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Theorem frgpuptinv 16916
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 5026 . . 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 16857 . . . . . . . . 9  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( a M b )  =  <. a ,  ( 1o  \ 
b ) >. )
43adantl 466 . . . . . . . 8  |-  ( (
ph  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( a M b )  =  <. a ,  ( 1o  \ 
b ) >. )
54fveq2d 5876 . . . . . . 7  |-  ( (
ph  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T `  (
a M b ) )  =  ( T `
 <. a ,  ( 1o  \  b )
>. ) )
6 df-ov 6299 . . . . . . 7  |-  ( a T ( 1o  \ 
b ) )  =  ( T `  <. a ,  ( 1o  \ 
b ) >. )
75, 6syl6eqr 2516 . . . . . 6  |-  ( (
ph  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T `  (
a M b ) )  =  ( a T ( 1o  \ 
b ) ) )
8 elpri 4052 . . . . . . . . 9  |-  ( b  e.  { (/) ,  1o }  ->  ( b  =  (/)  \/  b  =  1o ) )
9 df2o3 7161 . . . . . . . . 9  |-  2o  =  { (/) ,  1o }
108, 9eleq2s 2565 . . . . . . . 8  |-  ( b  e.  2o  ->  (
b  =  (/)  \/  b  =  1o ) )
11 simpr 461 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  I )  ->  a  e.  I )
12 1on 7155 . . . . . . . . . . . . . . 15  |-  1o  e.  On
1312elexi 3119 . . . . . . . . . . . . . 14  |-  1o  e.  _V
1413prid2 4141 . . . . . . . . . . . . 13  |-  1o  e.  {
(/) ,  1o }
1514, 9eleqtrri 2544 . . . . . . . . . . . 12  |-  1o  e.  2o
16 1n0 7163 . . . . . . . . . . . . . . . 16  |-  1o  =/=  (/)
17 neeq1 2738 . . . . . . . . . . . . . . . 16  |-  ( z  =  1o  ->  (
z  =/=  (/)  <->  1o  =/=  (/) ) )
1816, 17mpbiri 233 . . . . . . . . . . . . . . 15  |-  ( z  =  1o  ->  z  =/=  (/) )
19 ifnefalse 3956 . . . . . . . . . . . . . . 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 5872 . . . . . . . . . . . . . . 15  |-  ( y  =  a  ->  ( F `  y )  =  ( F `  a ) )
2221fveq2d 5876 . . . . . . . . . . . . . 14  |-  ( y  =  a  ->  ( N `  ( F `  y ) )  =  ( N `  ( F `  a )
) )
2320, 22sylan9eqr 2520 . . . . . . . . . . . . 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 5882 . . . . . . . . . . . . 13  |-  ( N `
 ( F `  a ) )  e. 
_V
2623, 24, 25ovmpt2a 6432 . . . . . . . . . . . 12  |-  ( ( a  e.  I  /\  1o  e.  2o )  -> 
( a T 1o )  =  ( N `
 ( F `  a ) ) )
2711, 15, 26sylancl 662 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  I )  ->  (
a T 1o )  =  ( N `  ( F `  a ) ) )
28 0ex 4587 . . . . . . . . . . . . . . 15  |-  (/)  e.  _V
2928prid1 4140 . . . . . . . . . . . . . 14  |-  (/)  e.  { (/)
,  1o }
3029, 9eleqtrri 2544 . . . . . . . . . . . . 13  |-  (/)  e.  2o
31 iftrue 3950 . . . . . . . . . . . . . . 15  |-  ( z  =  (/)  ->  if ( z  =  (/) ,  ( F `  y ) ,  ( N `  ( F `  y ) ) )  =  ( F `  y ) )
3231, 21sylan9eqr 2520 . . . . . . . . . . . . . 14  |-  ( ( y  =  a  /\  z  =  (/) )  ->  if ( z  =  (/) ,  ( F `  y
) ,  ( N `
 ( F `  y ) ) )  =  ( F `  a ) )
33 fvex 5882 . . . . . . . . . . . . . 14  |-  ( F `
 a )  e. 
_V
3432, 24, 33ovmpt2a 6432 . . . . . . . . . . . . 13  |-  ( ( a  e.  I  /\  (/) 
e.  2o )  -> 
( a T (/) )  =  ( F `  a ) )
3511, 30, 34sylancl 662 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  I )  ->  (
a T (/) )  =  ( F `  a
) )
3635fveq2d 5876 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  I )  ->  ( N `  ( a T (/) ) )  =  ( N `  ( F `  a )
) )
3727, 36eqtr4d 2501 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  I )  ->  (
a T 1o )  =  ( N `  ( a T (/) ) ) )
38 difeq2 3612 . . . . . . . . . . . . 13  |-  ( b  =  (/)  ->  ( 1o 
\  b )  =  ( 1o  \  (/) ) )
39 dif0 3901 . . . . . . . . . . . . 13  |-  ( 1o 
\  (/) )  =  1o
4038, 39syl6eq 2514 . . . . . . . . . . . 12  |-  ( b  =  (/)  ->  ( 1o 
\  b )  =  1o )
4140oveq2d 6312 . . . . . . . . . . 11  |-  ( b  =  (/)  ->  ( a T ( 1o  \ 
b ) )  =  ( a T 1o ) )
42 oveq2 6304 . . . . . . . . . . . 12  |-  ( b  =  (/)  ->  ( a T b )  =  ( a T (/) ) )
4342fveq2d 5876 . . . . . . . . . . 11  |-  ( b  =  (/)  ->  ( N `
 ( a T b ) )  =  ( N `  (
a T (/) ) ) )
4441, 43eqeq12d 2479 . . . . . . . . . 10  |-  ( b  =  (/)  ->  ( ( a T ( 1o 
\  b ) )  =  ( N `  ( a T b ) )  <->  ( a T 1o )  =  ( N `  ( a T (/) ) ) ) )
4537, 44syl5ibrcom 222 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  I )  ->  (
b  =  (/)  ->  (
a T ( 1o 
\  b ) )  =  ( N `  ( a T b ) ) ) )
4637fveq2d 5876 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  I )  ->  ( N `  ( a T 1o ) )  =  ( N `  ( N `  ( a T (/) ) ) ) )
47 frgpup.h . . . . . . . . . . . . 13  |-  ( ph  ->  H  e.  Grp )
4847adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  I )  ->  H  e.  Grp )
49 frgpup.a . . . . . . . . . . . . . 14  |-  ( ph  ->  F : I --> B )
5049ffvelrnda 6032 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  I )  ->  ( F `  a )  e.  B )
5135, 50eqeltrd 2545 . . . . . . . . . . . 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 16232 . . . . . . . . . . . 12  |-  ( ( H  e.  Grp  /\  ( a T (/) )  e.  B )  ->  ( N `  ( N `  ( a T (/) ) ) )  =  ( a T
(/) ) )
5548, 51, 54syl2anc 661 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  I )  ->  ( N `  ( N `  ( a T (/) ) ) )  =  ( a T (/) ) )
5646, 55eqtr2d 2499 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  I )  ->  (
a T (/) )  =  ( N `  (
a T 1o ) ) )
57 difeq2 3612 . . . . . . . . . . . . 13  |-  ( b  =  1o  ->  ( 1o  \  b )  =  ( 1o  \  1o ) )
58 difid 3899 . . . . . . . . . . . . 13  |-  ( 1o 
\  1o )  =  (/)
5957, 58syl6eq 2514 . . . . . . . . . . . 12  |-  ( b  =  1o  ->  ( 1o  \  b )  =  (/) )
6059oveq2d 6312 . . . . . . . . . . 11  |-  ( b  =  1o  ->  (
a T ( 1o 
\  b ) )  =  ( a T
(/) ) )
61 oveq2 6304 . . . . . . . . . . . 12  |-  ( b  =  1o  ->  (
a T b )  =  ( a T 1o ) )
6261fveq2d 5876 . . . . . . . . . . 11  |-  ( b  =  1o  ->  ( N `  ( a T b ) )  =  ( N `  ( a T 1o ) ) )
6360, 62eqeq12d 2479 . . . . . . . . . 10  |-  ( b  =  1o  ->  (
( a T ( 1o  \  b ) )  =  ( N `
 ( a T b ) )  <->  ( a T (/) )  =  ( N `  ( a T 1o ) ) ) )
6456, 63syl5ibrcom 222 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  I )  ->  (
b  =  1o  ->  ( a T ( 1o 
\  b ) )  =  ( N `  ( a T b ) ) ) )
6545, 64jaod 380 . . . . . . . 8  |-  ( (
ph  /\  a  e.  I )  ->  (
( b  =  (/)  \/  b  =  1o )  ->  ( a T ( 1o  \  b
) )  =  ( N `  ( a T b ) ) ) )
6610, 65syl5 32 . . . . . . 7  |-  ( (
ph  /\  a  e.  I )  ->  (
b  e.  2o  ->  ( a T ( 1o 
\  b ) )  =  ( N `  ( a T b ) ) ) )
6766impr 619 . . . . . 6  |-  ( (
ph  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( a T ( 1o  \  b ) )  =  ( N `
 ( a T b ) ) )
687, 67eqtrd 2498 . . . . 5  |-  ( (
ph  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( T `  (
a M b ) )  =  ( N `
 ( a T b ) ) )
69 fveq2 5872 . . . . . . . 8  |-  ( A  =  <. a ,  b
>.  ->  ( M `  A )  =  ( M `  <. a ,  b >. )
)
70 df-ov 6299 . . . . . . . 8  |-  ( a M b )  =  ( M `  <. a ,  b >. )
7169, 70syl6eqr 2516 . . . . . . 7  |-  ( A  =  <. a ,  b
>.  ->  ( M `  A )  =  ( a M b ) )
7271fveq2d 5876 . . . . . 6  |-  ( A  =  <. a ,  b
>.  ->  ( T `  ( M `  A ) )  =  ( T `
 ( a M b ) ) )
73 fveq2 5872 . . . . . . . 8  |-  ( A  =  <. a ,  b
>.  ->  ( T `  A )  =  ( T `  <. a ,  b >. )
)
74 df-ov 6299 . . . . . . . 8  |-  ( a T b )  =  ( T `  <. a ,  b >. )
7573, 74syl6eqr 2516 . . . . . . 7  |-  ( A  =  <. a ,  b
>.  ->  ( T `  A )  =  ( a T b ) )
7675fveq2d 5876 . . . . . 6  |-  ( A  =  <. a ,  b
>.  ->  ( N `  ( T `  A ) )  =  ( N `
 ( a T b ) ) )
7772, 76eqeq12d 2479 . . . . 5  |-  ( A  =  <. a ,  b
>.  ->  ( ( T `
 ( M `  A ) )  =  ( N `  ( T `  A )
)  <->  ( T `  ( a M b ) )  =  ( N `  ( a T b ) ) ) )
7868, 77syl5ibrcom 222 . . . 4  |-  ( (
ph  /\  ( a  e.  I  /\  b  e.  2o ) )  -> 
( A  =  <. a ,  b >.  ->  ( T `  ( M `  A ) )  =  ( N `  ( T `  A )
) ) )
7978rexlimdvva 2956 . . 3  |-  ( ph  ->  ( E. a  e.  I  E. b  e.  2o  A  =  <. a ,  b >.  ->  ( T `  ( M `  A ) )  =  ( N `  ( T `  A )
) ) )
801, 79syl5bi 217 . 2  |-  ( ph  ->  ( A  e.  ( I  X.  2o )  ->  ( T `  ( M `  A ) )  =  ( N `
 ( T `  A ) ) ) )
8180imp 429 1  |-  ( (
ph  /\  A  e.  ( I  X.  2o ) )  ->  ( T `  ( M `  A ) )  =  ( N `  ( T `  A )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   E.wrex 2808    \ cdif 3468   (/)c0 3793   ifcif 3944   {cpr 4034   <.cop 4038   Oncon0 4887    X. cxp 5006   -->wf 5590   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   1oc1o 7141   2oc2o 7142   Basecbs 14644   Grpcgrp 16180   invgcminusg 16181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1o 7148  df-2o 7149  df-0g 14859  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-grp 16184  df-minusg 16185
This theorem is referenced by:  frgpuplem  16917
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