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Theorem frgpuptinv 15080
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 4707 . . 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 15021 . . . . . . . . 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 5529 . . . . . . 7  |-  ( (
ph  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T `  (
a M b ) )  =  ( T `
 <. a ,  ( 1o  \  b )
>. ) )
6 df-ov 5861 . . . . . . 7  |-  ( a T ( 1o  \ 
b ) )  =  ( T `  <. a ,  ( 1o  \ 
b ) >. )
75, 6syl6eqr 2333 . . . . . 6  |-  ( (
ph  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T `  (
a M b ) )  =  ( a T ( 1o  \ 
b ) ) )
8 elpri 3660 . . . . . . . . 9  |-  ( b  e.  { (/) ,  1o }  ->  ( b  =  (/)  \/  b  =  1o ) )
9 df2o3 6492 . . . . . . . . 9  |-  2o  =  { (/) ,  1o }
108, 9eleq2s 2375 . . . . . . . 8  |-  ( b  e.  2o  ->  (
b  =  (/)  \/  b  =  1o ) )
11 simpr 447 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  I )  ->  a  e.  I )
12 1on 6486 . . . . . . . . . . . . . . 15  |-  1o  e.  On
1312elexi 2797 . . . . . . . . . . . . . 14  |-  1o  e.  _V
1413prid2 3735 . . . . . . . . . . . . 13  |-  1o  e.  {
(/) ,  1o }
1514, 9eleqtrri 2356 . . . . . . . . . . . 12  |-  1o  e.  2o
16 1n0 6494 . . . . . . . . . . . . . . . 16  |-  1o  =/=  (/)
17 neeq1 2454 . . . . . . . . . . . . . . . 16  |-  ( z  =  1o  ->  (
z  =/=  (/)  <->  1o  =/=  (/) ) )
1816, 17mpbiri 224 . . . . . . . . . . . . . . 15  |-  ( z  =  1o  ->  z  =/=  (/) )
19 ifnefalse 3573 . . . . . . . . . . . . . . 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 5525 . . . . . . . . . . . . . . 15  |-  ( y  =  a  ->  ( F `  y )  =  ( F `  a ) )
2221fveq2d 5529 . . . . . . . . . . . . . 14  |-  ( y  =  a  ->  ( N `  ( F `  y ) )  =  ( N `  ( F `  a )
) )
2320, 22sylan9eqr 2337 . . . . . . . . . . . . 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 5539 . . . . . . . . . . . . 13  |-  ( N `
 ( F `  a ) )  e. 
_V
2623, 24, 25ovmpt2a 5978 . . . . . . . . . . . 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 4150 . . . . . . . . . . . . . . 15  |-  (/)  e.  _V
2928prid1 3734 . . . . . . . . . . . . . 14  |-  (/)  e.  { (/)
,  1o }
3029, 9eleqtrri 2356 . . . . . . . . . . . . 13  |-  (/)  e.  2o
31 iftrue 3571 . . . . . . . . . . . . . . 15  |-  ( z  =  (/)  ->  if ( z  =  (/) ,  ( F `  y ) ,  ( N `  ( F `  y ) ) )  =  ( F `  y ) )
3231, 21sylan9eqr 2337 . . . . . . . . . . . . . 14  |-  ( ( y  =  a  /\  z  =  (/) )  ->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) )  =  ( F `  a ) )
33 fvex 5539 . . . . . . . . . . . . . 14  |-  ( F `
 a )  e. 
_V
3432, 24, 33ovmpt2a 5978 . . . . . . . . . . . . 13  |-  ( ( a  e.  I  /\  (/) 
e.  2o )  -> 
( a T (/) )  =  ( F `  a ) )
3511, 30, 34sylancl 643 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  I )  ->  (
a T (/) )  =  ( F `  a
) )
3635fveq2d 5529 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  I )  ->  ( N `  ( a T (/) ) )  =  ( N `  ( F `  a )
) )
3727, 36eqtr4d 2318 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  I )  ->  (
a T 1o )  =  ( N `  ( a T (/) ) ) )
38 difeq2 3288 . . . . . . . . . . . . 13  |-  ( b  =  (/)  ->  ( 1o 
\  b )  =  ( 1o  \  (/) ) )
39 dif0 3524 . . . . . . . . . . . . 13  |-  ( 1o 
\  (/) )  =  1o
4038, 39syl6eq 2331 . . . . . . . . . . . 12  |-  ( b  =  (/)  ->  ( 1o 
\  b )  =  1o )
4140oveq2d 5874 . . . . . . . . . . 11  |-  ( b  =  (/)  ->  ( a T ( 1o  \ 
b ) )  =  ( a T 1o ) )
42 oveq2 5866 . . . . . . . . . . . 12  |-  ( b  =  (/)  ->  ( a T b )  =  ( a T (/) ) )
4342fveq2d 5529 . . . . . . . . . . 11  |-  ( b  =  (/)  ->  ( N `
 ( a T b ) )  =  ( N `  (
a T (/) ) ) )
4441, 43eqeq12d 2297 . . . . . . . . . 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 5529 . . . . . . . . . . 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 5663 . . . . . . . . . . . . . 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 2357 . . . . . . . . . . . 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 14535 . . . . . . . . . . . 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 2316 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  I )  ->  (
a T (/) )  =  ( N `  (
a T 1o ) ) )
58 difeq2 3288 . . . . . . . . . . . . 13  |-  ( b  =  1o  ->  ( 1o  \  b )  =  ( 1o  \  1o ) )
59 difid 3522 . . . . . . . . . . . . 13  |-  ( 1o 
\  1o )  =  (/)
6058, 59syl6eq 2331 . . . . . . . . . . . 12  |-  ( b  =  1o  ->  ( 1o  \  b )  =  (/) )
6160oveq2d 5874 . . . . . . . . . . 11  |-  ( b  =  1o  ->  (
a T ( 1o 
\  b ) )  =  ( a T
(/) ) )
62 oveq2 5866 . . . . . . . . . . . 12  |-  ( b  =  1o  ->  (
a T b )  =  ( a T 1o ) )
6362fveq2d 5529 . . . . . . . . . . 11  |-  ( b  =  1o  ->  ( N `  ( a T b ) )  =  ( N `  ( a T 1o ) ) )
6461, 63eqeq12d 2297 . . . . . . . . . 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 2315 . . . . 5  |-  ( (
ph  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T `  (
a M b ) )  =  ( N `
 ( a T b ) ) )
70 fveq2 5525 . . . . . . . 8  |-  ( A  =  <. a ,  b
>.  ->  ( M `  A )  =  ( M `  <. a ,  b >. )
)
71 df-ov 5861 . . . . . . . 8  |-  ( a M b )  =  ( M `  <. a ,  b >. )
7270, 71syl6eqr 2333 . . . . . . 7  |-  ( A  =  <. a ,  b
>.  ->  ( M `  A )  =  ( a M b ) )
7372fveq2d 5529 . . . . . 6  |-  ( A  =  <. a ,  b
>.  ->  ( T `  ( M `  A ) )  =  ( T `
 ( a M b ) ) )
74 fveq2 5525 . . . . . . . 8  |-  ( A  =  <. a ,  b
>.  ->  ( T `  A )  =  ( T `  <. a ,  b >. )
)
75 df-ov 5861 . . . . . . . 8  |-  ( a T b )  =  ( T `  <. a ,  b >. )
7674, 75syl6eqr 2333 . . . . . . 7  |-  ( A  =  <. a ,  b
>.  ->  ( T `  A )  =  ( a T b ) )
7776fveq2d 5529 . . . . . 6  |-  ( A  =  <. a ,  b
>.  ->  ( N `  ( T `  A ) )  =  ( N `
 ( a T b ) ) )
7873, 77eqeq12d 2297 . . . . 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 2674 . . 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 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544    \ cdif 3149   (/)c0 3455   ifcif 3565   {cpr 3641   <.cop 3643   Oncon0 4392    X. cxp 4687   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1oc1o 6472   2oc2o 6473   Basecbs 13148   Grpcgrp 14362   inv gcminusg 14363
This theorem is referenced by:  frgpuplem  15081
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
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-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  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-uni 3828  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-suc 4398  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-riota 6304  df-1o 6479  df-2o 6480  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490
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