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Theorem efgred 16572
Description: The reduced word that forms the base of the sequence in efgsval 16555 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 5925 . . . . . . . 8  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
31, 2eqsstri 3534 . . . . . . 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 16553 . . . . . . . . . 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 5736 . . . . . . . . . . 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 5724 . . . . . . . . . 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 209 . . . . . . . . 9  |-  S : dom  S --> W
1312ffvelrni 6020 . . . . . . . 8  |-  ( A  e.  dom  S  -> 
( S `  A
)  e.  W )
1413adantr 465 . . . . . . 7  |-  ( ( A  e.  dom  S  /\  B  e.  dom  S )  ->  ( S `  A )  e.  W
)
153, 14sseldi 3502 . . . . . 6  |-  ( ( A  e.  dom  S  /\  B  e.  dom  S )  ->  ( S `  A )  e. Word  (
I  X.  2o ) )
16 lencl 12528 . . . . . 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 10836 . . . . 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 4451 . . . . . . 7  |-  ( c  =  0  ->  (
( # `  ( S `
 a ) )  <  c  <->  ( # `  ( S `  a )
)  <  0 ) )
2120imbi1d 317 . . . . . 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 2908 . . . . 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 4451 . . . . . . 7  |-  ( c  =  i  ->  (
( # `  ( S `
 a ) )  <  c  <->  ( # `  ( S `  a )
)  <  i )
)
2423imbi1d 317 . . . . . 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 2908 . . . . 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 4451 . . . . . . 7  |-  ( c  =  ( i  +  1 )  ->  (
( # `  ( S `
 a ) )  <  c  <->  ( # `  ( S `  a )
)  <  ( i  +  1 ) ) )
2726imbi1d 317 . . . . . 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 2908 . . . . 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 4451 . . . . . . 7  |-  ( c  =  ( ( # `  ( S `  A
) )  +  1 )  ->  ( ( # `
 ( S `  a ) )  < 
c  <->  ( # `  ( S `  a )
)  <  ( ( # `
 ( S `  A ) )  +  1 ) ) )
3029imbi1d 317 . . . . . 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 2908 . . . . 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 6020 . . . . . . . . . . 11  |-  ( a  e.  dom  S  -> 
( S `  a
)  e.  W )
333, 32sseldi 3502 . . . . . . . . . 10  |-  ( a  e.  dom  S  -> 
( S `  a
)  e. Word  ( I  X.  2o ) )
34 lencl 12528 . . . . . . . . . 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 10822 . . . . . . . . 9  |-  ( (
# `  ( S `  a ) )  e. 
NN0  ->  -.  ( # `  ( S `  a )
)  <  0 )
3735, 36syl 16 . . . . . . . 8  |-  ( a  e.  dom  S  ->  -.  ( # `  ( S `  a )
)  <  0 )
3837pm2.21d 106 . . . . . . 7  |-  ( a  e.  dom  S  -> 
( ( # `  ( S `  a )
)  <  0  ->  ( ( S `  a
)  =  ( S `
 b )  -> 
( a `  0
)  =  ( b `
 0 ) ) ) )
3938adantr 465 . . . . . 6  |-  ( ( a  e.  dom  S  /\  b  e.  dom  S )  ->  ( ( # `
 ( S `  a ) )  <  0  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) ) )
4039rgen2a 2891 . . . . 5  |-  A. a  e.  dom  S A. b  e.  dom  S ( (
# `  ( S `  a ) )  <  0  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) )
41 simpl1 999 . . . . . . . . . . . . . 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 1051 . . . . . . . . . . . . . . 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 4451 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  ( S `  c ) )  =  i  ->  ( ( # `
 ( S `  a ) )  < 
( # `  ( S `
 c ) )  <-> 
( # `  ( S `
 a ) )  <  i ) )
4443imbi1d 317 . . . . . . . . . . . . . . . 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 2908 . . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . 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 1049 . . . . . . . . . . . . 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 1050 . . . . . . . . . . . . 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 1052 . . . . . . . . . . . . 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 461 . . . . . . . . . . . . 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 16571 . . . . . . . . . . . 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 424 . . . . . . . . . . . 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 209 . . . . . . . . . . 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 1198 . . . . . . . . . 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 ) ) )
5655expd 436 . . . . . . . . 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 2885 . . . . . . . 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 5866 . . . . . . . . . . . 12  |-  ( c  =  a  ->  ( S `  c )  =  ( S `  a ) )
5958fveq2d 5870 . . . . . . . . . . 11  |-  ( c  =  a  ->  ( # `
 ( S `  c ) )  =  ( # `  ( S `  a )
) )
6059eqeq1d 2469 . . . . . . . . . 10  |-  ( c  =  a  ->  (
( # `  ( S `
 c ) )  =  i  <->  ( # `  ( S `  a )
)  =  i ) )
6158eqeq1d 2469 . . . . . . . . . . 11  |-  ( c  =  a  ->  (
( S `  c
)  =  ( S `
 d )  <->  ( S `  a )  =  ( S `  d ) ) )
62 fveq1 5865 . . . . . . . . . . . 12  |-  ( c  =  a  ->  (
c `  0 )  =  ( a ` 
0 ) )
6362eqeq1d 2469 . . . . . . . . . . 11  |-  ( c  =  a  ->  (
( c `  0
)  =  ( d `
 0 )  <->  ( a `  0 )  =  ( d `  0
) ) )
6461, 63imbi12d 320 . . . . . . . . . 10  |-  ( c  =  a  ->  (
( ( S `  c )  =  ( S `  d )  ->  ( c ` 
0 )  =  ( d `  0 ) )  <->  ( ( S `
 a )  =  ( S `  d
)  ->  ( a `  0 )  =  ( d `  0
) ) ) )
6560, 64imbi12d 320 . . . . . . . . 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 5866 . . . . . . . . . . . 12  |-  ( d  =  b  ->  ( S `  d )  =  ( S `  b ) )
6766eqeq2d 2481 . . . . . . . . . . 11  |-  ( d  =  b  ->  (
( S `  a
)  =  ( S `
 d )  <->  ( S `  a )  =  ( S `  b ) ) )
68 fveq1 5865 . . . . . . . . . . . 12  |-  ( d  =  b  ->  (
d `  0 )  =  ( b ` 
0 ) )
6968eqeq2d 2481 . . . . . . . . . . 11  |-  ( d  =  b  ->  (
( a `  0
)  =  ( d `
 0 )  <->  ( a `  0 )  =  ( b `  0
) ) )
7067, 69imbi12d 320 . . . . . . . . . 10  |-  ( d  =  b  ->  (
( ( S `  a )  =  ( S `  d )  ->  ( a ` 
0 )  =  ( d `  0 ) )  <->  ( ( S `
 a )  =  ( S `  b
)  ->  ( a `  0 )  =  ( b `  0
) ) ) )
7170imbi2d 316 . . . . . . . . 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 3096 . . . . . . . 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 196 . . . . . . 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 551 . . . . . 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 465 . . . . . . . . . . 11  |-  ( ( a  e.  dom  S  /\  b  e.  dom  S )  ->  ( # `  ( S `  a )
)  e.  NN0 )
76 nn0leltp1 10921 . . . . . . . . . . . . 13  |-  ( ( ( # `  ( S `  a )
)  e.  NN0  /\  i  e.  NN0 )  -> 
( ( # `  ( S `  a )
)  <_  i  <->  ( # `  ( S `  a )
)  <  ( i  +  1 ) ) )
77 nn0re 10804 . . . . . . . . . . . . . 14  |-  ( (
# `  ( S `  a ) )  e. 
NN0  ->  ( # `  ( S `  a )
)  e.  RR )
78 nn0re 10804 . . . . . . . . . . . . . 14  |-  ( i  e.  NN0  ->  i  e.  RR )
79 leloe 9671 . . . . . . . . . . . . . 14  |-  ( ( ( # `  ( S `  a )
)  e.  RR  /\  i  e.  RR )  ->  ( ( # `  ( S `  a )
)  <_  i  <->  ( ( # `
 ( S `  a ) )  < 
i  \/  ( # `  ( S `  a
) )  =  i ) ) )
8077, 78, 79syl2an 477 . . . . . . . . . . . . 13  |-  ( ( ( # `  ( S `  a )
)  e.  NN0  /\  i  e.  NN0 )  -> 
( ( # `  ( S `  a )
)  <_  i  <->  ( ( # `
 ( S `  a ) )  < 
i  \/  ( # `  ( S `  a
) )  =  i ) ) )
8176, 80bitr3d 255 . . . . . . . . . . . 12  |-  ( ( ( # `  ( S `  a )
)  e.  NN0  /\  i  e.  NN0 )  -> 
( ( # `  ( S `  a )
)  <  ( i  +  1 )  <->  ( ( # `
 ( S `  a ) )  < 
i  \/  ( # `  ( S `  a
) )  =  i ) ) )
8281ancoms 453 . . . . . . . . . . 11  |-  ( ( i  e.  NN0  /\  ( # `  ( S `
 a ) )  e.  NN0 )  -> 
( ( # `  ( S `  a )
)  <  ( i  +  1 )  <->  ( ( # `
 ( S `  a ) )  < 
i  \/  ( # `  ( S `  a
) )  =  i ) ) )
8375, 82sylan2 474 . . . . . . . . . 10  |-  ( ( i  e.  NN0  /\  ( a  e.  dom  S  /\  b  e.  dom  S ) )  ->  (
( # `  ( S `
 a ) )  <  ( i  +  1 )  <->  ( ( # `
 ( S `  a ) )  < 
i  \/  ( # `  ( S `  a
) )  =  i ) ) )
8483imbi1d 317 . . . . . . . . 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 781 . . . . . . . . 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 261 . . . . . . . 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 2906 . . . . . . 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 2990 . . . . . . 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 261 . . . . . 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 221 . . . . 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 10957 . . . 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 10853 . . . 4  |-  ( ( A  e.  dom  S  /\  B  e.  dom  S )  ->  ( # `  ( S `  A )
)  e.  RR )
9493ltp1d 10476 . . 3  |-  ( ( A  e.  dom  S  /\  B  e.  dom  S )  ->  ( # `  ( S `  A )
)  <  ( ( # `
 ( S `  A ) )  +  1 ) )
95 fveq2 5866 . . . . . . 7  |-  ( a  =  A  ->  ( S `  a )  =  ( S `  A ) )
9695fveq2d 5870 . . . . . 6  |-  ( a  =  A  ->  ( # `
 ( S `  a ) )  =  ( # `  ( S `  A )
) )
9796breq1d 4457 . . . . 5  |-  ( a  =  A  ->  (
( # `  ( S `
 a ) )  <  ( ( # `  ( S `  A
) )  +  1 )  <->  ( # `  ( S `  A )
)  <  ( ( # `
 ( S `  A ) )  +  1 ) ) )
9895eqeq1d 2469 . . . . . 6  |-  ( a  =  A  ->  (
( S `  a
)  =  ( S `
 b )  <->  ( S `  A )  =  ( S `  b ) ) )
99 fveq1 5865 . . . . . . 7  |-  ( a  =  A  ->  (
a `  0 )  =  ( A ` 
0 ) )
10099eqeq1d 2469 . . . . . 6  |-  ( a  =  A  ->  (
( a `  0
)  =  ( b `
 0 )  <->  ( A `  0 )  =  ( b `  0
) ) )
10198, 100imbi12d 320 . . . . 5  |-  ( a  =  A  ->  (
( ( S `  a )  =  ( S `  b )  ->  ( a ` 
0 )  =  ( b `  0 ) )  <->  ( ( S `
 A )  =  ( S `  b
)  ->  ( A `  0 )  =  ( b `  0
) ) ) )
10297, 101imbi12d 320 . . . 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 5866 . . . . . . 7  |-  ( b  =  B  ->  ( S `  b )  =  ( S `  B ) )
104103eqeq2d 2481 . . . . . 6  |-  ( b  =  B  ->  (
( S `  A
)  =  ( S `
 b )  <->  ( S `  A )  =  ( S `  B ) ) )
105 fveq1 5865 . . . . . . 7  |-  ( b  =  B  ->  (
b `  0 )  =  ( B ` 
0 ) )
106105eqeq2d 2481 . . . . . 6  |-  ( b  =  B  ->  (
( A `  0
)  =  ( b `
 0 )  <->  ( A `  0 )  =  ( B `  0
) ) )
107104, 106imbi12d 320 . . . . 5  |-  ( b  =  B  ->  (
( ( S `  A )  =  ( S `  b )  ->  ( A ` 
0 )  =  ( b `  0 ) )  <->  ( ( S `
 A )  =  ( S `  B
)  ->  ( A `  0 )  =  ( B `  0
) ) ) )
108107imbi2d 316 . . . 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 3223 . . 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 45 . 2  |-  ( ( A  e.  dom  S  /\  B  e.  dom  S )  ->  ( ( S `  A )  =  ( S `  B )  ->  ( A `  0 )  =  ( B ` 
0 ) ) )
1111103impia 1193 1  |-  ( ( A  e.  dom  S  /\  B  e.  dom  S  /\  ( S `  A )  =  ( S `  B ) )  ->  ( A `  0 )  =  ( B `  0
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   {crab 2818    \ cdif 3473   (/)c0 3785   {csn 4027   <.cop 4033   <.cotp 4035   U_ciun 4325   class class class wbr 4447    |-> cmpt 4505    _I cid 4790    X. cxp 4997   dom cdm 4999   ran crn 5000   -->wf 5584   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286   1oc1o 7123   2oc2o 7124   RRcr 9491   0cc0 9492   1c1 9493    + caddc 9495    < clt 9628    <_ cle 9629    - cmin 9805   NN0cn0 10795   ...cfz 11672  ..^cfzo 11792   #chash 12373  Word cword 12500   splice csplice 12505   <"cs2 12769   ~FG cefg 16530
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-ot 4036  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-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  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-2 10594  df-n0 10796  df-z 10865  df-uz 11083  df-rp 11221  df-fz 11673  df-fzo 11793  df-hash 12374  df-word 12508  df-concat 12510  df-s1 12511  df-substr 12512  df-splice 12513  df-s2 12776
This theorem is referenced by:  efgrelexlemb  16574
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