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Theorem imasgrp 14935
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 979 . . . 4  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  R  e.  Grp )
8 simp2 959 . . . . 5  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  x  e.  V )
923ad2ant1 979 . . . . 5  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  V  =  ( Base `  R )
)
108, 9eleqtrd 2513 . . . 4  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  x  e.  ( Base `  R )
)
11 simp3 960 . . . . 5  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  y  e.  V )
1211, 9eleqtrd 2513 . . . 4  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  y  e.  ( Base `  R )
)
13 eqid 2437 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
14 eqid 2437 . . . . 5  |-  ( +g  `  R )  =  ( +g  `  R )
1513, 14grpcl 14819 . . . 4  |-  ( ( R  e.  Grp  /\  x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) )  ->  (
x ( +g  `  R
) y )  e.  ( Base `  R
) )
167, 10, 12, 15syl3anc 1185 . . 3  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  ( x
( +g  `  R ) y )  e.  (
Base `  R )
)
1733ad2ant1 979 . . . 4  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  .+  =  ( +g  `  R ) )
1817oveqd 6099 . . 3  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  ( x  .+  y )  =  ( x ( +g  `  R
) y ) )
1916, 18, 93eltr4d 2518 . 2  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  ( x  .+  y )  e.  V
)
206adantr 453 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  R  e.  Grp )
21103adant3r3 1165 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  x  e.  ( Base `  R ) )
22123adant3r3 1165 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
y  e.  ( Base `  R ) )
23 simpr3 966 . . . . . 6  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
z  e.  V )
242adantr 453 . . . . . 6  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  V  =  ( Base `  R ) )
2523, 24eleqtrd 2513 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
z  e.  ( Base `  R ) )
2613, 14grpass 14820 . . . . 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 1187 . . . 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 453 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  .+  =  ( +g  `  R ) )
29183adant3r3 1165 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( x  .+  y
)  =  ( x ( +g  `  R
) y ) )
30 eqidd 2438 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
z  =  z )
3128, 29, 30oveq123d 6103 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( ( x  .+  y )  .+  z
)  =  ( ( x ( +g  `  R
) y ) ( +g  `  R ) z ) )
32 eqidd 2438 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  x  =  x )
3328oveqd 6099 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( y  .+  z
)  =  ( y ( +g  `  R
) z ) )
3428, 32, 33oveq123d 6103 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( x  .+  (
y  .+  z )
)  =  ( x ( +g  `  R
) ( y ( +g  `  R ) z ) ) )
3527, 31, 343eqtr4d 2479 . . 3  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( ( x  .+  y )  .+  z
)  =  ( x 
.+  ( y  .+  z ) ) )
3635fveq2d 5733 . 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 14834 . . . 4  |-  ( R  e.  Grp  ->  .0.  e.  ( Base `  R
) )
396, 38syl 16 . . 3  |-  ( ph  ->  .0.  e.  ( Base `  R ) )
4039, 2eleqtrrd 2514 . 2  |-  ( ph  ->  .0.  e.  V )
413adantr 453 . . . . 5  |-  ( (
ph  /\  x  e.  V )  ->  .+  =  ( +g  `  R ) )
4241oveqd 6099 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (  .0.  .+  x )  =  (  .0.  ( +g  `  R ) x ) )
436adantr 453 . . . . 5  |-  ( (
ph  /\  x  e.  V )  ->  R  e.  Grp )
442eleq2d 2504 . . . . . 6  |-  ( ph  ->  ( x  e.  V  <->  x  e.  ( Base `  R
) ) )
4544biimpa 472 . . . . 5  |-  ( (
ph  /\  x  e.  V )  ->  x  e.  ( Base `  R
) )
4613, 14, 37grplid 14836 . . . . 5  |-  ( ( R  e.  Grp  /\  x  e.  ( Base `  R ) )  -> 
(  .0.  ( +g  `  R ) x )  =  x )
4743, 45, 46syl2anc 644 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (  .0.  ( +g  `  R
) x )  =  x )
4842, 47eqtrd 2469 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  (  .0.  .+  x )  =  x )
4948fveq2d 5733 . 2  |-  ( (
ph  /\  x  e.  V )  ->  ( F `  (  .0.  .+  x ) )  =  ( F `  x
) )
50 eqid 2437 . . . . 5  |-  ( inv g `  R )  =  ( inv g `  R )
5113, 50grpinvcl 14851 . . . 4  |-  ( ( R  e.  Grp  /\  x  e.  ( Base `  R ) )  -> 
( ( inv g `  R ) `  x
)  e.  ( Base `  R ) )
5243, 45, 51syl2anc 644 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  (
( inv g `  R ) `  x
)  e.  ( Base `  R ) )
532adantr 453 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  V  =  ( Base `  R
) )
5452, 53eleqtrrd 2514 . 2  |-  ( (
ph  /\  x  e.  V )  ->  (
( inv g `  R ) `  x
)  e.  V )
5541oveqd 6099 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (
( ( inv g `  R ) `  x
)  .+  x )  =  ( ( ( inv g `  R
) `  x )
( +g  `  R ) x ) )
5613, 14, 37, 50grplinv 14852 . . . . 5  |-  ( ( R  e.  Grp  /\  x  e.  ( Base `  R ) )  -> 
( ( ( inv g `  R ) `
 x ) ( +g  `  R ) x )  =  .0.  )
5743, 45, 56syl2anc 644 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (
( ( inv g `  R ) `  x
) ( +g  `  R
) x )  =  .0.  )
5855, 57eqtrd 2469 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  (
( ( inv g `  R ) `  x
)  .+  x )  =  .0.  )
5958fveq2d 5733 . 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 14934 1  |-  ( ph  ->  ( U  e.  Grp  /\  ( F `  .0.  )  =  ( 0g `  U ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   -onto->wfo 5453   ` cfv 5455  (class class class)co 6082   Basecbs 13470   +g cplusg 13530   0gc0g 13724    "s cimas 13731   Grpcgrp 14686   inv gcminusg 14687
This theorem is referenced by:  imasgrpf1  14936  imasrng  15726  imasgim  27242
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-oadd 6729  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-sup 7447  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-nn 10002  df-2 10059  df-3 10060  df-4 10061  df-5 10062  df-6 10063  df-7 10064  df-8 10065  df-9 10066  df-10 10067  df-n0 10223  df-z 10284  df-dec 10384  df-uz 10490  df-fz 11045  df-struct 13472  df-ndx 13473  df-slot 13474  df-base 13475  df-plusg 13543  df-mulr 13544  df-sca 13546  df-vsca 13547  df-tset 13549  df-ple 13550  df-ds 13552  df-0g 13728  df-imas 13735  df-mnd 14691  df-grp 14813  df-minusg 14814
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