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Theorem divsgrp2 14823
Description: Prove that a quotient structure is a group. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
divsgrp2.u  |-  ( ph  ->  U  =  ( R 
/.s  .~  ) )
divsgrp2.v  |-  ( ph  ->  V  =  ( Base `  R ) )
divsgrp2.p  |-  ( ph  ->  .+  =  ( +g  `  R ) )
divsgrp2.r  |-  ( ph  ->  .~  Er  V )
divsgrp2.x  |-  ( ph  ->  R  e.  X )
divsgrp2.e  |-  ( ph  ->  ( ( a  .~  p  /\  b  .~  q
)  ->  ( a  .+  b )  .~  (
p  .+  q )
) )
divsgrp2.1  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  ( x  .+  y )  e.  V
)
divsgrp2.2  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( ( x  .+  y )  .+  z
)  .~  ( x  .+  ( y  .+  z
) ) )
divsgrp2.3  |-  ( ph  ->  .0.  e.  V )
divsgrp2.4  |-  ( (
ph  /\  x  e.  V )  ->  (  .0.  .+  x )  .~  x )
divsgrp2.5  |-  ( (
ph  /\  x  e.  V )  ->  N  e.  V )
divsgrp2.6  |-  ( (
ph  /\  x  e.  V )  ->  ( N  .+  x )  .~  .0.  )
Assertion
Ref Expression
divsgrp2  |-  ( ph  ->  ( U  e.  Grp  /\ 
[  .0.  ]  .~  =  ( 0g `  U ) ) )
Distinct variable groups:    a, b, p, q, x, y, z, 
.~    .0. , a, b, p, q, x    N, p    R, p, q    .+ , a,
b, p, q, x, y    ph, a, b, p, q, x, y, z    V, a, b, p, q, x, y, z    U, a, b, p, q, x, y, z
Allowed substitution hints:    .+ ( z)    R( x, y, z, a, b)    N( x, y, z, q, a, b)    X( x, y, z, q, p, a, b)    .0. ( y,
z)

Proof of Theorem divsgrp2
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 divsgrp2.u . . . 4  |-  ( ph  ->  U  =  ( R 
/.s  .~  ) )
2 divsgrp2.v . . . 4  |-  ( ph  ->  V  =  ( Base `  R ) )
3 eqid 2366 . . . 4  |-  ( u  e.  V  |->  [ u ]  .~  )  =  ( u  e.  V  |->  [ u ]  .~  )
4 divsgrp2.r . . . . 5  |-  ( ph  ->  .~  Er  V )
5 fvex 5646 . . . . . 6  |-  ( Base `  R )  e.  _V
62, 5syl6eqel 2454 . . . . 5  |-  ( ph  ->  V  e.  _V )
7 erex 6826 . . . . 5  |-  (  .~  Er  V  ->  ( V  e.  _V  ->  .~  e.  _V ) )
84, 6, 7sylc 56 . . . 4  |-  ( ph  ->  .~  e.  _V )
9 divsgrp2.x . . . 4  |-  ( ph  ->  R  e.  X )
101, 2, 3, 8, 9divsval 13654 . . 3  |-  ( ph  ->  U  =  ( ( u  e.  V  |->  [ u ]  .~  )  "s  R ) )
11 divsgrp2.p . . 3  |-  ( ph  ->  .+  =  ( +g  `  R ) )
121, 2, 3, 8, 9divslem 13655 . . 3  |-  ( ph  ->  ( u  e.  V  |->  [ u ]  .~  ) : V -onto-> ( V /.  .~  ) )
13 divsgrp2.1 . . . . 5  |-  ( (
ph  /\  x  e.  V  /\  y  e.  V
)  ->  ( x  .+  y )  e.  V
)
14133expb 1153 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V ) )  -> 
( x  .+  y
)  e.  V )
15 divsgrp2.e . . . 4  |-  ( ph  ->  ( ( a  .~  p  /\  b  .~  q
)  ->  ( a  .+  b )  .~  (
p  .+  q )
) )
164, 6, 3, 14, 15ercpbl 13661 . . 3  |-  ( (
ph  /\  ( a  e.  V  /\  b  e.  V )  /\  (
p  e.  V  /\  q  e.  V )
)  ->  ( (
( ( u  e.  V  |->  [ u ]  .~  ) `  a )  =  ( ( u  e.  V  |->  [ u ]  .~  ) `  p
)  /\  ( (
u  e.  V  |->  [ u ]  .~  ) `  b )  =  ( ( u  e.  V  |->  [ u ]  .~  ) `  q )
)  ->  ( (
u  e.  V  |->  [ u ]  .~  ) `  ( a  .+  b
) )  =  ( ( u  e.  V  |->  [ u ]  .~  ) `  ( p  .+  q ) ) ) )
174adantr 451 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  .~  Er  V )
18 divsgrp2.2 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( ( x  .+  y )  .+  z
)  .~  ( x  .+  ( y  .+  z
) ) )
1917, 18erthi 6848 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  [ ( ( x 
.+  y )  .+  z ) ]  .~  =  [ ( x  .+  ( y  .+  z
) ) ]  .~  )
206adantr 451 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  V  e.  _V )
2117, 20, 3divsfval 13659 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( ( u  e.  V  |->  [ u ]  .~  ) `  ( ( x  .+  y ) 
.+  z ) )  =  [ ( ( x  .+  y ) 
.+  z ) ]  .~  )
2217, 20, 3divsfval 13659 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( ( u  e.  V  |->  [ u ]  .~  ) `  ( x 
.+  ( y  .+  z ) ) )  =  [ ( x 
.+  ( y  .+  z ) ) ]  .~  )
2319, 21, 223eqtr4d 2408 . . 3  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( ( u  e.  V  |->  [ u ]  .~  ) `  ( ( x  .+  y ) 
.+  z ) )  =  ( ( u  e.  V  |->  [ u ]  .~  ) `  (
x  .+  ( y  .+  z ) ) ) )
24 divsgrp2.3 . . 3  |-  ( ph  ->  .0.  e.  V )
254adantr 451 . . . . 5  |-  ( (
ph  /\  x  e.  V )  ->  .~  Er  V )
26 divsgrp2.4 . . . . 5  |-  ( (
ph  /\  x  e.  V )  ->  (  .0.  .+  x )  .~  x )
2725, 26erthi 6848 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  [ (  .0.  .+  x ) ]  .~  =  [ x ]  .~  )
286adantr 451 . . . . 5  |-  ( (
ph  /\  x  e.  V )  ->  V  e.  _V )
2925, 28, 3divsfval 13659 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (
( u  e.  V  |->  [ u ]  .~  ) `  (  .0.  .+  x ) )  =  [ (  .0.  .+  x ) ]  .~  )
3025, 28, 3divsfval 13659 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (
( u  e.  V  |->  [ u ]  .~  ) `  x )  =  [ x ]  .~  )
3127, 29, 303eqtr4d 2408 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  (
( u  e.  V  |->  [ u ]  .~  ) `  (  .0.  .+  x ) )  =  ( ( u  e.  V  |->  [ u ]  .~  ) `  x ) )
32 divsgrp2.5 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  N  e.  V )
33 divsgrp2.6 . . . . . 6  |-  ( (
ph  /\  x  e.  V )  ->  ( N  .+  x )  .~  .0.  )
3425, 33ersym 6814 . . . . 5  |-  ( (
ph  /\  x  e.  V )  ->  .0.  .~  ( N  .+  x
) )
3525, 34erthi 6848 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  [  .0.  ]  .~  =  [ ( N  .+  x ) ]  .~  )
3625, 28, 3divsfval 13659 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (
( u  e.  V  |->  [ u ]  .~  ) `  .0.  )  =  [  .0.  ]  .~  )
3725, 28, 3divsfval 13659 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (
( u  e.  V  |->  [ u ]  .~  ) `  ( N  .+  x ) )  =  [ ( N  .+  x ) ]  .~  )
3835, 36, 373eqtr4rd 2409 . . 3  |-  ( (
ph  /\  x  e.  V )  ->  (
( u  e.  V  |->  [ u ]  .~  ) `  ( N  .+  x ) )  =  ( ( u  e.  V  |->  [ u ]  .~  ) `  .0.  )
)
3910, 2, 11, 12, 16, 9, 13, 23, 24, 31, 32, 38imasgrp2 14820 . 2  |-  ( ph  ->  ( U  e.  Grp  /\  ( ( u  e.  V  |->  [ u ]  .~  ) `  .0.  )  =  ( 0g `  U ) ) )
404, 6, 3divsfval 13659 . . . . 5  |-  ( ph  ->  ( ( u  e.  V  |->  [ u ]  .~  ) `  .0.  )  =  [  .0.  ]  .~  )
4140eqcomd 2371 . . . 4  |-  ( ph  ->  [  .0.  ]  .~  =  ( ( u  e.  V  |->  [ u ]  .~  ) `  .0.  ) )
4241eqeq1d 2374 . . 3  |-  ( ph  ->  ( [  .0.  ]  .~  =  ( 0g `  U )  <->  ( (
u  e.  V  |->  [ u ]  .~  ) `  .0.  )  =  ( 0g `  U ) ) )
4342anbi2d 684 . 2  |-  ( ph  ->  ( ( U  e. 
Grp  /\  [  .0.  ]  .~  =  ( 0g
`  U ) )  <-> 
( U  e.  Grp  /\  ( ( u  e.  V  |->  [ u ]  .~  ) `  .0.  )  =  ( 0g `  U ) ) ) )
4439, 43mpbird 223 1  |-  ( ph  ->  ( U  e.  Grp  /\ 
[  .0.  ]  .~  =  ( 0g `  U ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715   _Vcvv 2873   class class class wbr 4125    e. cmpt 4179   ` cfv 5358  (class class class)co 5981    Er wer 6799   [cec 6800   /.cqs 6801   Basecbs 13356   +g cplusg 13416   0gc0g 13610    /.s cqus 13618   Grpcgrp 14572
This theorem is referenced by:  divsgrp  14882  frgp0  15279  pi1grplem  18762
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-oadd 6625  df-er 6802  df-ec 6804  df-qs 6808  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-sup 7341  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-nn 9894  df-2 9951  df-3 9952  df-4 9953  df-5 9954  df-6 9955  df-7 9956  df-8 9957  df-9 9958  df-10 9959  df-n0 10115  df-z 10176  df-dec 10276  df-uz 10382  df-fz 10936  df-struct 13358  df-ndx 13359  df-slot 13360  df-base 13361  df-plusg 13429  df-mulr 13430  df-sca 13432  df-vsca 13433  df-tset 13435  df-ple 13436  df-ds 13438  df-0g 13614  df-imas 13621  df-divs 13622  df-mnd 14577  df-grp 14699
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