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Theorem frgpuptinv 15331
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 4837 . . 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 15272 . . . . . . . . 9  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( a M b )  =  <. a ,  ( 1o  \ 
b ) >. )
43adantl 453 . . . . . . . 8  |-  ( (
ph  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( a M b )  =  <. a ,  ( 1o  \ 
b ) >. )
54fveq2d 5673 . . . . . . 7  |-  ( (
ph  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T `  (
a M b ) )  =  ( T `
 <. a ,  ( 1o  \  b )
>. ) )
6 df-ov 6024 . . . . . . 7  |-  ( a T ( 1o  \ 
b ) )  =  ( T `  <. a ,  ( 1o  \ 
b ) >. )
75, 6syl6eqr 2438 . . . . . 6  |-  ( (
ph  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T `  (
a M b ) )  =  ( a T ( 1o  \ 
b ) ) )
8 elpri 3778 . . . . . . . . 9  |-  ( b  e.  { (/) ,  1o }  ->  ( b  =  (/)  \/  b  =  1o ) )
9 df2o3 6674 . . . . . . . . 9  |-  2o  =  { (/) ,  1o }
108, 9eleq2s 2480 . . . . . . . 8  |-  ( b  e.  2o  ->  (
b  =  (/)  \/  b  =  1o ) )
11 simpr 448 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  I )  ->  a  e.  I )
12 1on 6668 . . . . . . . . . . . . . . 15  |-  1o  e.  On
1312elexi 2909 . . . . . . . . . . . . . 14  |-  1o  e.  _V
1413prid2 3857 . . . . . . . . . . . . 13  |-  1o  e.  {
(/) ,  1o }
1514, 9eleqtrri 2461 . . . . . . . . . . . 12  |-  1o  e.  2o
16 1n0 6676 . . . . . . . . . . . . . . . 16  |-  1o  =/=  (/)
17 neeq1 2559 . . . . . . . . . . . . . . . 16  |-  ( z  =  1o  ->  (
z  =/=  (/)  <->  1o  =/=  (/) ) )
1816, 17mpbiri 225 . . . . . . . . . . . . . . 15  |-  ( z  =  1o  ->  z  =/=  (/) )
19 ifnefalse 3691 . . . . . . . . . . . . . . 15  |-  ( z  =/=  (/)  ->  if (
z  =  (/) ,  ( F `  y ) ,  ( N `  ( F `  y ) ) )  =  ( N `  ( F `
 y ) ) )
2018, 19syl 16 . . . . . . . . . . . . . 14  |-  ( z  =  1o  ->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) )  =  ( N `  ( F `  y ) ) )
21 fveq2 5669 . . . . . . . . . . . . . . 15  |-  ( y  =  a  ->  ( F `  y )  =  ( F `  a ) )
2221fveq2d 5673 . . . . . . . . . . . . . 14  |-  ( y  =  a  ->  ( N `  ( F `  y ) )  =  ( N `  ( F `  a )
) )
2320, 22sylan9eqr 2442 . . . . . . . . . . . . 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 5683 . . . . . . . . . . . . 13  |-  ( N `
 ( F `  a ) )  e. 
_V
2623, 24, 25ovmpt2a 6144 . . . . . . . . . . . 12  |-  ( ( a  e.  I  /\  1o  e.  2o )  -> 
( a T 1o )  =  ( N `
 ( F `  a ) ) )
2711, 15, 26sylancl 644 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  I )  ->  (
a T 1o )  =  ( N `  ( F `  a ) ) )
28 0ex 4281 . . . . . . . . . . . . . . 15  |-  (/)  e.  _V
2928prid1 3856 . . . . . . . . . . . . . 14  |-  (/)  e.  { (/)
,  1o }
3029, 9eleqtrri 2461 . . . . . . . . . . . . 13  |-  (/)  e.  2o
31 iftrue 3689 . . . . . . . . . . . . . . 15  |-  ( z  =  (/)  ->  if ( z  =  (/) ,  ( F `  y ) ,  ( N `  ( F `  y ) ) )  =  ( F `  y ) )
3231, 21sylan9eqr 2442 . . . . . . . . . . . . . 14  |-  ( ( y  =  a  /\  z  =  (/) )  ->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) )  =  ( F `  a ) )
33 fvex 5683 . . . . . . . . . . . . . 14  |-  ( F `
 a )  e. 
_V
3432, 24, 33ovmpt2a 6144 . . . . . . . . . . . . 13  |-  ( ( a  e.  I  /\  (/) 
e.  2o )  -> 
( a T (/) )  =  ( F `  a ) )
3511, 30, 34sylancl 644 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  I )  ->  (
a T (/) )  =  ( F `  a
) )
3635fveq2d 5673 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  I )  ->  ( N `  ( a T (/) ) )  =  ( N `  ( F `  a )
) )
3727, 36eqtr4d 2423 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  I )  ->  (
a T 1o )  =  ( N `  ( a T (/) ) ) )
38 difeq2 3403 . . . . . . . . . . . . 13  |-  ( b  =  (/)  ->  ( 1o 
\  b )  =  ( 1o  \  (/) ) )
39 dif0 3642 . . . . . . . . . . . . 13  |-  ( 1o 
\  (/) )  =  1o
4038, 39syl6eq 2436 . . . . . . . . . . . 12  |-  ( b  =  (/)  ->  ( 1o 
\  b )  =  1o )
4140oveq2d 6037 . . . . . . . . . . 11  |-  ( b  =  (/)  ->  ( a T ( 1o  \ 
b ) )  =  ( a T 1o ) )
42 oveq2 6029 . . . . . . . . . . . 12  |-  ( b  =  (/)  ->  ( a T b )  =  ( a T (/) ) )
4342fveq2d 5673 . . . . . . . . . . 11  |-  ( b  =  (/)  ->  ( N `
 ( a T b ) )  =  ( N `  (
a T (/) ) ) )
4441, 43eqeq12d 2402 . . . . . . . . . 10  |-  ( b  =  (/)  ->  ( ( a T ( 1o 
\  b ) )  =  ( N `  ( a T b ) )  <->  ( a T 1o )  =  ( N `  ( a T (/) ) ) ) )
4537, 44syl5ibrcom 214 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  I )  ->  (
b  =  (/)  ->  (
a T ( 1o 
\  b ) )  =  ( N `  ( a T b ) ) ) )
4637fveq2d 5673 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  I )  ->  ( N `  ( a T 1o ) )  =  ( N `  ( N `  ( a T (/) ) ) ) )
47 frgpup.h . . . . . . . . . . . . 13  |-  ( ph  ->  H  e.  Grp )
4847adantr 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  I )  ->  H  e.  Grp )
49 frgpup.a . . . . . . . . . . . . . 14  |-  ( ph  ->  F : I --> B )
5049ffvelrnda 5810 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  I )  ->  ( F `  a )  e.  B )
5135, 50eqeltrd 2462 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  I )  ->  (
a T (/) )  e.  B )
52 frgpup.b . . . . . . . . . . . . 13  |-  B  =  ( Base `  H
)
53 frgpup.n . . . . . . . . . . . . 13  |-  N  =  ( inv g `  H )
5452, 53grpinvinv 14786 . . . . . . . . . . . 12  |-  ( ( H  e.  Grp  /\  ( a T (/) )  e.  B )  ->  ( N `  ( N `  ( a T (/) ) ) )  =  ( a T
(/) ) )
5548, 51, 54syl2anc 643 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  I )  ->  ( N `  ( N `  ( a T (/) ) ) )  =  ( a T (/) ) )
5646, 55eqtr2d 2421 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  I )  ->  (
a T (/) )  =  ( N `  (
a T 1o ) ) )
57 difeq2 3403 . . . . . . . . . . . . 13  |-  ( b  =  1o  ->  ( 1o  \  b )  =  ( 1o  \  1o ) )
58 difid 3640 . . . . . . . . . . . . 13  |-  ( 1o 
\  1o )  =  (/)
5957, 58syl6eq 2436 . . . . . . . . . . . 12  |-  ( b  =  1o  ->  ( 1o  \  b )  =  (/) )
6059oveq2d 6037 . . . . . . . . . . 11  |-  ( b  =  1o  ->  (
a T ( 1o 
\  b ) )  =  ( a T
(/) ) )
61 oveq2 6029 . . . . . . . . . . . 12  |-  ( b  =  1o  ->  (
a T b )  =  ( a T 1o ) )
6261fveq2d 5673 . . . . . . . . . . 11  |-  ( b  =  1o  ->  ( N `  ( a T b ) )  =  ( N `  ( a T 1o ) ) )
6360, 62eqeq12d 2402 . . . . . . . . . 10  |-  ( b  =  1o  ->  (
( a T ( 1o  \  b ) )  =  ( N `
 ( a T b ) )  <->  ( a T (/) )  =  ( N `  ( a T 1o ) ) ) )
6456, 63syl5ibrcom 214 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  I )  ->  (
b  =  1o  ->  ( a T ( 1o 
\  b ) )  =  ( N `  ( a T b ) ) ) )
6545, 64jaod 370 . . . . . . . 8  |-  ( (
ph  /\  a  e.  I )  ->  (
( b  =  (/)  \/  b  =  1o )  ->  ( a T ( 1o  \  b
) )  =  ( N `  ( a T b ) ) ) )
6610, 65syl5 30 . . . . . . 7  |-  ( (
ph  /\  a  e.  I )  ->  (
b  e.  2o  ->  ( a T ( 1o 
\  b ) )  =  ( N `  ( a T b ) ) ) )
6766impr 603 . . . . . 6  |-  ( (
ph  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( a T ( 1o  \  b ) )  =  ( N `
 ( a T b ) ) )
687, 67eqtrd 2420 . . . . 5  |-  ( (
ph  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T `  (
a M b ) )  =  ( N `
 ( a T b ) ) )
69 fveq2 5669 . . . . . . . 8  |-  ( A  =  <. a ,  b
>.  ->  ( M `  A )  =  ( M `  <. a ,  b >. )
)
70 df-ov 6024 . . . . . . . 8  |-  ( a M b )  =  ( M `  <. a ,  b >. )
7169, 70syl6eqr 2438 . . . . . . 7  |-  ( A  =  <. a ,  b
>.  ->  ( M `  A )  =  ( a M b ) )
7271fveq2d 5673 . . . . . 6  |-  ( A  =  <. a ,  b
>.  ->  ( T `  ( M `  A ) )  =  ( T `
 ( a M b ) ) )
73 fveq2 5669 . . . . . . . 8  |-  ( A  =  <. a ,  b
>.  ->  ( T `  A )  =  ( T `  <. a ,  b >. )
)
74 df-ov 6024 . . . . . . . 8  |-  ( a T b )  =  ( T `  <. a ,  b >. )
7573, 74syl6eqr 2438 . . . . . . 7  |-  ( A  =  <. a ,  b
>.  ->  ( T `  A )  =  ( a T b ) )
7675fveq2d 5673 . . . . . 6  |-  ( A  =  <. a ,  b
>.  ->  ( N `  ( T `  A ) )  =  ( N `
 ( a T b ) ) )
7772, 76eqeq12d 2402 . . . . 5  |-  ( A  =  <. a ,  b
>.  ->  ( ( T `
 ( M `  A ) )  =  ( N `  ( T `  A )
)  <->  ( T `  ( a M b ) )  =  ( N `  ( a T b ) ) ) )
7868, 77syl5ibrcom 214 . . . 4  |-  ( (
ph  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( A  =  <. a ,  b >.  ->  ( T `  ( M `  A ) )  =  ( N `  ( T `  A )
) ) )
7978rexlimdvva 2781 . . 3  |-  ( ph  ->  ( E. a  e.  I  E. b  e.  2o  A  =  <. a ,  b >.  ->  ( T `  ( M `  A ) )  =  ( N `  ( T `  A )
) ) )
801, 79syl5bi 209 . 2  |-  ( ph  ->  ( A  e.  ( I  X.  2o )  ->  ( T `  ( M `  A ) )  =  ( N `
 ( T `  A ) ) ) )
8180imp 419 1  |-  ( (
ph  /\  A  e.  ( I  X.  2o ) )  ->  ( T `  ( M `  A ) )  =  ( N `  ( T `  A )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2551   E.wrex 2651    \ cdif 3261   (/)c0 3572   ifcif 3683   {cpr 3759   <.cop 3761   Oncon0 4523    X. cxp 4817   -->wf 5391   ` cfv 5395  (class class class)co 6021    e. cmpt2 6023   1oc1o 6654   2oc2o 6655   Basecbs 13397   Grpcgrp 14613   inv gcminusg 14614
This theorem is referenced by:  frgpuplem  15332
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-suc 4529  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-riota 6486  df-1o 6661  df-2o 6662  df-0g 13655  df-mnd 14618  df-grp 14740  df-minusg 14741
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