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Theorem frgpuptinv 15096
Description: Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
frgpup.b  |-  B  =  ( Base `  H
)
frgpup.n  |-  N  =  ( inv g `  H )
frgpup.t  |-  T  =  ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) ) )
frgpup.h  |-  ( ph  ->  H  e.  Grp )
frgpup.i  |-  ( ph  ->  I  e.  V )
frgpup.a  |-  ( ph  ->  F : I --> B )
frgpuptinv.m  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
Assertion
Ref Expression
frgpuptinv  |-  ( (
ph  /\  A  e.  ( I  X.  2o ) )  ->  ( T `  ( M `  A ) )  =  ( N `  ( T `  A )
) )
Distinct variable groups:    y, z, A    y, F, z    y, N, z    y, B, z    ph, y, z    y, I, z
Allowed substitution hints:    T( y, z)    H( y, z)    M( y, z)    V( y, z)

Proof of Theorem frgpuptinv
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elxp2 4723 . . 3  |-  ( A  e.  ( I  X.  2o )  <->  E. a  e.  I  E. b  e.  2o  A  =  <. a ,  b >. )
2 frgpuptinv.m . . . . . . . . . 10  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
32efgmval 15037 . . . . . . . . 9  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( a M b )  =  <. a ,  ( 1o  \ 
b ) >. )
43adantl 452 . . . . . . . 8  |-  ( (
ph  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( a M b )  =  <. a ,  ( 1o  \ 
b ) >. )
54fveq2d 5545 . . . . . . 7  |-  ( (
ph  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T `  (
a M b ) )  =  ( T `
 <. a ,  ( 1o  \  b )
>. ) )
6 df-ov 5877 . . . . . . 7  |-  ( a T ( 1o  \ 
b ) )  =  ( T `  <. a ,  ( 1o  \ 
b ) >. )
75, 6syl6eqr 2346 . . . . . 6  |-  ( (
ph  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T `  (
a M b ) )  =  ( a T ( 1o  \ 
b ) ) )
8 elpri 3673 . . . . . . . . 9  |-  ( b  e.  { (/) ,  1o }  ->  ( b  =  (/)  \/  b  =  1o ) )
9 df2o3 6508 . . . . . . . . 9  |-  2o  =  { (/) ,  1o }
108, 9eleq2s 2388 . . . . . . . 8  |-  ( b  e.  2o  ->  (
b  =  (/)  \/  b  =  1o ) )
11 simpr 447 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  I )  ->  a  e.  I )
12 1on 6502 . . . . . . . . . . . . . . 15  |-  1o  e.  On
1312elexi 2810 . . . . . . . . . . . . . 14  |-  1o  e.  _V
1413prid2 3748 . . . . . . . . . . . . 13  |-  1o  e.  {
(/) ,  1o }
1514, 9eleqtrri 2369 . . . . . . . . . . . 12  |-  1o  e.  2o
16 1n0 6510 . . . . . . . . . . . . . . . 16  |-  1o  =/=  (/)
17 neeq1 2467 . . . . . . . . . . . . . . . 16  |-  ( z  =  1o  ->  (
z  =/=  (/)  <->  1o  =/=  (/) ) )
1816, 17mpbiri 224 . . . . . . . . . . . . . . 15  |-  ( z  =  1o  ->  z  =/=  (/) )
19 ifnefalse 3586 . . . . . . . . . . . . . . 15  |-  ( z  =/=  (/)  ->  if (
z  =  (/) ,  ( F `  y ) ,  ( N `  ( F `  y ) ) )  =  ( N `  ( F `
 y ) ) )
2018, 19syl 15 . . . . . . . . . . . . . 14  |-  ( z  =  1o  ->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) )  =  ( N `  ( F `  y ) ) )
21 fveq2 5541 . . . . . . . . . . . . . . 15  |-  ( y  =  a  ->  ( F `  y )  =  ( F `  a ) )
2221fveq2d 5545 . . . . . . . . . . . . . 14  |-  ( y  =  a  ->  ( N `  ( F `  y ) )  =  ( N `  ( F `  a )
) )
2320, 22sylan9eqr 2350 . . . . . . . . . . . . 13  |-  ( ( y  =  a  /\  z  =  1o )  ->  if ( z  =  (/) ,  ( F `  y ) ,  ( N `  ( F `
 y ) ) )  =  ( N `
 ( F `  a ) ) )
24 frgpup.t . . . . . . . . . . . . 13  |-  T  =  ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) ) )
25 fvex 5555 . . . . . . . . . . . . 13  |-  ( N `
 ( F `  a ) )  e. 
_V
2623, 24, 25ovmpt2a 5994 . . . . . . . . . . . 12  |-  ( ( a  e.  I  /\  1o  e.  2o )  -> 
( a T 1o )  =  ( N `
 ( F `  a ) ) )
2711, 15, 26sylancl 643 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  I )  ->  (
a T 1o )  =  ( N `  ( F `  a ) ) )
28 0ex 4166 . . . . . . . . . . . . . . 15  |-  (/)  e.  _V
2928prid1 3747 . . . . . . . . . . . . . 14  |-  (/)  e.  { (/)
,  1o }
3029, 9eleqtrri 2369 . . . . . . . . . . . . 13  |-  (/)  e.  2o
31 iftrue 3584 . . . . . . . . . . . . . . 15  |-  ( z  =  (/)  ->  if ( z  =  (/) ,  ( F `  y ) ,  ( N `  ( F `  y ) ) )  =  ( F `  y ) )
3231, 21sylan9eqr 2350 . . . . . . . . . . . . . 14  |-  ( ( y  =  a  /\  z  =  (/) )  ->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) )  =  ( F `  a ) )
33 fvex 5555 . . . . . . . . . . . . . 14  |-  ( F `
 a )  e. 
_V
3432, 24, 33ovmpt2a 5994 . . . . . . . . . . . . 13  |-  ( ( a  e.  I  /\  (/) 
e.  2o )  -> 
( a T (/) )  =  ( F `  a ) )
3511, 30, 34sylancl 643 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  I )  ->  (
a T (/) )  =  ( F `  a
) )
3635fveq2d 5545 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  I )  ->  ( N `  ( a T (/) ) )  =  ( N `  ( F `  a )
) )
3727, 36eqtr4d 2331 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  I )  ->  (
a T 1o )  =  ( N `  ( a T (/) ) ) )
38 difeq2 3301 . . . . . . . . . . . . 13  |-  ( b  =  (/)  ->  ( 1o 
\  b )  =  ( 1o  \  (/) ) )
39 dif0 3537 . . . . . . . . . . . . 13  |-  ( 1o 
\  (/) )  =  1o
4038, 39syl6eq 2344 . . . . . . . . . . . 12  |-  ( b  =  (/)  ->  ( 1o 
\  b )  =  1o )
4140oveq2d 5890 . . . . . . . . . . 11  |-  ( b  =  (/)  ->  ( a T ( 1o  \ 
b ) )  =  ( a T 1o ) )
42 oveq2 5882 . . . . . . . . . . . 12  |-  ( b  =  (/)  ->  ( a T b )  =  ( a T (/) ) )
4342fveq2d 5545 . . . . . . . . . . 11  |-  ( b  =  (/)  ->  ( N `
 ( a T b ) )  =  ( N `  (
a T (/) ) ) )
4441, 43eqeq12d 2310 . . . . . . . . . 10  |-  ( b  =  (/)  ->  ( ( a T ( 1o 
\  b ) )  =  ( N `  ( a T b ) )  <->  ( a T 1o )  =  ( N `  ( a T (/) ) ) ) )
4537, 44syl5ibrcom 213 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  I )  ->  (
b  =  (/)  ->  (
a T ( 1o 
\  b ) )  =  ( N `  ( a T b ) ) ) )
4637fveq2d 5545 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  I )  ->  ( N `  ( a T 1o ) )  =  ( N `  ( N `  ( a T (/) ) ) ) )
47 frgpup.h . . . . . . . . . . . . 13  |-  ( ph  ->  H  e.  Grp )
4847adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  I )  ->  H  e.  Grp )
49 frgpup.a . . . . . . . . . . . . . 14  |-  ( ph  ->  F : I --> B )
50 ffvelrn 5679 . . . . . . . . . . . . . 14  |-  ( ( F : I --> B  /\  a  e.  I )  ->  ( F `  a
)  e.  B )
5149, 50sylan 457 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  I )  ->  ( F `  a )  e.  B )
5235, 51eqeltrd 2370 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  I )  ->  (
a T (/) )  e.  B )
53 frgpup.b . . . . . . . . . . . . 13  |-  B  =  ( Base `  H
)
54 frgpup.n . . . . . . . . . . . . 13  |-  N  =  ( inv g `  H )
5553, 54grpinvinv 14551 . . . . . . . . . . . 12  |-  ( ( H  e.  Grp  /\  ( a T (/) )  e.  B )  ->  ( N `  ( N `  ( a T (/) ) ) )  =  ( a T
(/) ) )
5648, 52, 55syl2anc 642 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  I )  ->  ( N `  ( N `  ( a T (/) ) ) )  =  ( a T (/) ) )
5746, 56eqtr2d 2329 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  I )  ->  (
a T (/) )  =  ( N `  (
a T 1o ) ) )
58 difeq2 3301 . . . . . . . . . . . . 13  |-  ( b  =  1o  ->  ( 1o  \  b )  =  ( 1o  \  1o ) )
59 difid 3535 . . . . . . . . . . . . 13  |-  ( 1o 
\  1o )  =  (/)
6058, 59syl6eq 2344 . . . . . . . . . . . 12  |-  ( b  =  1o  ->  ( 1o  \  b )  =  (/) )
6160oveq2d 5890 . . . . . . . . . . 11  |-  ( b  =  1o  ->  (
a T ( 1o 
\  b ) )  =  ( a T
(/) ) )
62 oveq2 5882 . . . . . . . . . . . 12  |-  ( b  =  1o  ->  (
a T b )  =  ( a T 1o ) )
6362fveq2d 5545 . . . . . . . . . . 11  |-  ( b  =  1o  ->  ( N `  ( a T b ) )  =  ( N `  ( a T 1o ) ) )
6461, 63eqeq12d 2310 . . . . . . . . . 10  |-  ( b  =  1o  ->  (
( a T ( 1o  \  b ) )  =  ( N `
 ( a T b ) )  <->  ( a T (/) )  =  ( N `  ( a T 1o ) ) ) )
6557, 64syl5ibrcom 213 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  I )  ->  (
b  =  1o  ->  ( a T ( 1o 
\  b ) )  =  ( N `  ( a T b ) ) ) )
6645, 65jaod 369 . . . . . . . 8  |-  ( (
ph  /\  a  e.  I )  ->  (
( b  =  (/)  \/  b  =  1o )  ->  ( a T ( 1o  \  b
) )  =  ( N `  ( a T b ) ) ) )
6710, 66syl5 28 . . . . . . 7  |-  ( (
ph  /\  a  e.  I )  ->  (
b  e.  2o  ->  ( a T ( 1o 
\  b ) )  =  ( N `  ( a T b ) ) ) )
6867impr 602 . . . . . 6  |-  ( (
ph  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( a T ( 1o  \  b ) )  =  ( N `
 ( a T b ) ) )
697, 68eqtrd 2328 . . . . 5  |-  ( (
ph  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T `  (
a M b ) )  =  ( N `
 ( a T b ) ) )
70 fveq2 5541 . . . . . . . 8  |-  ( A  =  <. a ,  b
>.  ->  ( M `  A )  =  ( M `  <. a ,  b >. )
)
71 df-ov 5877 . . . . . . . 8  |-  ( a M b )  =  ( M `  <. a ,  b >. )
7270, 71syl6eqr 2346 . . . . . . 7  |-  ( A  =  <. a ,  b
>.  ->  ( M `  A )  =  ( a M b ) )
7372fveq2d 5545 . . . . . 6  |-  ( A  =  <. a ,  b
>.  ->  ( T `  ( M `  A ) )  =  ( T `
 ( a M b ) ) )
74 fveq2 5541 . . . . . . . 8  |-  ( A  =  <. a ,  b
>.  ->  ( T `  A )  =  ( T `  <. a ,  b >. )
)
75 df-ov 5877 . . . . . . . 8  |-  ( a T b )  =  ( T `  <. a ,  b >. )
7674, 75syl6eqr 2346 . . . . . . 7  |-  ( A  =  <. a ,  b
>.  ->  ( T `  A )  =  ( a T b ) )
7776fveq2d 5545 . . . . . 6  |-  ( A  =  <. a ,  b
>.  ->  ( N `  ( T `  A ) )  =  ( N `
 ( a T b ) ) )
7873, 77eqeq12d 2310 . . . . 5  |-  ( A  =  <. a ,  b
>.  ->  ( ( T `
 ( M `  A ) )  =  ( N `  ( T `  A )
)  <->  ( T `  ( a M b ) )  =  ( N `  ( a T b ) ) ) )
7969, 78syl5ibrcom 213 . . . 4  |-  ( (
ph  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( A  =  <. a ,  b >.  ->  ( T `  ( M `  A ) )  =  ( N `  ( T `  A )
) ) )
8079rexlimdvva 2687 . . 3  |-  ( ph  ->  ( E. a  e.  I  E. b  e.  2o  A  =  <. a ,  b >.  ->  ( T `  ( M `  A ) )  =  ( N `  ( T `  A )
) ) )
811, 80syl5bi 208 . 2  |-  ( ph  ->  ( A  e.  ( I  X.  2o )  ->  ( T `  ( M `  A ) )  =  ( N `
 ( T `  A ) ) ) )
8281imp 418 1  |-  ( (
ph  /\  A  e.  ( I  X.  2o ) )  ->  ( T `  ( M `  A ) )  =  ( N `  ( T `  A )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557    \ cdif 3162   (/)c0 3468   ifcif 3578   {cpr 3654   <.cop 3656   Oncon0 4408    X. cxp 4703   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1oc1o 6488   2oc2o 6489   Basecbs 13164   Grpcgrp 14378   inv gcminusg 14379
This theorem is referenced by:  frgpuplem  15097
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-1o 6495  df-2o 6496  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506
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