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Theorem efgred2 15390
Description: Two extension sequences have related endpoints iff they have the same base. (Contributed by Mario Carneiro, 1-Oct-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
efgred2  |-  ( ( 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 efgred2
Dummy variables  d 
i are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 efgval.w . . . . . . . 8  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
2 efgval.r . . . . . . . 8  |-  .~  =  ( ~FG  `  I )
3 efgval2.m . . . . . . . 8  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
4 efgval2.t . . . . . . . 8  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
5 efgred.d . . . . . . . 8  |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `  x ) )
6 efgred.s . . . . . . . 8  |-  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 ) ) )
71, 2, 3, 4, 5, 6efgsfo 15376 . . . . . . 7  |-  S : dom  S -onto-> W
8 fof 5656 . . . . . . 7  |-  ( S : dom  S -onto-> W  ->  S : dom  S --> W )
97, 8ax-mp 5 . . . . . 6  |-  S : dom  S --> W
109ffvelrni 5872 . . . . 5  |-  ( B  e.  dom  S  -> 
( S `  B
)  e.  W )
1110ad2antlr 709 . . . 4  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( S `
 A )  .~  ( S `  B ) )  ->  ( S `  B )  e.  W
)
121, 2, 3, 4, 5, 6efgredeu 15389 . . . 4  |-  ( ( S `  B )  e.  W  ->  E! d  e.  D  d  .~  ( S `  B
) )
13 reurmo 2925 . . . 4  |-  ( E! d  e.  D  d  .~  ( S `  B )  ->  E* d  e.  D d  .~  ( S `  B
) )
1411, 12, 133syl 19 . . 3  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( S `
 A )  .~  ( S `  B ) )  ->  E* d  e.  D d  .~  ( S `  B )
)
151, 2, 3, 4, 5, 6efgsdm 15367 . . . . 5  |-  ( A  e.  dom  S  <->  ( A  e.  (Word  W  \  { (/)
} )  /\  ( A `  0 )  e.  D  /\  A. i  e.  ( 1..^ ( # `  A ) ) ( A `  i )  e.  ran  ( T `
 ( A `  ( i  -  1 ) ) ) ) )
1615simp2bi 974 . . . 4  |-  ( A  e.  dom  S  -> 
( A `  0
)  e.  D )
1716ad2antrr 708 . . 3  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( S `
 A )  .~  ( S `  B ) )  ->  ( A `  0 )  e.  D )
181, 2efger 15355 . . . . 5  |-  .~  Er  W
1918a1i 11 . . . 4  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( S `
 A )  .~  ( S `  B ) )  ->  .~  Er  W
)
201, 2, 3, 4, 5, 6efgsrel 15371 . . . . 5  |-  ( A  e.  dom  S  -> 
( A `  0
)  .~  ( S `  A ) )
2120ad2antrr 708 . . . 4  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( S `
 A )  .~  ( S `  B ) )  ->  ( A `  0 )  .~  ( S `  A ) )
22 simpr 449 . . . 4  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( S `
 A )  .~  ( S `  B ) )  ->  ( S `  A )  .~  ( S `  B )
)
2319, 21, 22ertrd 6924 . . 3  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( S `
 A )  .~  ( S `  B ) )  ->  ( A `  0 )  .~  ( S `  B ) )
241, 2, 3, 4, 5, 6efgsdm 15367 . . . . 5  |-  ( B  e.  dom  S  <->  ( B  e.  (Word  W  \  { (/)
} )  /\  ( B `  0 )  e.  D  /\  A. i  e.  ( 1..^ ( # `  B ) ) ( B `  i )  e.  ran  ( T `
 ( B `  ( i  -  1 ) ) ) ) )
2524simp2bi 974 . . . 4  |-  ( B  e.  dom  S  -> 
( B `  0
)  e.  D )
2625ad2antlr 709 . . 3  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( S `
 A )  .~  ( S `  B ) )  ->  ( B `  0 )  e.  D )
271, 2, 3, 4, 5, 6efgsrel 15371 . . . 4  |-  ( B  e.  dom  S  -> 
( B `  0
)  .~  ( S `  B ) )
2827ad2antlr 709 . . 3  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( S `
 A )  .~  ( S `  B ) )  ->  ( B `  0 )  .~  ( S `  B ) )
29 breq1 4218 . . . 4  |-  ( d  =  ( A ` 
0 )  ->  (
d  .~  ( S `  B )  <->  ( A `  0 )  .~  ( S `  B ) ) )
30 breq1 4218 . . . 4  |-  ( d  =  ( B ` 
0 )  ->  (
d  .~  ( S `  B )  <->  ( B `  0 )  .~  ( S `  B ) ) )
3129, 30rmoi 3252 . . 3  |-  ( ( E* d  e.  D
d  .~  ( S `  B )  /\  (
( A `  0
)  e.  D  /\  ( A `  0 )  .~  ( S `  B ) )  /\  ( ( B ` 
0 )  e.  D  /\  ( B `  0
)  .~  ( S `  B ) ) )  ->  ( A ` 
0 )  =  ( B `  0 ) )
3214, 17, 23, 26, 28, 31syl122anc 1194 . 2  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( S `
 A )  .~  ( S `  B ) )  ->  ( A `  0 )  =  ( B `  0
) )
3318a1i 11 . . 3  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( A `
 0 )  =  ( B `  0
) )  ->  .~  Er  W )
3420ad2antrr 708 . . 3  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( A `
 0 )  =  ( B `  0
) )  ->  ( A `  0 )  .~  ( S `  A
) )
35 simpr 449 . . . 4  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( A `
 0 )  =  ( B `  0
) )  ->  ( A `  0 )  =  ( B ` 
0 ) )
3627ad2antlr 709 . . . 4  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( A `
 0 )  =  ( B `  0
) )  ->  ( B `  0 )  .~  ( S `  B
) )
3735, 36eqbrtrd 4235 . . 3  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( A `
 0 )  =  ( B `  0
) )  ->  ( A `  0 )  .~  ( S `  B
) )
3833, 34, 37ertr3d 6926 . 2  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( A `
 0 )  =  ( B `  0
) )  ->  ( S `  A )  .~  ( S `  B
) )
3932, 38impbida 807 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:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   E!wreu 2709   E*wrmo 2710   {crab 2711    \ cdif 3319   (/)c0 3630   {csn 3816   <.cop 3819   <.cotp 3820   U_ciun 4095   class class class wbr 4215    e. cmpt 4269    _I cid 4496    X. cxp 4879   dom cdm 4881   ran crn 4882   -->wf 5453   -onto->wfo 5455   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086   1oc1o 6720   2oc2o 6721    Er wer 6905   0cc0 8995   1c1 8996    - cmin 9296   ...cfz 11048  ..^cfzo 11140   #chash 11623  Word cword 11722   splice csplice 11726   <"cs2 11810   ~FG cefg 15343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-ot 3826  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-ec 6910  df-map 7023  df-pm 7024  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-fz 11049  df-fzo 11141  df-hash 11624  df-word 11728  df-concat 11729  df-s1 11730  df-substr 11731  df-splice 11732  df-s2 11817  df-efg 15346
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