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Theorem imasgrp 14627
Description: The image structure of a group is a group. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.)
Hypotheses
Ref Expression
imasgrp.u  |-  ( ph  ->  U  =  ( F 
"s  R ) )
imasgrp.v  |-  ( ph  ->  V  =  ( Base `  R ) )
imasgrp.p  |-  ( ph  ->  .+  =  ( +g  `  R ) )
imasgrp.f  |-  ( ph  ->  F : V -onto-> B
)
imasgrp.e  |-  ( (
ph  /\  ( a  e.  V  /\  b  e.  V )  /\  (
p  e.  V  /\  q  e.  V )
)  ->  ( (
( F `  a
)  =  ( F `
 p )  /\  ( F `  b )  =  ( F `  q ) )  -> 
( F `  (
a  .+  b )
)  =  ( F `
 ( p  .+  q ) ) ) )
imasgrp.r  |-  ( ph  ->  R  e.  Grp )
imasgrp.z  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
imasgrp  |-  ( ph  ->  ( U  e.  Grp  /\  ( F `  .0.  )  =  ( 0g `  U ) ) )
Distinct variable groups:    q, p, B    a, b, p, q,
ph    R, p, q    F, a, b, p, q    .+ , p, q    U, a, b, p, q    V, a, b, p, q    .0. , p, q
Allowed substitution hints:    B( a, b)    .+ ( a, b)    R( a, b)    .0. ( a, b)

Proof of Theorem imasgrp
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imasgrp.u . 2  |-  ( ph  ->  U  =  ( F 
"s  R ) )
2 imasgrp.v . 2  |-  ( ph  ->  V  =  ( Base `  R ) )
3 imasgrp.p . 2  |-  ( ph  ->  .+  =  ( +g  `  R ) )
4 imasgrp.f . 2  |-  ( ph  ->  F : V -onto-> B
)
5 imasgrp.e . 2  |-  ( (
ph  /\  ( a  e.  V  /\  b  e.  V )  /\  (
p  e.  V  /\  q  e.  V )
)  ->  ( (
( F `  a
)  =  ( F `
 p )  /\  ( F `  b )  =  ( F `  q ) )  -> 
( F `  (
a  .+  b )
)  =  ( F `
 ( p  .+  q ) ) ) )
6 imasgrp.r . 2  |-  ( ph  ->  R  e.  Grp )
763ad2ant1 976 . . . 4  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  R  e.  Grp )
8 simp2 956 . . . . 5  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  x  e.  V )
923ad2ant1 976 . . . . 5  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  V  =  ( Base `  R )
)
108, 9eleqtrd 2372 . . . 4  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  x  e.  ( Base `  R )
)
11 simp3 957 . . . . 5  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  y  e.  V )
1211, 9eleqtrd 2372 . . . 4  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  y  e.  ( Base `  R )
)
13 eqid 2296 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
14 eqid 2296 . . . . 5  |-  ( +g  `  R )  =  ( +g  `  R )
1513, 14grpcl 14511 . . . 4  |-  ( ( R  e.  Grp  /\  x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) )  ->  (
x ( +g  `  R
) y )  e.  ( Base `  R
) )
167, 10, 12, 15syl3anc 1182 . . 3  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  ( x
( +g  `  R ) y )  e.  (
Base `  R )
)
1733ad2ant1 976 . . . . 5  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  .+  =  ( +g  `  R ) )
1817oveqd 5891 . . . 4  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  ( x  .+  y )  =  ( x ( +g  `  R
) y ) )
1918, 9eleq12d 2364 . . 3  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  ( (
x  .+  y )  e.  V  <->  ( x ( +g  `  R ) y )  e.  (
Base `  R )
) )
2016, 19mpbird 223 . 2  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  ( x  .+  y )  e.  V
)
216adantr 451 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  R  e.  Grp )
22103adant3r3 1162 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  x  e.  ( Base `  R ) )
23123adant3r3 1162 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
y  e.  ( Base `  R ) )
24 simpr3 963 . . . . . 6  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
z  e.  V )
252adantr 451 . . . . . 6  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  V  =  ( Base `  R ) )
2624, 25eleqtrd 2372 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
z  e.  ( Base `  R ) )
2713, 14grpass 14512 . . . . 5  |-  ( ( R  e.  Grp  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R
) ) )  -> 
( ( x ( +g  `  R ) y ) ( +g  `  R ) z )  =  ( x ( +g  `  R ) ( y ( +g  `  R ) z ) ) )
2821, 22, 23, 26, 27syl13anc 1184 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( ( x ( +g  `  R ) y ) ( +g  `  R ) z )  =  ( x ( +g  `  R ) ( y ( +g  `  R ) z ) ) )
293adantr 451 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  .+  =  ( +g  `  R ) )
30183adant3r3 1162 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( x  .+  y
)  =  ( x ( +g  `  R
) y ) )
31 eqidd 2297 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
z  =  z )
3229, 30, 31oveq123d 5895 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( ( x  .+  y )  .+  z
)  =  ( ( x ( +g  `  R
) y ) ( +g  `  R ) z ) )
33 eqidd 2297 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  x  =  x )
3429oveqd 5891 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( y  .+  z
)  =  ( y ( +g  `  R
) z ) )
3529, 33, 34oveq123d 5895 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( x  .+  (
y  .+  z )
)  =  ( x ( +g  `  R
) ( y ( +g  `  R ) z ) ) )
3628, 32, 353eqtr4d 2338 . . 3  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( ( x  .+  y )  .+  z
)  =  ( x 
.+  ( y  .+  z ) ) )
3736fveq2d 5545 . 2  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( F `  (
( x  .+  y
)  .+  z )
)  =  ( F `
 ( x  .+  ( y  .+  z
) ) ) )
38 imasgrp.z . . . . 5  |-  .0.  =  ( 0g `  R )
3913, 38grpidcl 14526 . . . 4  |-  ( R  e.  Grp  ->  .0.  e.  ( Base `  R
) )
406, 39syl 15 . . 3  |-  ( ph  ->  .0.  e.  ( Base `  R ) )
4140, 2eleqtrrd 2373 . 2  |-  ( ph  ->  .0.  e.  V )
423adantr 451 . . . . 5  |-  ( (
ph  /\  x  e.  V )  ->  .+  =  ( +g  `  R ) )
4342oveqd 5891 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (  .0.  .+  x )  =  (  .0.  ( +g  `  R ) x ) )
446adantr 451 . . . . 5  |-  ( (
ph  /\  x  e.  V )  ->  R  e.  Grp )
452eleq2d 2363 . . . . . 6  |-  ( ph  ->  ( x  e.  V  <->  x  e.  ( Base `  R
) ) )
4645biimpa 470 . . . . 5  |-  ( (
ph  /\  x  e.  V )  ->  x  e.  ( Base `  R
) )
4713, 14, 38grplid 14528 . . . . 5  |-  ( ( R  e.  Grp  /\  x  e.  ( Base `  R ) )  -> 
(  .0.  ( +g  `  R ) x )  =  x )
4844, 46, 47syl2anc 642 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (  .0.  ( +g  `  R
) x )  =  x )
4943, 48eqtrd 2328 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  (  .0.  .+  x )  =  x )
5049fveq2d 5545 . 2  |-  ( (
ph  /\  x  e.  V )  ->  ( F `  (  .0.  .+  x ) )  =  ( F `  x
) )
51 eqid 2296 . . . . 5  |-  ( inv g `  R )  =  ( inv g `  R )
5213, 51grpinvcl 14543 . . . 4  |-  ( ( R  e.  Grp  /\  x  e.  ( Base `  R ) )  -> 
( ( inv g `  R ) `  x
)  e.  ( Base `  R ) )
5344, 46, 52syl2anc 642 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  (
( inv g `  R ) `  x
)  e.  ( Base `  R ) )
542adantr 451 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  V  =  ( Base `  R
) )
5553, 54eleqtrrd 2373 . 2  |-  ( (
ph  /\  x  e.  V )  ->  (
( inv g `  R ) `  x
)  e.  V )
5642oveqd 5891 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (
( ( inv g `  R ) `  x
)  .+  x )  =  ( ( ( inv g `  R
) `  x )
( +g  `  R ) x ) )
5713, 14, 38, 51grplinv 14544 . . . . 5  |-  ( ( R  e.  Grp  /\  x  e.  ( Base `  R ) )  -> 
( ( ( inv g `  R ) `
 x ) ( +g  `  R ) x )  =  .0.  )
5844, 46, 57syl2anc 642 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (
( ( inv g `  R ) `  x
) ( +g  `  R
) x )  =  .0.  )
5956, 58eqtrd 2328 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  (
( ( inv g `  R ) `  x
)  .+  x )  =  .0.  )
6059fveq2d 5545 . 2  |-  ( (
ph  /\  x  e.  V )  ->  ( F `  ( (
( inv g `  R ) `  x
)  .+  x )
)  =  ( F `
 .0.  ) )
611, 2, 3, 4, 5, 6, 20, 37, 41, 50, 55, 60imasgrp2 14626 1  |-  ( ph  ->  ( U  e.  Grp  /\  ( F `  .0.  )  =  ( 0g `  U ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   -onto->wfo 5269   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   0gc0g 13416    "s cimas 13423   Grpcgrp 14378   inv gcminusg 14379
This theorem is referenced by:  imasgrpf1  14628  imasrng  15418  imasgim  27367
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  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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-nel 2462  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-int 3879  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-lim 4413  df-suc 4414  df-om 4673  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-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-0g 13420  df-imas 13427  df-mnd 14383  df-grp 14505  df-minusg 14506
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