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Theorem imasgrp 16744
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 1026 . . . 4  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  R  e.  Grp )
8 simp2 1006 . . . . 5  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  x  e.  V )
923ad2ant1 1026 . . . . 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 1007 . . . . 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 2429 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
14 eqid 2429 . . . . 5  |-  ( +g  `  R )  =  ( +g  `  R )
1513, 14grpcl 16621 . . . 4  |-  ( ( R  e.  Grp  /\  x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) )  ->  (
x ( +g  `  R
) y )  e.  ( Base `  R
) )
167, 10, 12, 15syl3anc 1264 . . 3  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  ( x
( +g  `  R ) y )  e.  (
Base `  R )
)
1733ad2ant1 1026 . . . 4  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  .+  =  ( +g  `  R ) )
1817oveqd 6322 . . 3  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  ( x  .+  y )  =  ( x ( +g  `  R
) y ) )
1916, 18, 93eltr4d 2532 . 2  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  ( x  .+  y )  e.  V
)
206adantr 466 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  R  e.  Grp )
21103adant3r3 1216 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  x  e.  ( Base `  R ) )
22123adant3r3 1216 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
y  e.  ( Base `  R ) )
23 simpr3 1013 . . . . . 6  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
z  e.  V )
242adantr 466 . . . . . 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 16622 . . . . 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 1266 . . . 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 466 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  .+  =  ( +g  `  R ) )
29183adant3r3 1216 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( x  .+  y
)  =  ( x ( +g  `  R
) y ) )
30 eqidd 2430 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
z  =  z )
3128, 29, 30oveq123d 6326 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( ( x  .+  y )  .+  z
)  =  ( ( x ( +g  `  R
) y ) ( +g  `  R ) z ) )
32 eqidd 2430 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  x  =  x )
3328oveqd 6322 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( y  .+  z
)  =  ( y ( +g  `  R
) z ) )
3428, 32, 33oveq123d 6326 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( x  .+  (
y  .+  z )
)  =  ( x ( +g  `  R
) ( y ( +g  `  R ) z ) ) )
3527, 31, 343eqtr4d 2480 . . 3  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( ( x  .+  y )  .+  z
)  =  ( x 
.+  ( y  .+  z ) ) )
3635fveq2d 5885 . 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 16636 . . . 4  |-  ( R  e.  Grp  ->  .0.  e.  ( Base `  R
) )
396, 38syl 17 . . 3  |-  ( ph  ->  .0.  e.  ( Base `  R ) )
4039, 2eleqtrrd 2520 . 2  |-  ( ph  ->  .0.  e.  V )
413adantr 466 . . . . 5  |-  ( (
ph  /\  x  e.  V )  ->  .+  =  ( +g  `  R ) )
4241oveqd 6322 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (  .0.  .+  x )  =  (  .0.  ( +g  `  R ) x ) )
436adantr 466 . . . . 5  |-  ( (
ph  /\  x  e.  V )  ->  R  e.  Grp )
442eleq2d 2499 . . . . . 6  |-  ( ph  ->  ( x  e.  V  <->  x  e.  ( Base `  R
) ) )
4544biimpa 486 . . . . 5  |-  ( (
ph  /\  x  e.  V )  ->  x  e.  ( Base `  R
) )
4613, 14, 37grplid 16638 . . . . 5  |-  ( ( R  e.  Grp  /\  x  e.  ( Base `  R ) )  -> 
(  .0.  ( +g  `  R ) x )  =  x )
4743, 45, 46syl2anc 665 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (  .0.  ( +g  `  R
) x )  =  x )
4842, 47eqtrd 2470 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  (  .0.  .+  x )  =  x )
4948fveq2d 5885 . 2  |-  ( (
ph  /\  x  e.  V )  ->  ( F `  (  .0.  .+  x ) )  =  ( F `  x
) )
50 eqid 2429 . . . . 5  |-  ( invg `  R )  =  ( invg `  R )
5113, 50grpinvcl 16653 . . . 4  |-  ( ( R  e.  Grp  /\  x  e.  ( Base `  R ) )  -> 
( ( invg `  R ) `  x
)  e.  ( Base `  R ) )
5243, 45, 51syl2anc 665 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  (
( invg `  R ) `  x
)  e.  ( Base `  R ) )
532adantr 466 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  V  =  ( Base `  R
) )
5452, 53eleqtrrd 2520 . 2  |-  ( (
ph  /\  x  e.  V )  ->  (
( invg `  R ) `  x
)  e.  V )
5541oveqd 6322 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (
( ( invg `  R ) `  x
)  .+  x )  =  ( ( ( invg `  R
) `  x )
( +g  `  R ) x ) )
5613, 14, 37, 50grplinv 16654 . . . . 5  |-  ( ( R  e.  Grp  /\  x  e.  ( Base `  R ) )  -> 
( ( ( invg `  R ) `
 x ) ( +g  `  R ) x )  =  .0.  )
5743, 45, 56syl2anc 665 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (
( ( invg `  R ) `  x
) ( +g  `  R
) x )  =  .0.  )
5855, 57eqtrd 2470 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  (
( ( invg `  R ) `  x
)  .+  x )  =  .0.  )
5958fveq2d 5885 . 2  |-  ( (
ph  /\  x  e.  V )  ->  ( F `  ( (
( invg `  R ) `  x
)  .+  x )
)  =  ( F `
 .0.  ) )
601, 2, 3, 4, 5, 6, 19, 36, 40, 49, 54, 59imasgrp2 16743 1  |-  ( ph  ->  ( U  e.  Grp  /\  ( F `  .0.  )  =  ( 0g `  U ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   -onto->wfo 5599   ` cfv 5601  (class class class)co 6305   Basecbs 15075   +g cplusg 15143   0gc0g 15288    "s cimas 15352   Grpcgrp 16611   invgcminusg 16612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-fz 11783  df-struct 15077  df-ndx 15078  df-slot 15079  df-base 15080  df-plusg 15156  df-mulr 15157  df-sca 15159  df-vsca 15160  df-ip 15161  df-tset 15162  df-ple 15163  df-ds 15165  df-0g 15290  df-imas 15356  df-mgm 16430  df-sgrp 16469  df-mnd 16479  df-grp 16615  df-minusg 16616
This theorem is referenced by:  imasgrpf1  16745  imasring  17773  imasgim  35654
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