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Theorem efgred 15057
Description: The reduced word that forms the base of the sequence in efgsval 15040 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 5580 . . . . . . . 8  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
31, 2eqsstri 3208 . . . . . . 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 15038 . . . . . . . . . 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 5394 . . . . . . . . . . 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 5384 . . . . . . . . . 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 200 . . . . . . . . 9  |-  S : dom  S --> W
1312ffvelrni 5664 . . . . . . . 8  |-  ( A  e.  dom  S  -> 
( S `  A
)  e.  W )
1413adantr 451 . . . . . . 7  |-  ( ( A  e.  dom  S  /\  B  e.  dom  S )  ->  ( S `  A )  e.  W
)
153, 14sseldi 3178 . . . . . 6  |-  ( ( A  e.  dom  S  /\  B  e.  dom  S )  ->  ( S `  A )  e. Word  (
I  X.  2o ) )
16 lencl 11421 . . . . . 6  |-  ( ( S `  A )  e. Word  ( I  X.  2o )  ->  ( # `  ( S `  A
) )  e.  NN0 )
1715, 16syl 15 . . . . 5  |-  ( ( A  e.  dom  S  /\  B  e.  dom  S )  ->  ( # `  ( S `  A )
)  e.  NN0 )
18 peano2nn0 10004 . . . . 5  |-  ( (
# `  ( S `  A ) )  e. 
NN0  ->  ( ( # `  ( S `  A
) )  +  1 )  e.  NN0 )
1917, 18syl 15 . . . 4  |-  ( ( A  e.  dom  S  /\  B  e.  dom  S )  ->  ( ( # `
 ( S `  A ) )  +  1 )  e.  NN0 )
20 breq2 4027 . . . . . . 7  |-  ( c  =  0  ->  (
( # `  ( S `
 a ) )  <  c  <->  ( # `  ( S `  a )
)  <  0 ) )
2120imbi1d 308 . . . . . 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 2585 . . . . 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 4027 . . . . . . 7  |-  ( c  =  i  ->  (
( # `  ( S `
 a ) )  <  c  <->  ( # `  ( S `  a )
)  <  i )
)
2423imbi1d 308 . . . . . 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 2585 . . . . 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 4027 . . . . . . 7  |-  ( c  =  ( i  +  1 )  ->  (
( # `  ( S `
 a ) )  <  c  <->  ( # `  ( S `  a )
)  <  ( i  +  1 ) ) )
2726imbi1d 308 . . . . . 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 2585 . . . . 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 4027 . . . . . . 7  |-  ( c  =  ( ( # `  ( S `  A
) )  +  1 )  ->  ( ( # `
 ( S `  a ) )  < 
c  <->  ( # `  ( S `  a )
)  <  ( ( # `
 ( S `  A ) )  +  1 ) ) )
3029imbi1d 308 . . . . . 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 2585 . . . . 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 5664 . . . . . . . . . . 11  |-  ( a  e.  dom  S  -> 
( S `  a
)  e.  W )
333, 32sseldi 3178 . . . . . . . . . 10  |-  ( a  e.  dom  S  -> 
( S `  a
)  e. Word  ( I  X.  2o ) )
34 lencl 11421 . . . . . . . . . 10  |-  ( ( S `  a )  e. Word  ( I  X.  2o )  ->  ( # `  ( S `  a
) )  e.  NN0 )
3533, 34syl 15 . . . . . . . . 9  |-  ( a  e.  dom  S  -> 
( # `  ( S `
 a ) )  e.  NN0 )
36 nn0nlt0 9992 . . . . . . . . 9  |-  ( (
# `  ( S `  a ) )  e. 
NN0  ->  -.  ( # `  ( S `  a )
)  <  0 )
3735, 36syl 15 . . . . . . . 8  |-  ( a  e.  dom  S  ->  -.  ( # `  ( S `  a )
)  <  0 )
3837pm2.21d 98 . . . . . . 7  |-  ( a  e.  dom  S  -> 
( ( # `  ( S `  a )
)  <  0  ->  ( ( S `  a
)  =  ( S `
 b )  -> 
( a `  0
)  =  ( b `
 0 ) ) ) )
3938adantr 451 . . . . . 6  |-  ( ( a  e.  dom  S  /\  b  e.  dom  S )  ->  ( ( # `
 ( S `  a ) )  <  0  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) ) )
4039rgen2a 2609 . . . . 5  |-  A. a  e.  dom  S A. b  e.  dom  S ( (
# `  ( S `  a ) )  <  0  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) )
41 simpl1 958 . . . . . . . . . . . . . 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 1010 . . . . . . . . . . . . . . 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 4027 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  ( S `  c ) )  =  i  ->  ( ( # `
 ( S `  a ) )  < 
( # `  ( S `
 c ) )  <-> 
( # `  ( S `
 a ) )  <  i ) )
4443imbi1d 308 . . . . . . . . . . . . . . . 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 2585 . . . . . . . . . . . . . . 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 15 . . . . . . . . . . . . . 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 223 . . . . . . . . . . . . 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 1008 . . . . . . . . . . . . 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 1009 . . . . . . . . . . . . 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 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
) )  ->  ( S `  c )  =  ( S `  d ) )
51 simpr 447 . . . . . . . . . . . . 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 15056 . . . . . . . . . . . 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 413 . . . . . . . . . . . 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 200 . . . . . . . . . . 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 1153 . . . . . . . . . 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 425 . . . . . . . . 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 2635 . . . . . . . 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 5525 . . . . . . . . . . . 12  |-  ( c  =  a  ->  ( S `  c )  =  ( S `  a ) )
5958fveq2d 5529 . . . . . . . . . . 11  |-  ( c  =  a  ->  ( # `
 ( S `  c ) )  =  ( # `  ( S `  a )
) )
6059eqeq1d 2291 . . . . . . . . . 10  |-  ( c  =  a  ->  (
( # `  ( S `
 c ) )  =  i  <->  ( # `  ( S `  a )
)  =  i ) )
6158eqeq1d 2291 . . . . . . . . . . 11  |-  ( c  =  a  ->  (
( S `  c
)  =  ( S `
 d )  <->  ( S `  a )  =  ( S `  d ) ) )
62 fveq1 5524 . . . . . . . . . . . 12  |-  ( c  =  a  ->  (
c `  0 )  =  ( a ` 
0 ) )
6362eqeq1d 2291 . . . . . . . . . . 11  |-  ( c  =  a  ->  (
( c `  0
)  =  ( d `
 0 )  <->  ( a `  0 )  =  ( d `  0
) ) )
6461, 63imbi12d 311 . . . . . . . . . 10  |-  ( c  =  a  ->  (
( ( S `  c )  =  ( S `  d )  ->  ( c ` 
0 )  =  ( d `  0 ) )  <->  ( ( S `
 a )  =  ( S `  d
)  ->  ( a `  0 )  =  ( d `  0
) ) ) )
6560, 64imbi12d 311 . . . . . . . . 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 5525 . . . . . . . . . . . 12  |-  ( d  =  b  ->  ( S `  d )  =  ( S `  b ) )
6766eqeq2d 2294 . . . . . . . . . . 11  |-  ( d  =  b  ->  (
( S `  a
)  =  ( S `
 d )  <->  ( S `  a )  =  ( S `  b ) ) )
68 fveq1 5524 . . . . . . . . . . . 12  |-  ( d  =  b  ->  (
d `  0 )  =  ( b ` 
0 ) )
6968eqeq2d 2294 . . . . . . . . . . 11  |-  ( d  =  b  ->  (
( a `  0
)  =  ( d `
 0 )  <->  ( a `  0 )  =  ( b `  0
) ) )
7067, 69imbi12d 311 . . . . . . . . . 10  |-  ( d  =  b  ->  (
( ( S `  a )  =  ( S `  d )  ->  ( a ` 
0 )  =  ( d `  0 ) )  <->  ( ( S `
 a )  =  ( S `  b
)  ->  ( a `  0 )  =  ( b `  0
) ) ) )
7170imbi2d 307 . . . . . . . . 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 2772 . . . . . . . 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 188 . . . . . . 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 534 . . . . . 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 451 . . . . . . . . . . 11  |-  ( ( a  e.  dom  S  /\  b  e.  dom  S )  ->  ( # `  ( S `  a )
)  e.  NN0 )
76 nn0leltp1 10075 . . . . . . . . . . . . 13  |-  ( ( ( # `  ( S `  a )
)  e.  NN0  /\  i  e.  NN0 )  -> 
( ( # `  ( S `  a )
)  <_  i  <->  ( # `  ( S `  a )
)  <  ( i  +  1 ) ) )
77 nn0re 9974 . . . . . . . . . . . . . 14  |-  ( (
# `  ( S `  a ) )  e. 
NN0  ->  ( # `  ( S `  a )
)  e.  RR )
78 nn0re 9974 . . . . . . . . . . . . . 14  |-  ( i  e.  NN0  ->  i  e.  RR )
79 leloe 8908 . . . . . . . . . . . . . 14  |-  ( ( ( # `  ( S `  a )
)  e.  RR  /\  i  e.  RR )  ->  ( ( # `  ( S `  a )
)  <_  i  <->  ( ( # `
 ( S `  a ) )  < 
i  \/  ( # `  ( S `  a
) )  =  i ) ) )
8077, 78, 79syl2an 463 . . . . . . . . . . . . 13  |-  ( ( ( # `  ( S `  a )
)  e.  NN0  /\  i  e.  NN0 )  -> 
( ( # `  ( S `  a )
)  <_  i  <->  ( ( # `
 ( S `  a ) )  < 
i  \/  ( # `  ( S `  a
) )  =  i ) ) )
8176, 80bitr3d 246 . . . . . . . . . . . 12  |-  ( ( ( # `  ( S `  a )
)  e.  NN0  /\  i  e.  NN0 )  -> 
( ( # `  ( S `  a )
)  <  ( i  +  1 )  <->  ( ( # `
 ( S `  a ) )  < 
i  \/  ( # `  ( S `  a
) )  =  i ) ) )
8281ancoms 439 . . . . . . . . . . 11  |-  ( ( i  e.  NN0  /\  ( # `  ( S `
 a ) )  e.  NN0 )  -> 
( ( # `  ( S `  a )
)  <  ( i  +  1 )  <->  ( ( # `
 ( S `  a ) )  < 
i  \/  ( # `  ( S `  a
) )  =  i ) ) )
8375, 82sylan2 460 . . . . . . . . . 10  |-  ( ( i  e.  NN0  /\  ( a  e.  dom  S  /\  b  e.  dom  S ) )  ->  (
( # `  ( S `
 a ) )  <  ( i  +  1 )  <->  ( ( # `
 ( S `  a ) )  < 
i  \/  ( # `  ( S `  a
) )  =  i ) ) )
8483imbi1d 308 . . . . . . . . 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 758 . . . . . . . . 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 252 . . . . . . . 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 2583 . . . . . . 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 2676 . . . . . . 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 252 . . . . . 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 212 . . . . 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 10108 . . . 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 15 . . 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 10019 . . . 4  |-  ( ( A  e.  dom  S  /\  B  e.  dom  S )  ->  ( # `  ( S `  A )
)  e.  RR )
9493ltp1d 9687 . . 3  |-  ( ( A  e.  dom  S  /\  B  e.  dom  S )  ->  ( # `  ( S `  A )
)  <  ( ( # `
 ( S `  A ) )  +  1 ) )
95 fveq2 5525 . . . . . . 7  |-  ( a  =  A  ->  ( S `  a )  =  ( S `  A ) )
9695fveq2d 5529 . . . . . 6  |-  ( a  =  A  ->  ( # `
 ( S `  a ) )  =  ( # `  ( S `  A )
) )
9796breq1d 4033 . . . . 5  |-  ( a  =  A  ->  (
( # `  ( S `
 a ) )  <  ( ( # `  ( S `  A
) )  +  1 )  <->  ( # `  ( S `  A )
)  <  ( ( # `
 ( S `  A ) )  +  1 ) ) )
9895eqeq1d 2291 . . . . . 6  |-  ( a  =  A  ->  (
( S `  a
)  =  ( S `
 b )  <->  ( S `  A )  =  ( S `  b ) ) )
99 fveq1 5524 . . . . . . 7  |-  ( a  =  A  ->  (
a `  0 )  =  ( A ` 
0 ) )
10099eqeq1d 2291 . . . . . 6  |-  ( a  =  A  ->  (
( a `  0
)  =  ( b `
 0 )  <->  ( A `  0 )  =  ( b `  0
) ) )
10198, 100imbi12d 311 . . . . 5  |-  ( a  =  A  ->  (
( ( S `  a )  =  ( S `  b )  ->  ( a ` 
0 )  =  ( b `  0 ) )  <->  ( ( S `
 A )  =  ( S `  b
)  ->  ( A `  0 )  =  ( b `  0
) ) ) )
10297, 101imbi12d 311 . . . 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 5525 . . . . . . 7  |-  ( b  =  B  ->  ( S `  b )  =  ( S `  B ) )
104103eqeq2d 2294 . . . . . 6  |-  ( b  =  B  ->  (
( S `  A
)  =  ( S `
 b )  <->  ( S `  A )  =  ( S `  B ) ) )
105 fveq1 5524 . . . . . . 7  |-  ( b  =  B  ->  (
b `  0 )  =  ( B ` 
0 ) )
106105eqeq2d 2294 . . . . . 6  |-  ( b  =  B  ->  (
( A `  0
)  =  ( b `
 0 )  <->  ( A `  0 )  =  ( B `  0
) ) )
107104, 106imbi12d 311 . . . . 5  |-  ( b  =  B  ->  (
( ( S `  A )  =  ( S `  b )  ->  ( A ` 
0 )  =  ( b `  0 ) )  <->  ( ( S `
 A )  =  ( S `  B
)  ->  ( A `  0 )  =  ( B `  0
) ) ) )
108107imbi2d 307 . . . 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 2890 . . 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 41 . 2  |-  ( ( A  e.  dom  S  /\  B  e.  dom  S )  ->  ( ( S `  A )  =  ( S `  B )  ->  ( A `  0 )  =  ( B ` 
0 ) ) )
1111103impia 1148 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 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547    \ cdif 3149   (/)c0 3455   {csn 3640   <.cop 3643   <.cotp 3644   U_ciun 3905   class class class wbr 4023    e. cmpt 4077    _I cid 4304    X. cxp 4687   dom cdm 4689   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1oc1o 6472   2oc2o 6473   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    < clt 8867    <_ cle 8868    - cmin 9037   NN0cn0 9965   ...cfz 10782  ..^cfzo 10870   #chash 11337  Word cword 11403   splice csplice 11407   <"cs2 11491   ~FG cefg 15015
This theorem is referenced by:  efgrelexlemb  15059
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-ot 3650  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-hash 11338  df-word 11409  df-concat 11410  df-s1 11411  df-substr 11412  df-splice 11413  df-s2 11498
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