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Theorem efgi 16329
Description: Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
Hypotheses
Ref Expression
efgval.w  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
efgval.r  |-  .~  =  ( ~FG  `  I )
Assertion
Ref Expression
efgi  |-  ( ( ( A  e.  W  /\  N  e.  (
0 ... ( # `  A
) ) )  /\  ( J  e.  I  /\  K  e.  2o ) )  ->  A  .~  ( A splice  <. N ,  N ,  <" <. J ,  K >. <. J , 
( 1o  \  K
) >. "> >. )
)

Proof of Theorem efgi
Dummy variables  a 
b  i  r  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5792 . . . . . . . . . . 11  |-  ( u  =  A  ->  ( # `
 u )  =  ( # `  A
) )
21oveq2d 6209 . . . . . . . . . 10  |-  ( u  =  A  ->  (
0 ... ( # `  u
) )  =  ( 0 ... ( # `  A ) ) )
3 id 22 . . . . . . . . . . . 12  |-  ( u  =  A  ->  u  =  A )
4 oveq1 6200 . . . . . . . . . . . 12  |-  ( u  =  A  ->  (
u splice  <. i ,  i ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  =  ( A splice  <. i ,  i , 
<" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> >. )
)
53, 4breq12d 4406 . . . . . . . . . . 11  |-  ( u  =  A  ->  (
u r ( u splice  <. i ,  i , 
<" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> >. )  <->  A r ( A splice  <. i ,  i ,  <"
<. a ,  b >. <. a ,  ( 1o 
\  b ) >. "> >. ) ) )
652ralbidv 2871 . . . . . . . . . 10  |-  ( u  =  A  ->  ( A. a  e.  I  A. b  e.  2o  u r ( u splice  <. i ,  i , 
<" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> >. )  <->  A. a  e.  I  A. b  e.  2o  A
r ( A splice  <. i ,  i ,  <"
<. a ,  b >. <. a ,  ( 1o 
\  b ) >. "> >. ) ) )
72, 6raleqbidv 3030 . . . . . . . . 9  |-  ( u  =  A  ->  ( A. i  e.  (
0 ... ( # `  u
) ) A. a  e.  I  A. b  e.  2o  u r ( u splice  <. i ,  i ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  <->  A. i  e.  ( 0 ... ( # `  A ) ) A. a  e.  I  A. b  e.  2o  A
r ( A splice  <. i ,  i ,  <"
<. a ,  b >. <. a ,  ( 1o 
\  b ) >. "> >. ) ) )
87rspcv 3168 . . . . . . . 8  |-  ( A  e.  W  ->  ( A. u  e.  W  A. i  e.  (
0 ... ( # `  u
) ) A. a  e.  I  A. b  e.  2o  u r ( u splice  <. i ,  i ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  ->  A. i  e.  ( 0 ... ( # `
 A ) ) A. a  e.  I  A. b  e.  2o  A r ( A splice  <. i ,  i , 
<" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> >. )
) )
9 oteq1 4169 . . . . . . . . . . . . 13  |-  ( i  =  N  ->  <. i ,  i ,  <"
<. a ,  b >. <. a ,  ( 1o 
\  b ) >. "> >.  =  <. N , 
i ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)
10 oteq2 4170 . . . . . . . . . . . . 13  |-  ( i  =  N  ->  <. N , 
i ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.  =  <. N ,  N ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)
119, 10eqtrd 2492 . . . . . . . . . . . 12  |-  ( i  =  N  ->  <. i ,  i ,  <"
<. a ,  b >. <. a ,  ( 1o 
\  b ) >. "> >.  =  <. N ,  N ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)
1211oveq2d 6209 . . . . . . . . . . 11  |-  ( i  =  N  ->  ( A splice  <. i ,  i ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  =  ( A splice  <. N ,  N ,  <" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> >. )
)
1312breq2d 4405 . . . . . . . . . 10  |-  ( i  =  N  ->  ( A r ( A splice  <. i ,  i , 
<" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> >. )  <->  A r ( A splice  <. N ,  N ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) )
14132ralbidv 2871 . . . . . . . . 9  |-  ( i  =  N  ->  ( A. a  e.  I  A. b  e.  2o  A r ( A splice  <. i ,  i , 
<" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> >. )  <->  A. a  e.  I  A. b  e.  2o  A
r ( A splice  <. N ,  N ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) )
1514rspcv 3168 . . . . . . . 8  |-  ( N  e.  ( 0 ... ( # `  A
) )  ->  ( A. i  e.  (
0 ... ( # `  A
) ) A. a  e.  I  A. b  e.  2o  A r ( A splice  <. i ,  i ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  ->  A. a  e.  I  A. b  e.  2o  A r ( A splice  <. N ,  N ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) )
168, 15sylan9 657 . . . . . . 7  |-  ( ( A  e.  W  /\  N  e.  ( 0 ... ( # `  A
) ) )  -> 
( A. u  e.  W  A. i  e.  ( 0 ... ( # `
 u ) ) A. a  e.  I  A. b  e.  2o  u r ( u splice  <. i ,  i , 
<" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> >. )  ->  A. a  e.  I  A. b  e.  2o  A r ( A splice  <. N ,  N ,  <" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> >. )
) )
17 opeq1 4160 . . . . . . . . . . . 12  |-  ( a  =  J  ->  <. a ,  b >.  =  <. J ,  b >. )
18 opeq1 4160 . . . . . . . . . . . 12  |-  ( a  =  J  ->  <. a ,  ( 1o  \ 
b ) >.  =  <. J ,  ( 1o  \ 
b ) >. )
1917, 18s2eqd 12600 . . . . . . . . . . 11  |-  ( a  =  J  ->  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. ">  =  <" <. J , 
b >. <. J ,  ( 1o  \  b )
>. "> )
2019oteq3d 4174 . . . . . . . . . 10  |-  ( a  =  J  ->  <. N ,  N ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.  =  <. N ,  N ,  <" <. J , 
b >. <. J ,  ( 1o  \  b )
>. "> >. )
2120oveq2d 6209 . . . . . . . . 9  |-  ( a  =  J  ->  ( A splice  <. N ,  N ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  =  ( A splice  <. N ,  N ,  <" <. J ,  b
>. <. J ,  ( 1o  \  b )
>. "> >. )
)
2221breq2d 4405 . . . . . . . 8  |-  ( a  =  J  ->  ( A r ( A splice  <. N ,  N ,  <" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> >. )  <->  A r ( A splice  <. N ,  N ,  <" <. J ,  b >. <. J , 
( 1o  \  b
) >. "> >. )
) )
23 opeq2 4161 . . . . . . . . . . . . 13  |-  ( b  =  K  ->  <. J , 
b >.  =  <. J ,  K >. )
24 difeq2 3569 . . . . . . . . . . . . . 14  |-  ( b  =  K  ->  ( 1o  \  b )  =  ( 1o  \  K
) )
2524opeq2d 4167 . . . . . . . . . . . . 13  |-  ( b  =  K  ->  <. J , 
( 1o  \  b
) >.  =  <. J , 
( 1o  \  K
) >. )
2623, 25s2eqd 12600 . . . . . . . . . . . 12  |-  ( b  =  K  ->  <" <. J ,  b >. <. J , 
( 1o  \  b
) >. ">  =  <" <. J ,  K >. <. J ,  ( 1o  \  K )
>. "> )
2726oteq3d 4174 . . . . . . . . . . 11  |-  ( b  =  K  ->  <. N ,  N ,  <" <. J ,  b >. <. J , 
( 1o  \  b
) >. "> >.  =  <. N ,  N ,  <"
<. J ,  K >. <. J ,  ( 1o  \  K ) >. "> >.
)
2827oveq2d 6209 . . . . . . . . . 10  |-  ( b  =  K  ->  ( A splice  <. N ,  N ,  <" <. J , 
b >. <. J ,  ( 1o  \  b )
>. "> >. )  =  ( A splice  <. N ,  N ,  <" <. J ,  K >. <. J , 
( 1o  \  K
) >. "> >. )
)
2928breq2d 4405 . . . . . . . . 9  |-  ( b  =  K  ->  ( A r ( A splice  <. N ,  N ,  <" <. J ,  b
>. <. J ,  ( 1o  \  b )
>. "> >. )  <->  A r ( A splice  <. N ,  N ,  <" <. J ,  K >. <. J , 
( 1o  \  K
) >. "> >. )
) )
30 df-br 4394 . . . . . . . . 9  |-  ( A r ( A splice  <. N ,  N ,  <" <. J ,  K >. <. J , 
( 1o  \  K
) >. "> >. )  <->  <. A ,  ( A splice  <. N ,  N ,  <" <. J ,  K >. <. J ,  ( 1o  \  K )
>. "> >. ) >.  e.  r )
3129, 30syl6bb 261 . . . . . . . 8  |-  ( b  =  K  ->  ( A r ( A splice  <. N ,  N ,  <" <. J ,  b
>. <. J ,  ( 1o  \  b )
>. "> >. )  <->  <. A ,  ( A splice  <. N ,  N ,  <" <. J ,  K >. <. J ,  ( 1o  \  K )
>. "> >. ) >.  e.  r ) )
3222, 31rspc2v 3179 . . . . . . 7  |-  ( ( J  e.  I  /\  K  e.  2o )  ->  ( A. a  e.  I  A. b  e.  2o  A r ( A splice  <. N ,  N ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  ->  <. A , 
( A splice  <. N ,  N ,  <" <. J ,  K >. <. J , 
( 1o  \  K
) >. "> >. ) >.  e.  r ) )
3316, 32sylan9 657 . . . . . 6  |-  ( ( ( A  e.  W  /\  N  e.  (
0 ... ( # `  A
) ) )  /\  ( J  e.  I  /\  K  e.  2o ) )  ->  ( A. u  e.  W  A. i  e.  (
0 ... ( # `  u
) ) A. a  e.  I  A. b  e.  2o  u r ( u splice  <. i ,  i ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
)  ->  <. A , 
( A splice  <. N ,  N ,  <" <. J ,  K >. <. J , 
( 1o  \  K
) >. "> >. ) >.  e.  r ) )
3433adantld 467 . . . . 5  |-  ( ( ( A  e.  W  /\  N  e.  (
0 ... ( # `  A
) ) )  /\  ( J  e.  I  /\  K  e.  2o ) )  ->  (
( r  Er  W  /\  A. u  e.  W  A. i  e.  (
0 ... ( # `  u
) ) A. a  e.  I  A. b  e.  2o  u r ( u splice  <. i ,  i ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) )  ->  <. A , 
( A splice  <. N ,  N ,  <" <. J ,  K >. <. J , 
( 1o  \  K
) >. "> >. ) >.  e.  r ) )
3534alrimiv 1686 . . . 4  |-  ( ( ( A  e.  W  /\  N  e.  (
0 ... ( # `  A
) ) )  /\  ( J  e.  I  /\  K  e.  2o ) )  ->  A. r
( ( r  Er  W  /\  A. u  e.  W  A. i  e.  ( 0 ... ( # `
 u ) ) A. a  e.  I  A. b  e.  2o  u r ( u splice  <. i ,  i , 
<" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> >. )
)  ->  <. A , 
( A splice  <. N ,  N ,  <" <. J ,  K >. <. J , 
( 1o  \  K
) >. "> >. ) >.  e.  r ) )
36 opex 4657 . . . . 5  |-  <. A , 
( A splice  <. N ,  N ,  <" <. J ,  K >. <. J , 
( 1o  \  K
) >. "> >. ) >.  e.  _V
3736elintab 4240 . . . 4  |-  ( <. A ,  ( A splice  <. N ,  N ,  <" <. J ,  K >. <. J ,  ( 1o  \  K )
>. "> >. ) >.  e.  |^| { r  |  ( r  Er  W  /\  A. u  e.  W  A. i  e.  (
0 ... ( # `  u
) ) A. a  e.  I  A. b  e.  2o  u r ( u splice  <. i ,  i ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) }  <->  A. r
( ( r  Er  W  /\  A. u  e.  W  A. i  e.  ( 0 ... ( # `
 u ) ) A. a  e.  I  A. b  e.  2o  u r ( u splice  <. i ,  i , 
<" <. a ,  b
>. <. a ,  ( 1o  \  b )
>. "> >. )
)  ->  <. A , 
( A splice  <. N ,  N ,  <" <. J ,  K >. <. J , 
( 1o  \  K
) >. "> >. ) >.  e.  r ) )
3835, 37sylibr 212 . . 3  |-  ( ( ( A  e.  W  /\  N  e.  (
0 ... ( # `  A
) ) )  /\  ( J  e.  I  /\  K  e.  2o ) )  ->  <. A , 
( A splice  <. N ,  N ,  <" <. J ,  K >. <. J , 
( 1o  \  K
) >. "> >. ) >.  e.  |^| { r  |  ( r  Er  W  /\  A. u  e.  W  A. i  e.  (
0 ... ( # `  u
) ) A. a  e.  I  A. b  e.  2o  u r ( u splice  <. i ,  i ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) } )
39 efgval.w . . . 4  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
40 efgval.r . . . 4  |-  .~  =  ( ~FG  `  I )
4139, 40efgval 16327 . . 3  |-  .~  =  |^| { r  |  ( r  Er  W  /\  A. u  e.  W  A. i  e.  ( 0 ... ( # `  u
) ) A. a  e.  I  A. b  e.  2o  u r ( u splice  <. i ,  i ,  <" <. a ,  b >. <. a ,  ( 1o  \ 
b ) >. "> >.
) ) }
4238, 41syl6eleqr 2550 . 2  |-  ( ( ( A  e.  W  /\  N  e.  (
0 ... ( # `  A
) ) )  /\  ( J  e.  I  /\  K  e.  2o ) )  ->  <. A , 
( A splice  <. N ,  N ,  <" <. J ,  K >. <. J , 
( 1o  \  K
) >. "> >. ) >.  e.  .~  )
43 df-br 4394 . 2  |-  ( A  .~  ( A splice  <. N ,  N ,  <" <. J ,  K >. <. J , 
( 1o  \  K
) >. "> >. )  <->  <. A ,  ( A splice  <. N ,  N ,  <" <. J ,  K >. <. J ,  ( 1o  \  K )
>. "> >. ) >.  e.  .~  )
4442, 43sylibr 212 1  |-  ( ( ( A  e.  W  /\  N  e.  (
0 ... ( # `  A
) ) )  /\  ( J  e.  I  /\  K  e.  2o ) )  ->  A  .~  ( A splice  <. N ,  N ,  <" <. J ,  K >. <. J , 
( 1o  \  K
) >. "> >. )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1368    = wceq 1370    e. wcel 1758   {cab 2436   A.wral 2795    \ cdif 3426   <.cop 3984   <.cotp 3986   |^|cint 4229   class class class wbr 4393    _I cid 4732    X. cxp 4939   ` cfv 5519  (class class class)co 6193   1oc1o 7016   2oc2o 7017    Er wer 7201   0cc0 9386   ...cfz 11547   #chash 12213  Word cword 12332   splice csplice 12337   <"cs2 12579   ~FG cefg 16316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-ot 3987  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-2o 7024  df-oadd 7027  df-er 7204  df-map 7319  df-pm 7320  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-card 8213  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-n0 10684  df-z 10751  df-uz 10966  df-fz 11548  df-fzo 11659  df-hash 12214  df-word 12340  df-concat 12342  df-s1 12343  df-substr 12344  df-splice 12345  df-s2 12586  df-efg 16319
This theorem is referenced by:  efgi0  16330  efgi1  16331
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