . . . . . . . . . 11  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
6 efgval2.t . . . . . . . . . . 11  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
7 efgred.d . . . . . . . . . . 11  |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `  x ) )
8 efgred.s . . . . . . . . . . 11  |-  S  =  ( m  e.  {
t  e.  (Word  W  \  { (/) } )  |  ( ( t ` 
0 )  e.  D  /\  A. k  e.  ( 1..^ ( # `  t
) ) ( t `
 k )  e. 
ran  ( T `  ( t `  (
k  -  1 ) ) ) ) } 
|->  ( m `  (
( # `  m )  -  1 ) ) )
91, 4, 5, 6, 7, 8efgsf 15316 . . . . . . . . . 10  |-  S : { t  e.  (Word 
W  \  { (/) } )  |  ( ( t `
 0 )  e.  D  /\  A. k  e.  ( 1..^ ( # `  t ) ) ( t `  k )  e.  ran  ( T `
 ( t `  ( k  -  1 ) ) ) ) } --> W
109fdmi 5555 . . . . . . . . . . 11  |-  dom  S  =  { t  e.  (Word 
W  \  { (/) } )  |  ( ( t `
 0 )  e.  D  /\  A. k  e.  ( 1..^ ( # `  t ) ) ( t `  k )  e.  ran  ( T `
 ( t `  ( k  -  1 ) ) ) ) }
1110feq2i 5545 . . . . . . . . . 10  |-  ( S : dom  S --> W  <->  S : { t  e.  (Word 
W  \  { (/) } )  |  ( ( t `
 0 )  e.  D  /\  A. k  e.  ( 1..^ ( # `  t ) ) ( t `  k )  e.  ran  ( T `
 ( t `  ( k  -  1 ) ) ) ) } --> W )
129, 11mpbir 201 . . . . . . . . 9  |-  S : dom  S --> W
1312ffvelrni 5828 . . . . . . . 8  |-  ( A  e.  dom  S  -> 
( S `  A
)  e.  W )
1413adantr 452 . . . . . . 7  |-  ( ( A  e.  dom  S  /\  B  e.  dom  S )  ->  ( S `  A )  e.  W
)
153, 14sseldi 3306 . . . . . 6  |-  ( ( A  e.  dom  S  /\  B  e.  dom  S )  ->  ( S `  A )  e. Word  (
I  X.  2o ) )
16 lencl 11690 . . . . . 6  |-  ( ( S `  A )  e. Word  ( I  X.  2o )  ->  ( # `  ( S `  A
) )  e.  NN0 )
1715, 16syl 16 . . . . 5  |-  ( ( A  e.  dom  S  /\  B  e.  dom  S )  ->  ( # `  ( S `  A )
)  e.  NN0 )
18 peano2nn0 10216 . . . . 5  |-  ( (
# `  ( S `  A ) )  e. 
NN0  ->  ( ( # `  ( S `  A
) )  +  1 )  e.  NN0 )
1917, 18syl 16 . . . 4  |-  ( ( A  e.  dom  S  /\  B  e.  dom  S )  ->  ( ( # `
 ( S `  A ) )  +  1 )  e.  NN0 )
20 breq2 4176 . . . . . . 7  |-  ( c  =  0  ->  (
( # `  ( S `
 a ) )  <  c  <->  ( # `  ( S `  a )
)  <  0 ) )
2120imbi1d 309 . . . . . 6  |-  ( c  =  0  ->  (
( ( # `  ( S `  a )
)  <  c  ->  ( ( S `  a
)  =  ( S `
 b )  -> 
( a `  0
)  =  ( b `
 0 ) ) )  <->  ( ( # `  ( S `  a
) )  <  0  ->  ( ( S `  a )  =  ( S `  b )  ->  ( a ` 
0 )  =  ( b `  0 ) ) ) ) )
22212ralbidv 2708 . . . . 5  |-  ( c  =  0  ->  ( A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a )
)  <  c  ->  ( ( S `  a
)  =  ( S `
 b )  -> 
( a `  0
)  =  ( b `
 0 ) ) )  <->  A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a )
)  <  0  ->  ( ( S `  a
)  =  ( S `
 b )  -> 
( a `  0
)  =  ( b `
 0 ) ) ) ) )
23 breq2 4176 . . . . . . 7  |-  ( c  =  i  ->  (
( # `  ( S `
 a ) )  <  c  <->  ( # `  ( S `  a )
)  <  i )
)
2423imbi1d 309 . . . . . 6  |-  ( c  =  i  ->  (
( ( # `  ( S `  a )
)  <  c  ->  ( ( S `  a
)  =  ( S `
 b )  -> 
( a `  0
)  =  ( b `
 0 ) ) )  <->  ( ( # `  ( S `  a
) )  <  i  ->  ( ( S `  a )  =  ( S `  b )  ->  ( a ` 
0 )  =  ( b `  0 ) ) ) ) )
25242ralbidv 2708 . . . . 5  |-  ( c  =  i  ->  ( A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a )
)  <  c  ->  ( ( S `  a
)  =  ( S `
 b )  -> 
( a `  0
)  =  ( b `
 0 ) ) )  <->  A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a )
)  <  i  ->  ( ( S `  a
)  =  ( S `
 b )  -> 
( a `  0
)  =  ( b `
 0 ) ) ) ) )
26 breq2 4176 . . . . . . 7  |-  ( c  =  ( i  +  1 )  ->  (
( # `  ( S `
 a ) )  <  c  <->  ( # `  ( S `  a )
)  <  ( i  +  1 ) ) )
2726imbi1d 309 . . . . . 6  |-  ( c  =  ( i  +  1 )  ->  (
( ( # `  ( S `  a )
)  <  c  ->  ( ( S `  a
)  =  ( S `
 b )  -> 
( a `  0
)  =  ( b `
 0 ) ) )  <->  ( ( # `  ( S `  a
) )  <  (
i  +  1 )  ->  ( ( S `
 a )  =  ( S `  b
)  ->  ( a `  0 )  =  ( b `  0
) ) ) ) )
28272ralbidv 2708 . . . . 5  |-  ( c  =  ( i  +  1 )  ->  ( A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a )
)  <  c  ->  ( ( S `  a
)  =  ( S `
 b )  -> 
( a `  0
)  =  ( b `
 0 ) ) )  <->  A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a )
)  <  ( i  +  1 )  -> 
( ( S `  a )  =  ( S `  b )  ->  ( a ` 
0 )  =  ( b `  0 ) ) ) ) )
29 breq2 4176 . . . . . . 7  |-  ( c  =  ( ( # `  ( S `  A
) )  +  1 )  ->  ( ( # `
 ( S `  a ) )  < 
c  <->  ( # `  ( S `  a )
)  <  ( ( # `
 ( S `  A ) )  +  1 ) ) )
3029imbi1d 309 . . . . . 6  |-  ( c  =  ( ( # `  ( S `  A
) )  +  1 )  ->  ( (
( # `  ( S `
 a ) )  <  c  ->  (
( S `  a
)  =  ( S `
 b )  -> 
( a `  0
)  =  ( b `
 0 ) ) )  <->  ( ( # `  ( S `  a
) )  <  (
( # `  ( S `
 A ) )  +  1 )  -> 
( ( S `  a )  =  ( S `  b )  ->  ( a ` 
0 )  =  ( b `  0 ) ) ) ) )
31302ralbidv 2708 . . . . 5  |-  ( c  =  ( ( # `  ( S `  A
) )  +  1 )  ->  ( A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a )
)  <  c  ->  ( ( S `  a
)  =  ( S `
 b )  -> 
( a `  0
)  =  ( b `
 0 ) ) )  <->  A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a )
)  <  ( ( # `
 ( S `  A ) )  +  1 )  ->  (
( S `  a
)  =  ( S `
 b )  -> 
( a `  0
)  =  ( b `
 0 ) ) ) ) )
3212ffvelrni 5828 . . . . . . . . . . 11  |-  ( a  e.  dom  S  -> 
( S `  a
)  e.  W )
333, 32sseldi 3306 . . . . . . . . . 10  |-  ( a  e.  dom  S  -> 
( S `  a
)  e. Word  ( I  X.  2o ) )
34 lencl 11690 . . . . . . . . . 10  |-  ( ( S `  a )  e. Word  ( I  X.  2o )  ->  ( # `  ( S `  a
) )  e.  NN0 )
3533, 34syl 16 . . . . . . . . 9  |-  ( a  e.  dom  S  -> 
( # `  ( S `
 a ) )  e.  NN0 )
36 nn0nlt0 10204 . . . . . . . . 9  |-  ( (
# `  ( S `  a ) )  e. 
NN0  ->  -.  ( # `  ( S `  a )
)  <  0 )
3735, 36syl 16 . . . . . . . 8  |-  ( a  e.  dom  S  ->  -.  ( # `  ( S `  a )
)  <  0 )
3837pm2.21d 100 . . . . . . 7  |-  ( a  e.  dom  S  -> 
( ( # `  ( S `  a )
)  <  0  ->  ( ( S `  a
)  =  ( S `
 b )  -> 
( a `  0
)  =  ( b `
 0 ) ) ) )
3938adantr 452 . . . . . 6  |-  ( ( a  e.  dom  S  /\  b  e.  dom  S )  ->  ( ( # `
 ( S `  a ) )  <  0  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) ) )
4039rgen2a 2732 . . . . 5  |-  A. a  e.  dom  S A. b  e.  dom  S ( (
# `  ( S `  a ) )  <  0  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) )
41 simpl1 960 . . . . . . . . . . . . . 14  |-  ( ( ( A. a  e. 
dom  S A. b  e.  dom  S ( (
# `  ( S `  a ) )  < 
i  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) )  /\  ( c  e. 
dom  S  /\  d  e.  dom  S )  /\  ( ( # `  ( S `  c )
)  =  i  /\  ( S `  c )  =  ( S `  d ) ) )  /\  -.  ( c `
 0 )  =  ( d `  0
) )  ->  A. a  e.  dom  S A. b  e.  dom  S ( (
# `  ( S `  a ) )  < 
i  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) ) )
42 simpl3l 1012 . . . . . . . . . . . . . . 15  |-  ( ( ( A. a  e. 
dom  S A. b  e.  dom  S ( (
# `  ( S `  a ) )  < 
i  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) )  /\  ( c  e. 
dom  S  /\  d  e.  dom  S )  /\  ( ( # `  ( S `  c )
)  =  i  /\  ( S `  c )  =  ( S `  d ) ) )  /\  -.  ( c `
 0 )  =  ( d `  0
) )  ->  ( # `
 ( S `  c ) )  =  i )
43 breq2 4176 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  ( S `  c ) )  =  i  ->  ( ( # `
 ( S `  a ) )  < 
( # `  ( S `
 c ) )  <-> 
( # `  ( S `
 a ) )  <  i ) )
4443imbi1d 309 . . . . . . . . . . . . . . . 16  |-  ( (
# `  ( S `  c ) )  =  i  ->  ( (
( # `  ( S `
 a ) )  <  ( # `  ( S `  c )
)  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) )  <-> 
( ( # `  ( S `  a )
)  <  i  ->  ( ( S `  a
)  =  ( S `
 b )  -> 
( a `  0
)  =  ( b `
 0 ) ) ) ) )
45442ralbidv 2708 . . . . . . . . . . . . . . 15  |-  ( (
# `  ( S `  c ) )  =  i  ->  ( A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a )
)  <  ( # `  ( S `  c )
)  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) )  <->  A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a )
)  <  i  ->  ( ( S `  a
)  =  ( S `
 b )  -> 
( a `  0
)  =  ( b `
 0 ) ) ) ) )
4642, 45syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( A. a  e. 
dom  S A. b  e.  dom  S ( (
# `  ( S `  a ) )  < 
i  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) )  /\  ( c  e. 
dom  S  /\  d  e.  dom  S )  /\  ( ( # `  ( S `  c )
)  =  i  /\  ( S `  c )  =  ( S `  d ) ) )  /\  -.  ( c `
 0 )  =  ( d `  0
) )  ->  ( A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a )
)  <  ( # `  ( S `  c )
)  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) )  <->  A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a )
)  <  i  ->  ( ( S `  a
)  =  ( S `
 b )  -> 
( a `  0
)  =  ( b `
 0 ) ) ) ) )
4741, 46mpbird 224 . . . . . . . . . . . . 13  |-  ( ( ( A. a  e. 
dom  S A. b  e.  dom  S ( (
# `  ( S `  a ) )  < 
i  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) )  /\  ( c  e. 
dom  S  /\  d  e.  dom  S )  /\  ( ( # `  ( S `  c )
)  =  i  /\  ( S `  c )  =  ( S `  d ) ) )  /\  -.  ( c `
 0 )  =  ( d `  0
) )  ->  A. a  e.  dom  S A. b  e.  dom  S ( (
# `  ( S `  a ) )  < 
( # `  ( S `
 c ) )  ->  ( ( S `
 a )  =  ( S `  b
)  ->  ( a `  0 )  =  ( b `  0
) ) ) )
48 simpl2l 1010 . . . . . . . . . . . . 13  |-  ( ( ( A. a  e. 
dom  S A. b  e.  dom  S ( (
# `  ( S `  a ) )  < 
i  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) )  /\  ( c  e. 
dom  S  /\  d  e.  dom  S )  /\  ( ( # `  ( S `  c )
)  =  i  /\  ( S `  c )  =  ( S `  d ) ) )  /\  -.  ( c `
 0 )  =  ( d `  0
) )  ->  c  e.  dom  S )
49 simpl2r 1011 . . . . . . . . . . . . 13  |-  ( ( ( A. a  e. 
dom  S A. b  e.  dom  S ( (
# `  ( S `  a ) )  < 
i  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) )  /\  ( c  e. 
dom  S  /\  d  e.  dom  S )  /\  ( ( # `  ( S `  c )
)  =  i  /\  ( S `  c )  =  ( S `  d ) ) )  /\  -.  ( c `
 0 )  =  ( d `  0
) )  ->  d  e.  dom  S )
50 simpl3r 1013 . . . . . . . . . . . . 13  |-  ( ( ( A. a  e. 
dom  S A. b  e.  dom  S ( (
# `  ( S `  a ) )  < 
i  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) )  /\  ( c  e. 
dom  S  /\  d  e.  dom  S )  /\  ( ( # `  ( S `  c )
)  =  i  /\  ( S `  c )  =  ( S `  d ) ) )  /\  -.  ( c `
 0 )  =  ( d `  0
) )  ->  ( S `  c )  =  ( S `  d ) )
51 simpr 448 . . . . . . . . . . . . 13  |-  ( ( ( A. a  e. 
dom  S A. b  e.  dom  S ( (
# `  ( S `  a ) )  < 
i  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) )  /\  ( c  e. 
dom  S  /\  d  e.  dom  S )  /\  ( ( # `  ( S `  c )
)  =  i  /\  ( S `  c )  =  ( S `  d ) ) )  /\  -.  ( c `
 0 )  =  ( d `  0
) )  ->  -.  ( c `  0
)  =  ( d `
 0 ) )
521, 4, 5, 6, 7, 8, 47, 48, 49, 50, 51efgredlem 15334 . . . . . . . . . . . 12  |-  -.  (
( A. a  e. 
dom  S A. b  e.  dom  S ( (
# `  ( S `  a ) )  < 
i  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) )  /\  ( c  e. 
dom  S  /\  d  e.  dom  S )  /\  ( ( # `  ( S `  c )
)  =  i  /\  ( S `  c )  =  ( S `  d ) ) )  /\  -.  ( c `
 0 )  =  ( d `  0
) )
53 iman 414 . . . . . . . . . . . 12  |-  ( ( ( A. a  e. 
dom  S A. b  e.  dom  S ( (
# `  ( S `  a ) )  < 
i  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) )  /\  ( c  e. 
dom  S  /\  d  e.  dom  S )  /\  ( ( # `  ( S `  c )
)  =  i  /\  ( S `  c )  =  ( S `  d ) ) )  ->  ( c ` 
0 )  =  ( d `  0 ) )  <->  -.  ( ( A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a )
)  <  i  ->  ( ( S `  a
)  =  ( S `
 b )  -> 
( a `  0
)  =  ( b `
 0 ) ) )  /\  ( c  e.  dom  S  /\  d  e.  dom  S )  /\  ( ( # `  ( S `  c
) )  =  i  /\  ( S `  c )  =  ( S `  d ) ) )  /\  -.  ( c `  0
)  =  ( d `
 0 ) ) )
5452, 53mpbir 201 . . . . . . . . . . 11  |-  ( ( A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a )
)  <  i  ->  ( ( S `  a
)  =  ( S `
 b )  -> 
( a `  0
)  =  ( b `
 0 ) ) )  /\  ( c  e.  dom  S  /\  d  e.  dom  S )  /\  ( ( # `  ( S `  c
) )  =  i  /\  ( S `  c )  =  ( S `  d ) ) )  ->  (
c `  0 )  =  ( d ` 
0 ) )
55543expia 1155 . . . . . . . . . 10  |-  ( ( A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a )
)  <  i  ->  ( ( S `  a
)  =  ( S `
 b )  -> 
( a `  0
)  =  ( b `
 0 ) ) )  /\  ( c  e.  dom  S  /\  d  e.  dom  S ) )  ->  ( (
( # `  ( S `
 c ) )  =  i  /\  ( S `  c )  =  ( S `  d ) )  -> 
( c `  0
)  =  ( d `
 0 ) ) )
5655exp3a 426 . . . . . . . . 9  |-  ( ( A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a )
)  <  i  ->  ( ( S `  a
)  =  ( S `
 b )  -> 
( a `  0
)  =  ( b `
 0 ) ) )  /\  ( c  e.  dom  S  /\  d  e.  dom  S ) )  ->  ( ( # `
 ( S `  c ) )  =  i  ->  ( ( S `  c )  =  ( S `  d )  ->  (
c `  0 )  =  ( d ` 
0 ) ) ) )
5756ralrimivva 2758 . . . . . . . 8  |-  ( A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a )
)  <  i  ->  ( ( S `  a
)  =  ( S `
 b )  -> 
( a `  0
)  =  ( b `
 0 ) ) )  ->  A. c  e.  dom  S A. d  e.  dom  S ( (
# `  ( S `  c ) )  =  i  ->  ( ( S `  c )  =  ( S `  d )  ->  (
c `  0 )  =  ( d ` 
0 ) ) ) )
58 fveq2 5687 . . . . . . . . . . . 12  |-  ( c  =  a  ->  ( S `  c )  =  ( S `  a ) )
5958fveq2d 5691 . . . . . . . . . . 11  |-  ( c  =  a  ->  ( # `
 ( S `  c ) )  =  ( # `  ( S `  a )
) )
6059eqeq1d 2412 . . . . . . . . . 10  |-  ( c  =  a  ->  (
( # `  ( S `
 c ) )  =  i  <->  ( # `  ( S `  a )
)  =  i ) )
6158eqeq1d 2412 . . . . . . . . . . 11  |-  ( c  =  a  ->  (
( S `  c
)  =  ( S `
 d )  <->  ( S `  a )  =  ( S `  d ) ) )
62 fveq1 5686 . . . . . . . . . . . 12  |-  ( c  =  a  ->  (
c `  0 )  =  ( a ` 
0 ) )
6362eqeq1d 2412 . . . . . . . . . . 11  |-  ( c  =  a  ->  (
( c `  0
)  =  ( d `
 0 )  <->  ( a `  0 )  =  ( d `  0
) ) )
6461, 63imbi12d 312 . . . . . . . . . 10  |-  ( c  =  a  ->  (
( ( S `  c )  =  ( S `  d )  ->  ( c ` 
0 )  =  ( d `  0 ) )  <->  ( ( S `
 a )  =  ( S `  d
)  ->  ( a `  0 )  =  ( d `  0
) ) ) )
6560, 64imbi12d 312 . . . . . . . . 9  |-  ( c  =  a  ->  (
( ( # `  ( S `  c )
)  =  i  -> 
( ( S `  c )  =  ( S `  d )  ->  ( c ` 
0 )  =  ( d `  0 ) ) )  <->  ( ( # `
 ( S `  a ) )  =  i  ->  ( ( S `  a )  =  ( S `  d )  ->  (
a `  0 )  =  ( d ` 
0 ) ) ) ) )
66 fveq2 5687 . . . . . . . . . . . 12  |-  ( d  =  b  ->  ( S `  d )  =  ( S `  b ) )
6766eqeq2d 2415 . . . . . . . . . . 11  |-  ( d  =  b  ->  (
( S `  a
)  =  ( S `
 d )  <->  ( S `  a )  =  ( S `  b ) ) )
68 fveq1 5686 . . . . . . . . . . . 12  |-  ( d  =  b  ->  (
d `  0 )  =  ( b ` 
0 ) )
6968eqeq2d 2415 . . . . . . . . . . 11  |-  ( d  =  b  ->  (
( a `  0
)  =  ( d `
 0 )  <->  ( a `  0 )  =  ( b `  0
) ) )
7067, 69imbi12d 312 . . . . . . . . . 10  |-  ( d  =  b  ->  (
( ( S `  a )  =  ( S `  d )  ->  ( a ` 
0 )  =  ( d `  0 ) )  <->  ( ( S `
 a )  =  ( S `  b
)  ->  ( a `  0 )  =  ( b `  0
) ) ) )
7170imbi2d 308 . . . . . . . . 9  |-  ( d  =  b  ->  (
( ( # `  ( S `  a )
)  =  i  -> 
( ( S `  a )  =  ( S `  d )  ->  ( a ` 
0 )  =  ( d `  0 ) ) )  <->  ( ( # `
 ( S `  a ) )  =  i  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) ) ) )
7265, 71cbvral2v 2900 . . . . . . . 8  |-  ( A. c  e.  dom  S A. d  e.  dom  S ( ( # `  ( S `  c )
)  =  i  -> 
( ( S `  c )  =  ( S `  d )  ->  ( c ` 
0 )  =  ( d `  0 ) ) )  <->  A. a  e.  dom  S A. b  e.  dom  S ( (
# `  ( S `  a ) )  =  i  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) ) )
7357, 72sylib 189 . . . . . . 7  |-  ( A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a )
)  <  i  ->  ( ( S `  a
)  =  ( S `
 b )  -> 
( a `  0
)  =  ( b `
 0 ) ) )  ->  A. a  e.  dom  S A. b  e.  dom  S ( (
# `  ( S `  a ) )  =  i  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) ) )
7473ancli 535 . . . . . 6  |-  ( A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a )
)  <  i  ->  ( ( S `  a
)  =  ( S `
 b )  -> 
( a `  0
)  =  ( b `
 0 ) ) )  ->  ( A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a )
)  <  i  ->  ( ( S `  a
)  =  ( S `
 b )  -> 
( a `  0
)  =  ( b `
 0 ) ) )  /\  A. a  e.  dom  S A. b  e.  dom  S ( (
# `  ( S `  a ) )  =  i  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) ) ) )
7535adantr 452 . . . . . . . . . . 11  |-  ( ( a  e.  dom  S  /\  b  e.  dom  S )  ->  ( # `  ( S `  a )
)  e.  NN0 )
76 nn0leltp1 10289 . . . . . . . . . . . . 13  |-  ( ( ( # `  ( S `  a )
)  e.  NN0  /\  i  e.  NN0 )  -> 
( ( # `  ( S `  a )
)  <_  i  <->  ( # `  ( S `  a )
)  <  ( i  +  1 ) ) )
77 nn0re 10186 . . . . . . . . . . . . . 14  |-  ( (
# `  ( S `  a ) )  e. 
NN0  ->  ( # `  ( S `  a )
)  e.  RR )
78 nn0re 10186 . . . . . . . . . . . . . 14  |-  ( i  e.  NN0  ->  i  e.  RR )
79 leloe 9117 . . . . . . . . . . . . . 14  |-  ( ( ( # `  ( S `  a )
)  e.  RR  /\  i  e.  RR )  ->  ( ( # `  ( S `  a )
)  <_  i  <->  ( ( # `
 ( S `  a ) )  < 
i  \/  ( # `  ( S `  a
) )  =  i ) ) )
8077, 78, 79syl2an 464 . . . . . . . . . . . . 13  |-  ( ( ( # `  ( S `  a )
)  e.  NN0  /\  i  e.  NN0 )  -> 
( ( # `  ( S `  a )
)  <_  i  <->  ( ( # `
 ( S `  a ) )  < 
i  \/  ( # `  ( S `  a
) )  =  i ) ) )
8176, 80bitr3d 247 . . . . . . . . . . . 12  |-  ( ( ( # `  ( S `  a )
)  e.  NN0  /\  i  e.  NN0 )  -> 
( ( # `  ( S `  a )
)  <  ( i  +  1 )  <->  ( ( # `
 ( S `  a ) )  < 
i  \/  ( # `  ( S `  a
) )  =  i ) ) )
8281ancoms 440 . . . . . . . . . . 11  |-  ( ( i  e.  NN0  /\  ( # `  ( S `
 a ) )  e.  NN0 )  -> 
( ( # `  ( S `  a )
)  <  ( i  +  1 )  <->  ( ( # `
 ( S `  a ) )  < 
i  \/  ( # `  ( S `  a
) )  =  i ) ) )
8375, 82sylan2 461 . . . . . . . . . 10  |-  ( ( i  e.  NN0  /\  ( a  e.  dom  S  /\  b  e.  dom  S ) )  ->  (
( # `  ( S `
 a ) )  <  ( i  +  1 )  <->  ( ( # `
 ( S `  a ) )  < 
i  \/  ( # `  ( S `  a
) )  =  i ) ) )
8483imbi1d 309 . . . . . . . . 9  |-  ( ( i  e.  NN0  /\  ( a  e.  dom  S  /\  b  e.  dom  S ) )  ->  (
( ( # `  ( S `  a )
)  <  ( i  +  1 )  -> 
( ( S `  a )  =  ( S `  b )  ->  ( a ` 
0 )  =  ( b `  0 ) ) )  <->  ( (
( # `  ( S `
 a ) )  <  i  \/  ( # `
 ( S `  a ) )  =  i )  ->  (
( S `  a
)  =  ( S `
 b )  -> 
( a `  0
)  =  ( b `
 0 ) ) ) ) )
85 jaob 759 . . . . . . . . 9  |-  ( ( ( ( # `  ( S `  a )
)  <  i  \/  ( # `  ( S `
 a ) )  =  i )  -> 
( ( S `  a )  =  ( S `  b )  ->  ( a ` 
0 )  =  ( b `  0 ) ) )  <->  ( (
( # `  ( S `
 a ) )  <  i  ->  (
( S `  a
)  =  ( S `
 b )  -> 
( a `  0
)  =  ( b `
 0 ) ) )  /\  ( (
# `  ( S `  a ) )  =  i  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) ) ) )
8684, 85syl6bb 253 . . . . . . . 8  |-  ( ( i  e.  NN0  /\  ( a  e.  dom  S  /\  b  e.  dom  S ) )  ->  (
( ( # `  ( S `  a )
)  <  ( i  +  1 )  -> 
( ( S `  a )  =  ( S `  b )  ->  ( a ` 
0 )  =  ( b `  0 ) ) )  <->  ( (
( # `  ( S `
 a ) )  <  i  ->  (
( S `  a
)  =  ( S `
 b )  -> 
( a `  0
)  =  ( b `
 0 ) ) )  /\  ( (
# `  ( S `  a ) )  =  i  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) ) ) ) )
87862ralbidva 2706 . . . . . . 7  |-  ( i  e.  NN0  ->  ( A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a )
)  <  ( i  +  1 )  -> 
( ( S `  a )  =  ( S `  b )  ->  ( a ` 
0 )  =  ( b `  0 ) ) )  <->  A. a  e.  dom  S A. b  e.  dom  S ( ( ( # `  ( S `  a )
)  <  i  ->  ( ( S `  a
)  =  ( S `
 b )  -> 
( a `  0
)  =  ( b `
 0 ) ) )  /\  ( (
# `  ( S `  a ) )  =  i  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) ) ) ) )
88 r19.26-2 2799 . . . . . . 7  |-  ( A. a  e.  dom  S A. b  e.  dom  S ( ( ( # `  ( S `  a )
)  <  i  ->  ( ( S `  a
)  =  ( S `
 b )  -> 
( a `  0
)  =  ( b `
 0 ) ) )  /\  ( (
# `  ( S `  a ) )  =  i  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) ) )  <->  ( A. a  e.  dom  S A. b  e.  dom  S ( (
# `  ( S `  a ) )  < 
i  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) )  /\  A. a  e. 
dom  S A. b  e.  dom  S ( (
# `  ( S `  a ) )  =  i  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) ) ) )
8987, 88syl6bb 253 . . . . . 6  |-  ( i  e.  NN0  ->  ( A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a )
)  <  ( i  +  1 )  -> 
( ( S `  a )  =  ( S `  b )  ->  ( a ` 
0 )  =  ( b `  0 ) ) )  <->  ( A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a )
)  <  i  ->  ( ( S `  a
)  =  ( S `
 b )  -> 
( a `  0
)  =  ( b `
 0 ) ) )  /\  A. a  e.  dom  S A. b  e.  dom  S ( (
# `  ( S `  a ) )  =  i  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) ) ) ) )
9074, 89syl5ibr 213 . . . . 5  |-  ( i  e.  NN0  ->  ( A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a )
)  <  i  ->  ( ( S `  a
)  =  ( S `
 b )  -> 
( a `  0
)  =  ( b `
 0 ) ) )  ->  A. a  e.  dom  S A. b  e.  dom  S ( (
# `  ( S `  a ) )  < 
( i  +  1 )  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) ) ) )
9122, 25, 28, 31, 40, 90nn0ind 10322 . . . 4  |-  ( ( ( # `  ( S `  A )
)  +  1 )  e.  NN0  ->  A. a  e.  dom  S A. b  e.  dom  S ( (
# `  ( S `  a ) )  < 
( ( # `  ( S `  A )
)  +  1 )  ->  ( ( S `
 a )  =  ( S `  b
)  ->  ( a `  0 )  =  ( b `  0
) ) ) )
9219, 91syl 16 . . 3  |-  ( ( A  e.  dom  S  /\  B  e.  dom  S )  ->  A. a  e.  dom  S A. b  e.  dom  S ( (
# `  ( S `  a ) )  < 
( ( # `  ( S `  A )
)  +  1 )  ->  ( ( S `
 a )  =  ( S `  b
)  ->  ( a `  0 )  =  ( b `  0
) ) ) )
9317nn0red 10231 . . . 4  |-  ( ( A  e.  dom  S  /\  B  e.  dom  S )  ->  ( # `  ( S `  A )
)  e.  RR )
9493ltp1d 9897 . . 3  |-  ( ( A  e.  dom  S  /\  B  e.  dom  S )  ->  ( # `  ( S `  A )
)  <  ( ( # `
 ( S `  A ) )  +  1 ) )
95 fveq2 5687 . . . . . . 7  |-  ( a  =  A  ->  ( S `  a )  =  ( S `  A ) )
9695fveq2d 5691 . . . . . 6  |-  ( a  =  A  ->  ( # `
 ( S `  a ) )  =  ( # `  ( S `  A )
) )
9796breq1d 4182 . . . . 5  |-  ( a  =  A  ->  (
( # `  ( S `
 a ) )  <  ( ( # `  ( S `  A
) )  +  1 )  <->  ( # `  ( S `  A )
)  <  ( ( # `
 ( S `  A ) )  +  1 ) ) )
9895eqeq1d 2412 . . . . . 6  |-  ( a  =  A  ->  (
( S `  a
)  =  ( S `
 b )  <->  ( S `  A )  =  ( S `  b ) ) )
99 fveq1 5686 . . . . . . 7  |-  ( a  =  A  ->  (
a `  0 )  =  ( A ` 
0 ) )
10099eqeq1d 2412 . . . . . 6  |-  ( a  =  A  ->  (
( a `  0
)  =  ( b `
 0 )  <->  ( A `  0 )  =  ( b `  0
) ) )
10198, 100imbi12d 312 . . . . 5  |-  ( a  =  A  ->  (
( ( S `  a )  =  ( S `  b )  ->  ( a ` 
0 )  =  ( b `  0 ) )  <->  ( ( S `
 A )  =  ( S `  b
)  ->  ( A `  0 )  =  ( b `  0
) ) ) )
10297, 101imbi12d 312 . . . 4  |-  ( a  =  A  ->  (
( ( # `  ( S `  a )
)  <  ( ( # `
 ( S `  A ) )  +  1 )  ->  (
( S `  a
)  =  ( S `
 b )  -> 
( a `  0
)  =  ( b `
 0 ) ) )  <->  ( ( # `  ( S `  A
) )  <  (
( # `  ( S `
 A ) )  +  1 )  -> 
( ( S `  A )  =  ( S `  b )  ->  ( A ` 
0 )  =  ( b `  0 ) ) ) ) )
103 fveq2 5687 . . . . . . 7  |-  ( b  =  B  ->  ( S `  b )  =  ( S `  B ) )
104103eqeq2d 2415 . . . . . 6  |-  ( b  =  B  ->  (
( S `  A
)  =  ( S `
 b )  <->  ( S `  A )  =  ( S `  B ) ) )
105 fveq1 5686 . . . . . . 7  |-  ( b  =  B  ->  (
b `  0 )  =  ( B ` 
0 ) )
106105eqeq2d 2415 . . . . . 6  |-  ( b  =  B  ->  (
( A `  0
)  =  ( b `
 0 )  <->  ( A `  0 )  =  ( B `  0
) ) )
107104, 106imbi12d 312 . . . . 5  |-  ( b  =  B  ->  (
( ( S `  A )  =  ( S `  b )  ->  ( A ` 
0 )  =  ( b `  0 ) )  <->  ( ( S `
 A )  =  ( S `  B
)  ->  ( A `  0 )  =  ( B `  0
) ) ) )
108107imbi2d 308 . . . 4  |-  ( b  =  B  ->  (
( ( # `  ( S `  A )
)  <  ( ( # `
 ( S `  A ) )  +  1 )  ->  (
( S `  A
)  =  ( S `
 b )  -> 
( A `  0
)  =  ( b `
 0 ) ) )  <->  ( ( # `  ( S `  A
) )  <  (
( # `  ( S `
 A ) )  +  1 )  -> 
( ( S `  A )  =  ( S `  B )  ->  ( A ` 
0 )  =  ( B `  0 ) ) ) ) )
109102, 108rspc2v 3018 . . 3  |-  ( ( A  e.  dom  S  /\  B  e.  dom  S )  ->  ( A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a )
)  <  ( ( # `
 ( S `  A ) )  +  1 )  ->  (
( S `  a
)  =  ( S `
 b )  -> 
( a `  0
)  =  ( b `
 0 ) ) )  ->  ( ( # `
 ( S `  A ) )  < 
( ( # `  ( S `  A )
)  +  1 )  ->  ( ( S `
 A )  =  ( S `  B
)  ->  ( A `  0 )  =  ( B `  0
) ) ) ) )
11092, 94, 109mp2d 43 . 2  |-  ( ( A  e.  dom  S  /\  B  e.  dom  S )  ->  ( ( S `  A )  =  ( S `  B )  ->  ( A `  0 )  =  ( B ` 
0 ) ) )
1111103impia 1150 1  |-  ( ( A  e.  dom  S  /\  B  e.  dom  S  /\  ( S `  A )  =  ( S `  B ) )  ->  ( A `  0 )  =  ( B `  0
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   {crab 2670    \ cdif 3277   (/)c0 3588   {csn 3774   <.cop 3777   <.cotp 3778   U_ciun 4053   class class class wbr 4172    e. cmpt 4226    _I cid 4453    X. cxp 4835   dom cdm 4837   ran crn 4838   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   1oc1o 6676   2oc2o 6677   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    < clt 9076    <_ cle 9077    - cmin 9247   NN0cn0 10177   ...cfz 10999  ..^cfzo 11090   #chash 11573  Word cword 11672   splice csplice 11676   <"cs2 11760   ~FG cefg 15293
This theorem is referenced by:  efgrelexlemb  15337
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-ot 3784  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-hash 11574  df-word 11678  df-concat 11679  df-s1 11680  df-substr 11681  df-splice 11682  df-s2 11767
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