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Theorem imasgrp 14889
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 978 . . . 4  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  R  e.  Grp )
8 simp2 958 . . . . 5  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  x  e.  V )
923ad2ant1 978 . . . . 5  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  V  =  ( Base `  R )
)
108, 9eleqtrd 2480 . . . 4  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  x  e.  ( Base `  R )
)
11 simp3 959 . . . . 5  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  y  e.  V )
1211, 9eleqtrd 2480 . . . 4  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  y  e.  ( Base `  R )
)
13 eqid 2404 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
14 eqid 2404 . . . . 5  |-  ( +g  `  R )  =  ( +g  `  R )
1513, 14grpcl 14773 . . . 4  |-  ( ( R  e.  Grp  /\  x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) )  ->  (
x ( +g  `  R
) y )  e.  ( Base `  R
) )
167, 10, 12, 15syl3anc 1184 . . 3  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  ( x
( +g  `  R ) y )  e.  (
Base `  R )
)
1733ad2ant1 978 . . . 4  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  .+  =  ( +g  `  R ) )
1817oveqd 6057 . . 3  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  ( x  .+  y )  =  ( x ( +g  `  R
) y ) )
1916, 18, 93eltr4d 2485 . 2  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  ( x  .+  y )  e.  V
)
206adantr 452 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  R  e.  Grp )
21103adant3r3 1164 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  x  e.  ( Base `  R ) )
22123adant3r3 1164 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
y  e.  ( Base `  R ) )
23 simpr3 965 . . . . . 6  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
z  e.  V )
242adantr 452 . . . . . 6  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  V  =  ( Base `  R ) )
2523, 24eleqtrd 2480 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
z  e.  ( Base `  R ) )
2613, 14grpass 14774 . . . . 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 ) ) )
2720, 21, 22, 25, 26syl13anc 1186 . . . 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 ) ) )
283adantr 452 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  .+  =  ( +g  `  R ) )
29183adant3r3 1164 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( x  .+  y
)  =  ( x ( +g  `  R
) y ) )
30 eqidd 2405 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
z  =  z )
3128, 29, 30oveq123d 6061 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( ( x  .+  y )  .+  z
)  =  ( ( x ( +g  `  R
) y ) ( +g  `  R ) z ) )
32 eqidd 2405 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  x  =  x )
3328oveqd 6057 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( y  .+  z
)  =  ( y ( +g  `  R
) z ) )
3428, 32, 33oveq123d 6061 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( x  .+  (
y  .+  z )
)  =  ( x ( +g  `  R
) ( y ( +g  `  R ) z ) ) )
3527, 31, 343eqtr4d 2446 . . 3  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( ( x  .+  y )  .+  z
)  =  ( x 
.+  ( y  .+  z ) ) )
3635fveq2d 5691 . 2  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( F `  (
( x  .+  y
)  .+  z )
)  =  ( F `
 ( x  .+  ( y  .+  z
) ) ) )
37 imasgrp.z . . . . 5  |-  .0.  =  ( 0g `  R )
3813, 37grpidcl 14788 . . . 4  |-  ( R  e.  Grp  ->  .0.  e.  ( Base `  R
) )
396, 38syl 16 . . 3  |-  ( ph  ->  .0.  e.  ( Base `  R ) )
4039, 2eleqtrrd 2481 . 2  |-  ( ph  ->  .0.  e.  V )
413adantr 452 . . . . 5  |-  ( (
ph  /\  x  e.  V )  ->  .+  =  ( +g  `  R ) )
4241oveqd 6057 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (  .0.  .+  x )  =  (  .0.  ( +g  `  R ) x ) )
436adantr 452 . . . . 5  |-  ( (
ph  /\  x  e.  V )  ->  R  e.  Grp )
442eleq2d 2471 . . . . . 6  |-  ( ph  ->  ( x  e.  V  <->  x  e.  ( Base `  R
) ) )
4544biimpa 471 . . . . 5  |-  ( (
ph  /\  x  e.  V )  ->  x  e.  ( Base `  R
) )
4613, 14, 37grplid 14790 . . . . 5  |-  ( ( R  e.  Grp  /\  x  e.  ( Base `  R ) )  -> 
(  .0.  ( +g  `  R ) x )  =  x )
4743, 45, 46syl2anc 643 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (  .0.  ( +g  `  R
) x )  =  x )
4842, 47eqtrd 2436 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  (  .0.  .+  x )  =  x )
4948fveq2d 5691 . 2  |-  ( (
ph  /\  x  e.  V )  ->  ( F `  (  .0.  .+  x ) )  =  ( F `  x
) )
50 eqid 2404 . . . . 5  |-  ( inv g `  R )  =  ( inv g `  R )
5113, 50grpinvcl 14805 . . . 4  |-  ( ( R  e.  Grp  /\  x  e.  ( Base `  R ) )  -> 
( ( inv g `  R ) `  x
)  e.  ( Base `  R ) )
5243, 45, 51syl2anc 643 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  (
( inv g `  R ) `  x
)  e.  ( Base `  R ) )
532adantr 452 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  V  =  ( Base `  R
) )
5452, 53eleqtrrd 2481 . 2  |-  ( (
ph  /\  x  e.  V )  ->  (
( inv g `  R ) `  x
)  e.  V )
5541oveqd 6057 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (
( ( inv g `  R ) `  x
)  .+  x )  =  ( ( ( inv g `  R
) `  x )
( +g  `  R ) x ) )
5613, 14, 37, 50grplinv 14806 . . . . 5  |-  ( ( R  e.  Grp  /\  x  e.  ( Base `  R ) )  -> 
( ( ( inv g `  R ) `
 x ) ( +g  `  R ) x )  =  .0.  )
5743, 45, 56syl2anc 643 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (
( ( inv g `  R ) `  x
) ( +g  `  R
) x )  =  .0.  )
5855, 57eqtrd 2436 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  (
( ( inv g `  R ) `  x
)  .+  x )  =  .0.  )
5958fveq2d 5691 . 2  |-  ( (
ph  /\  x  e.  V )  ->  ( F `  ( (
( inv g `  R ) `  x
)  .+  x )
)  =  ( F `
 .0.  ) )
601, 2, 3, 4, 5, 6, 19, 36, 40, 49, 54, 59imasgrp2 14888 1  |-  ( ph  ->  ( U  e.  Grp  /\  ( F `  .0.  )  =  ( 0g `  U ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   -onto->wfo 5411   ` cfv 5413  (class class class)co 6040   Basecbs 13424   +g cplusg 13484   0gc0g 13678    "s cimas 13685   Grpcgrp 14640   inv gcminusg 14641
This theorem is referenced by:  imasgrpf1  14890  imasrng  15680  imasgim  27132
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-plusg 13497  df-mulr 13498  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-0g 13682  df-imas 13689  df-mnd 14645  df-grp 14767  df-minusg 14768
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