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Theorem imasgrp 15676
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 1009 . . . 4  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  R  e.  Grp )
8 simp2 989 . . . . 5  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  x  e.  V )
923ad2ant1 1009 . . . . 5  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  V  =  ( Base `  R )
)
108, 9eleqtrd 2519 . . . 4  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  x  e.  ( Base `  R )
)
11 simp3 990 . . . . 5  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  y  e.  V )
1211, 9eleqtrd 2519 . . . 4  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  y  e.  ( Base `  R )
)
13 eqid 2443 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
14 eqid 2443 . . . . 5  |-  ( +g  `  R )  =  ( +g  `  R )
1513, 14grpcl 15556 . . . 4  |-  ( ( R  e.  Grp  /\  x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) )  ->  (
x ( +g  `  R
) y )  e.  ( Base `  R
) )
167, 10, 12, 15syl3anc 1218 . . 3  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  ( x
( +g  `  R ) y )  e.  (
Base `  R )
)
1733ad2ant1 1009 . . . 4  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  .+  =  ( +g  `  R ) )
1817oveqd 6113 . . 3  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  ( x  .+  y )  =  ( x ( +g  `  R
) y ) )
1916, 18, 93eltr4d 2524 . 2  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  ( x  .+  y )  e.  V
)
206adantr 465 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  R  e.  Grp )
21103adant3r3 1198 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  x  e.  ( Base `  R ) )
22123adant3r3 1198 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
y  e.  ( Base `  R ) )
23 simpr3 996 . . . . . 6  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
z  e.  V )
242adantr 465 . . . . . 6  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  V  =  ( Base `  R ) )
2523, 24eleqtrd 2519 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
z  e.  ( Base `  R ) )
2613, 14grpass 15557 . . . . 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 1220 . . . 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 465 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  .+  =  ( +g  `  R ) )
29183adant3r3 1198 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( x  .+  y
)  =  ( x ( +g  `  R
) y ) )
30 eqidd 2444 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
z  =  z )
3128, 29, 30oveq123d 6117 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( ( x  .+  y )  .+  z
)  =  ( ( x ( +g  `  R
) y ) ( +g  `  R ) z ) )
32 eqidd 2444 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  x  =  x )
3328oveqd 6113 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( y  .+  z
)  =  ( y ( +g  `  R
) z ) )
3428, 32, 33oveq123d 6117 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( x  .+  (
y  .+  z )
)  =  ( x ( +g  `  R
) ( y ( +g  `  R ) z ) ) )
3527, 31, 343eqtr4d 2485 . . 3  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( ( x  .+  y )  .+  z
)  =  ( x 
.+  ( y  .+  z ) ) )
3635fveq2d 5700 . 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 15571 . . . 4  |-  ( R  e.  Grp  ->  .0.  e.  ( Base `  R
) )
396, 38syl 16 . . 3  |-  ( ph  ->  .0.  e.  ( Base `  R ) )
4039, 2eleqtrrd 2520 . 2  |-  ( ph  ->  .0.  e.  V )
413adantr 465 . . . . 5  |-  ( (
ph  /\  x  e.  V )  ->  .+  =  ( +g  `  R ) )
4241oveqd 6113 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (  .0.  .+  x )  =  (  .0.  ( +g  `  R ) x ) )
436adantr 465 . . . . 5  |-  ( (
ph  /\  x  e.  V )  ->  R  e.  Grp )
442eleq2d 2510 . . . . . 6  |-  ( ph  ->  ( x  e.  V  <->  x  e.  ( Base `  R
) ) )
4544biimpa 484 . . . . 5  |-  ( (
ph  /\  x  e.  V )  ->  x  e.  ( Base `  R
) )
4613, 14, 37grplid 15573 . . . . 5  |-  ( ( R  e.  Grp  /\  x  e.  ( Base `  R ) )  -> 
(  .0.  ( +g  `  R ) x )  =  x )
4743, 45, 46syl2anc 661 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (  .0.  ( +g  `  R
) x )  =  x )
4842, 47eqtrd 2475 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  (  .0.  .+  x )  =  x )
4948fveq2d 5700 . 2  |-  ( (
ph  /\  x  e.  V )  ->  ( F `  (  .0.  .+  x ) )  =  ( F `  x
) )
50 eqid 2443 . . . . 5  |-  ( invg `  R )  =  ( invg `  R )
5113, 50grpinvcl 15588 . . . 4  |-  ( ( R  e.  Grp  /\  x  e.  ( Base `  R ) )  -> 
( ( invg `  R ) `  x
)  e.  ( Base `  R ) )
5243, 45, 51syl2anc 661 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  (
( invg `  R ) `  x
)  e.  ( Base `  R ) )
532adantr 465 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  V  =  ( Base `  R
) )
5452, 53eleqtrrd 2520 . 2  |-  ( (
ph  /\  x  e.  V )  ->  (
( invg `  R ) `  x
)  e.  V )
5541oveqd 6113 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (
( ( invg `  R ) `  x
)  .+  x )  =  ( ( ( invg `  R
) `  x )
( +g  `  R ) x ) )
5613, 14, 37, 50grplinv 15589 . . . . 5  |-  ( ( R  e.  Grp  /\  x  e.  ( Base `  R ) )  -> 
( ( ( invg `  R ) `
 x ) ( +g  `  R ) x )  =  .0.  )
5743, 45, 56syl2anc 661 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (
( ( invg `  R ) `  x
) ( +g  `  R
) x )  =  .0.  )
5855, 57eqtrd 2475 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  (
( ( invg `  R ) `  x
)  .+  x )  =  .0.  )
5958fveq2d 5700 . 2  |-  ( (
ph  /\  x  e.  V )  ->  ( F `  ( (
( invg `  R ) `  x
)  .+  x )
)  =  ( F `
 .0.  ) )
601, 2, 3, 4, 5, 6, 19, 36, 40, 49, 54, 59imasgrp2 15675 1  |-  ( ph  ->  ( U  e.  Grp  /\  ( F `  .0.  )  =  ( 0g `  U ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   -onto->wfo 5421   ` cfv 5423  (class class class)co 6096   Basecbs 14179   +g cplusg 14243   0gc0g 14383    "s cimas 14447   Grpcgrp 15415   invgcminusg 15416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-fz 11443  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-plusg 14256  df-mulr 14257  df-sca 14259  df-vsca 14260  df-ip 14261  df-tset 14262  df-ple 14263  df-ds 14265  df-0g 14385  df-imas 14451  df-mnd 15420  df-grp 15550  df-minusg 15551
This theorem is referenced by:  imasgrpf1  15677  imasrng  16716  imasgim  29460
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