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Theorem efgred 16245
Description: The reduced word that forms the base of the sequence in efgsval 16228 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 5749 . . . . . . . 8  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
31, 2eqsstri 3386 . . . . . . 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 16226 . . . . . . . . . 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 5564 . . . . . . . . . . 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 5552 . . . . . . . . . 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 5842 . . . . . . . 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 3354 . . . . . 6  |-  ( ( A  e.  dom  S  /\  B  e.  dom  S )  ->  ( S `  A )  e. Word  (
I  X.  2o ) )
16 lencl 12249 . . . . . 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 10620 . . . . 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 4296 . . . . . . 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 2757 . . . . 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 4296 . . . . . . 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 2757 . . . . 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 4296 . . . . . . 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 2757 . . . . 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 4296 . . . . . . 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 2757 . . . . 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 5842 . . . . . . . . . . 11  |-  ( a  e.  dom  S  -> 
( S `  a
)  e.  W )
333, 32sseldi 3354 . . . . . . . . . 10  |-  ( a  e.  dom  S  -> 
( S `  a
)  e. Word  ( I  X.  2o ) )
34 lencl 12249 . . . . . . . . . 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 10606 . . . . . . . . 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 2782 . . . . 5  |-  A. a  e.  dom  S A. b  e.  dom  S ( (
# `  ( S `  a ) )  <  0  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) )
41 simpl1 991 . . . . . . . . . . . . . 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 1043 . . . . . . . . . . . . . . 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 4296 . . . . . . . . . . . . . . . . 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 2757 . . . . . . . . . . . . . . 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 1041 . . . . . . . . . . . . 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 1042 . . . . . . . . . . . . 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 1044 . . . . . . . . . . . . 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 16244 . . . . . . . . . . . 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 1189 . . . . . . . . . 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 2808 . . . . . . . 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 5691 . . . . . . . . . . . 12  |-  ( c  =  a  ->  ( S `  c )  =  ( S `  a ) )
5958fveq2d 5695 . . . . . . . . . . 11  |-  ( c  =  a  ->  ( # `
 ( S `  c ) )  =  ( # `  ( S `  a )
) )
6059eqeq1d 2451 . . . . . . . . . 10  |-  ( c  =  a  ->  (
( # `  ( S `
 c ) )  =  i  <->  ( # `  ( S `  a )
)  =  i ) )
6158eqeq1d 2451 . . . . . . . . . . 11  |-  ( c  =  a  ->  (
( S `  c
)  =  ( S `
 d )  <->  ( S `  a )  =  ( S `  d ) ) )
62 fveq1 5690 . . . . . . . . . . . 12  |-  ( c  =  a  ->  (
c `  0 )  =  ( a ` 
0 ) )
6362eqeq1d 2451 . . . . . . . . . . 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 5691 . . . . . . . . . . . 12  |-  ( d  =  b  ->  ( S `  d )  =  ( S `  b ) )
6766eqeq2d 2454 . . . . . . . . . . 11  |-  ( d  =  b  ->  (
( S `  a
)  =  ( S `
 d )  <->  ( S `  a )  =  ( S `  b ) ) )
68 fveq1 5690 . . . . . . . . . . . 12  |-  ( d  =  b  ->  (
d `  0 )  =  ( b ` 
0 ) )
6968eqeq2d 2454 . . . . . . . . . . 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 2955 . . . . . . . 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 10703 . . . . . . . . . . . . 13  |-  ( ( ( # `  ( S `  a )
)  e.  NN0  /\  i  e.  NN0 )  -> 
( ( # `  ( S `  a )
)  <_  i  <->  ( # `  ( S `  a )
)  <  ( i  +  1 ) ) )
77 nn0re 10588 . . . . . . . . . . . . . 14  |-  ( (
# `  ( S `  a ) )  e. 
NN0  ->  ( # `  ( S `  a )
)  e.  RR )
78 nn0re 10588 . . . . . . . . . . . . . 14  |-  ( i  e.  NN0  ->  i  e.  RR )
79 leloe 9461 . . . . . . . . . . . . . 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 2755 . . . . . . 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 2850 . . . . . . 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 10738 . . . 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 10637 . . . 4  |-  ( ( A  e.  dom  S  /\  B  e.  dom  S )  ->  ( # `  ( S `  A )
)  e.  RR )
9493ltp1d 10263 . . 3  |-  ( ( A  e.  dom  S  /\  B  e.  dom  S )  ->  ( # `  ( S `  A )
)  <  ( ( # `
 ( S `  A ) )  +  1 ) )
95 fveq2 5691 . . . . . . 7  |-  ( a  =  A  ->  ( S `  a )  =  ( S `  A ) )
9695fveq2d 5695 . . . . . 6  |-  ( a  =  A  ->  ( # `
 ( S `  a ) )  =  ( # `  ( S `  A )
) )
9796breq1d 4302 . . . . 5  |-  ( a  =  A  ->  (
( # `  ( S `
 a ) )  <  ( ( # `  ( S `  A
) )  +  1 )  <->  ( # `  ( S `  A )
)  <  ( ( # `
 ( S `  A ) )  +  1 ) ) )
9895eqeq1d 2451 . . . . . 6  |-  ( a  =  A  ->  (
( S `  a
)  =  ( S `
 b )  <->  ( S `  A )  =  ( S `  b ) ) )
99 fveq1 5690 . . . . . . 7  |-  ( a  =  A  ->  (
a `  0 )  =  ( A ` 
0 ) )
10099eqeq1d 2451 . . . . . 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 5691 . . . . . . 7  |-  ( b  =  B  ->  ( S `  b )  =  ( S `  B ) )
104103eqeq2d 2454 . . . . . 6  |-  ( b  =  B  ->  (
( S `  A
)  =  ( S `
 b )  <->  ( S `  A )  =  ( S `  B ) ) )
105 fveq1 5690 . . . . . . 7  |-  ( b  =  B  ->  (
b `  0 )  =  ( B ` 
0 ) )
106105eqeq2d 2454 . . . . . 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 3079 . . 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 1184 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 965    = wceq 1369    e. wcel 1756   A.wral 2715   {crab 2719    \ cdif 3325   (/)c0 3637   {csn 3877   <.cop 3883   <.cotp 3885   U_ciun 4171   class class class wbr 4292    e. cmpt 4350    _I cid 4631    X. cxp 4838   dom cdm 4840   ran crn 4841   -->wf 5414   ` cfv 5418  (class class class)co 6091    e. cmpt2 6093   1oc1o 6913   2oc2o 6914   RRcr 9281   0cc0 9282   1c1 9283    + caddc 9285    < clt 9418    <_ cle 9419    - cmin 9595   NN0cn0 10579   ...cfz 11437  ..^cfzo 11548   #chash 12103  Word cword 12221   splice csplice 12226   <"cs2 12468   ~FG cefg 16203
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-ot 3886  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-fz 11438  df-fzo 11549  df-hash 12104  df-word 12229  df-concat 12231  df-s1 12232  df-substr 12233  df-splice 12234  df-s2 12475
This theorem is referenced by:  efgrelexlemb  16247
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