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Theorem frgpuptinv 17470
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  =  ( invg `  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 4871 . . 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 17411 . . . . . . . . 9  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( a M b )  =  <. a ,  ( 1o  \ 
b ) >. )
43adantl 472 . . . . . . . 8  |-  ( (
ph  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( a M b )  =  <. a ,  ( 1o  \ 
b ) >. )
54fveq2d 5892 . . . . . . 7  |-  ( (
ph  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T `  (
a M b ) )  =  ( T `
 <. a ,  ( 1o  \  b )
>. ) )
6 df-ov 6318 . . . . . . 7  |-  ( a T ( 1o  \ 
b ) )  =  ( T `  <. a ,  ( 1o  \ 
b ) >. )
75, 6syl6eqr 2514 . . . . . 6  |-  ( (
ph  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T `  (
a M b ) )  =  ( a T ( 1o  \ 
b ) ) )
8 elpri 3997 . . . . . . . . 9  |-  ( b  e.  { (/) ,  1o }  ->  ( b  =  (/)  \/  b  =  1o ) )
9 df2o3 7221 . . . . . . . . 9  |-  2o  =  { (/) ,  1o }
108, 9eleq2s 2558 . . . . . . . 8  |-  ( b  e.  2o  ->  (
b  =  (/)  \/  b  =  1o ) )
11 simpr 467 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  I )  ->  a  e.  I )
12 1on 7215 . . . . . . . . . . . . . . 15  |-  1o  e.  On
1312elexi 3067 . . . . . . . . . . . . . 14  |-  1o  e.  _V
1413prid2 4094 . . . . . . . . . . . . 13  |-  1o  e.  {
(/) ,  1o }
1514, 9eleqtrri 2539 . . . . . . . . . . . 12  |-  1o  e.  2o
16 1n0 7223 . . . . . . . . . . . . . . . 16  |-  1o  =/=  (/)
17 neeq1 2698 . . . . . . . . . . . . . . . 16  |-  ( z  =  1o  ->  (
z  =/=  (/)  <->  1o  =/=  (/) ) )
1816, 17mpbiri 241 . . . . . . . . . . . . . . 15  |-  ( z  =  1o  ->  z  =/=  (/) )
19 ifnefalse 3905 . . . . . . . . . . . . . . 15  |-  ( z  =/=  (/)  ->  if (
z  =  (/) ,  ( F `  y ) ,  ( N `  ( F `  y ) ) )  =  ( N `  ( F `
 y ) ) )
2018, 19syl 17 . . . . . . . . . . . . . 14  |-  ( z  =  1o  ->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) )  =  ( N `  ( F `  y ) ) )
21 fveq2 5888 . . . . . . . . . . . . . . 15  |-  ( y  =  a  ->  ( F `  y )  =  ( F `  a ) )
2221fveq2d 5892 . . . . . . . . . . . . . 14  |-  ( y  =  a  ->  ( N `  ( F `  y ) )  =  ( N `  ( F `  a )
) )
2320, 22sylan9eqr 2518 . . . . . . . . . . . . 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 5898 . . . . . . . . . . . . 13  |-  ( N `
 ( F `  a ) )  e. 
_V
2623, 24, 25ovmpt2a 6454 . . . . . . . . . . . 12  |-  ( ( a  e.  I  /\  1o  e.  2o )  -> 
( a T 1o )  =  ( N `
 ( F `  a ) ) )
2711, 15, 26sylancl 673 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  I )  ->  (
a T 1o )  =  ( N `  ( F `  a ) ) )
28 0ex 4549 . . . . . . . . . . . . . . 15  |-  (/)  e.  _V
2928prid1 4093 . . . . . . . . . . . . . 14  |-  (/)  e.  { (/)
,  1o }
3029, 9eleqtrri 2539 . . . . . . . . . . . . 13  |-  (/)  e.  2o
31 iftrue 3899 . . . . . . . . . . . . . . 15  |-  ( z  =  (/)  ->  if ( z  =  (/) ,  ( F `  y ) ,  ( N `  ( F `  y ) ) )  =  ( F `  y ) )
3231, 21sylan9eqr 2518 . . . . . . . . . . . . . 14  |-  ( ( y  =  a  /\  z  =  (/) )  ->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) )  =  ( F `  a ) )
33 fvex 5898 . . . . . . . . . . . . . 14  |-  ( F `
 a )  e. 
_V
3432, 24, 33ovmpt2a 6454 . . . . . . . . . . . . 13  |-  ( ( a  e.  I  /\  (/) 
e.  2o )  -> 
( a T (/) )  =  ( F `  a ) )
3511, 30, 34sylancl 673 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  I )  ->  (
a T (/) )  =  ( F `  a
) )
3635fveq2d 5892 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  I )  ->  ( N `  ( a T (/) ) )  =  ( N `  ( F `  a )
) )
3727, 36eqtr4d 2499 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  I )  ->  (
a T 1o )  =  ( N `  ( a T (/) ) ) )
38 difeq2 3557 . . . . . . . . . . . . 13  |-  ( b  =  (/)  ->  ( 1o 
\  b )  =  ( 1o  \  (/) ) )
39 dif0 3849 . . . . . . . . . . . . 13  |-  ( 1o 
\  (/) )  =  1o
4038, 39syl6eq 2512 . . . . . . . . . . . 12  |-  ( b  =  (/)  ->  ( 1o 
\  b )  =  1o )
4140oveq2d 6331 . . . . . . . . . . 11  |-  ( b  =  (/)  ->  ( a T ( 1o  \ 
b ) )  =  ( a T 1o ) )
42 oveq2 6323 . . . . . . . . . . . 12  |-  ( b  =  (/)  ->  ( a T b )  =  ( a T (/) ) )
4342fveq2d 5892 . . . . . . . . . . 11  |-  ( b  =  (/)  ->  ( N `
 ( a T b ) )  =  ( N `  (
a T (/) ) ) )
4441, 43eqeq12d 2477 . . . . . . . . . 10  |-  ( b  =  (/)  ->  ( ( a T ( 1o 
\  b ) )  =  ( N `  ( a T b ) )  <->  ( a T 1o )  =  ( N `  ( a T (/) ) ) ) )
4537, 44syl5ibrcom 230 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  I )  ->  (
b  =  (/)  ->  (
a T ( 1o 
\  b ) )  =  ( N `  ( a T b ) ) ) )
4637fveq2d 5892 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  I )  ->  ( N `  ( a T 1o ) )  =  ( N `  ( N `  ( a T (/) ) ) ) )
47 frgpup.h . . . . . . . . . . . . 13  |-  ( ph  ->  H  e.  Grp )
4847adantr 471 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  I )  ->  H  e.  Grp )
49 frgpup.a . . . . . . . . . . . . . 14  |-  ( ph  ->  F : I --> B )
5049ffvelrnda 6045 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  I )  ->  ( F `  a )  e.  B )
5135, 50eqeltrd 2540 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  I )  ->  (
a T (/) )  e.  B )
52 frgpup.b . . . . . . . . . . . . 13  |-  B  =  ( Base `  H
)
53 frgpup.n . . . . . . . . . . . . 13  |-  N  =  ( invg `  H )
5452, 53grpinvinv 16770 . . . . . . . . . . . 12  |-  ( ( H  e.  Grp  /\  ( a T (/) )  e.  B )  ->  ( N `  ( N `  ( a T (/) ) ) )  =  ( a T
(/) ) )
5548, 51, 54syl2anc 671 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  I )  ->  ( N `  ( N `  ( a T (/) ) ) )  =  ( a T (/) ) )
5646, 55eqtr2d 2497 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  I )  ->  (
a T (/) )  =  ( N `  (
a T 1o ) ) )
57 difeq2 3557 . . . . . . . . . . . . 13  |-  ( b  =  1o  ->  ( 1o  \  b )  =  ( 1o  \  1o ) )
58 difid 3847 . . . . . . . . . . . . 13  |-  ( 1o 
\  1o )  =  (/)
5957, 58syl6eq 2512 . . . . . . . . . . . 12  |-  ( b  =  1o  ->  ( 1o  \  b )  =  (/) )
6059oveq2d 6331 . . . . . . . . . . 11  |-  ( b  =  1o  ->  (
a T ( 1o 
\  b ) )  =  ( a T
(/) ) )
61 oveq2 6323 . . . . . . . . . . . 12  |-  ( b  =  1o  ->  (
a T b )  =  ( a T 1o ) )
6261fveq2d 5892 . . . . . . . . . . 11  |-  ( b  =  1o  ->  ( N `  ( a T b ) )  =  ( N `  ( a T 1o ) ) )
6360, 62eqeq12d 2477 . . . . . . . . . 10  |-  ( b  =  1o  ->  (
( a T ( 1o  \  b ) )  =  ( N `
 ( a T b ) )  <->  ( a T (/) )  =  ( N `  ( a T 1o ) ) ) )
6456, 63syl5ibrcom 230 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  I )  ->  (
b  =  1o  ->  ( a T ( 1o 
\  b ) )  =  ( N `  ( a T b ) ) ) )
6545, 64jaod 386 . . . . . . . 8  |-  ( (
ph  /\  a  e.  I )  ->  (
( b  =  (/)  \/  b  =  1o )  ->  ( a T ( 1o  \  b
) )  =  ( N `  ( a T b ) ) ) )
6610, 65syl5 33 . . . . . . 7  |-  ( (
ph  /\  a  e.  I )  ->  (
b  e.  2o  ->  ( a T ( 1o 
\  b ) )  =  ( N `  ( a T b ) ) ) )
6766impr 629 . . . . . 6  |-  ( (
ph  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( a T ( 1o  \  b ) )  =  ( N `
 ( a T b ) ) )
687, 67eqtrd 2496 . . . . 5  |-  ( (
ph  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T `  (
a M b ) )  =  ( N `
 ( a T b ) ) )
69 fveq2 5888 . . . . . . . 8  |-  ( A  =  <. a ,  b
>.  ->  ( M `  A )  =  ( M `  <. a ,  b >. )
)
70 df-ov 6318 . . . . . . . 8  |-  ( a M b )  =  ( M `  <. a ,  b >. )
7169, 70syl6eqr 2514 . . . . . . 7  |-  ( A  =  <. a ,  b
>.  ->  ( M `  A )  =  ( a M b ) )
7271fveq2d 5892 . . . . . 6  |-  ( A  =  <. a ,  b
>.  ->  ( T `  ( M `  A ) )  =  ( T `
 ( a M b ) ) )
73 fveq2 5888 . . . . . . . 8  |-  ( A  =  <. a ,  b
>.  ->  ( T `  A )  =  ( T `  <. a ,  b >. )
)
74 df-ov 6318 . . . . . . . 8  |-  ( a T b )  =  ( T `  <. a ,  b >. )
7573, 74syl6eqr 2514 . . . . . . 7  |-  ( A  =  <. a ,  b
>.  ->  ( T `  A )  =  ( a T b ) )
7675fveq2d 5892 . . . . . 6  |-  ( A  =  <. a ,  b
>.  ->  ( N `  ( T `  A ) )  =  ( N `
 ( a T b ) ) )
7772, 76eqeq12d 2477 . . . . 5  |-  ( A  =  <. a ,  b
>.  ->  ( ( T `
 ( M `  A ) )  =  ( N `  ( T `  A )
)  <->  ( T `  ( a M b ) )  =  ( N `  ( a T b ) ) ) )
7868, 77syl5ibrcom 230 . . . 4  |-  ( (
ph  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( A  =  <. a ,  b >.  ->  ( T `  ( M `  A ) )  =  ( N `  ( T `  A )
) ) )
7978rexlimdvva 2898 . . 3  |-  ( ph  ->  ( E. a  e.  I  E. b  e.  2o  A  =  <. a ,  b >.  ->  ( T `  ( M `  A ) )  =  ( N `  ( T `  A )
) ) )
801, 79syl5bi 225 . 2  |-  ( ph  ->  ( A  e.  ( I  X.  2o )  ->  ( T `  ( M `  A ) )  =  ( N `
 ( T `  A ) ) ) )
8180imp 435 1  |-  ( (
ph  /\  A  e.  ( I  X.  2o ) )  ->  ( T `  ( M `  A ) )  =  ( N `  ( T `  A )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 374    /\ wa 375    = wceq 1455    e. wcel 1898    =/= wne 2633   E.wrex 2750    \ cdif 3413   (/)c0 3743   ifcif 3893   {cpr 3982   <.cop 3986    X. cxp 4851   Oncon0 5442   -->wf 5597   ` cfv 5601  (class class class)co 6315    |-> cmpt2 6317   1oc1o 7201   2oc2o 7202   Basecbs 15170   Grpcgrp 16718   invgcminusg 16719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-ord 5445  df-on 5446  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-1o 7208  df-2o 7209  df-0g 15389  df-mgm 16537  df-sgrp 16576  df-mnd 16586  df-grp 16722  df-minusg 16723
This theorem is referenced by:  frgpuplem  17471
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