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Theorem divsgrp2 14613
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 2283 . . . 4  |-  ( u  e.  V  |->  [ u ]  .~  )  =  ( u  e.  V  |->  [ u ]  .~  )
4 divsgrp2.r . . . . 5  |-  ( ph  ->  .~  Er  V )
5 fvex 5539 . . . . . 6  |-  ( Base `  R )  e.  _V
62, 5syl6eqel 2371 . . . . 5  |-  ( ph  ->  V  e.  _V )
7 erex 6684 . . . . 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 13444 . . 3  |-  ( ph  ->  U  =  ( ( u  e.  V  |->  [ u ]  .~  )  "s  R ) )
11 divsgrp2.p . . 3  |-  ( ph  ->  .+  =  ( +g  `  R ) )
121, 2, 3, 8, 9divslem 13445 . . 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 1152 . . . 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 13451 . . 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 6706 . . . 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 13449 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( ( u  e.  V  |->  [ u ]  .~  ) `  ( ( x  .+  y ) 
.+  z ) )  =  [ ( ( x  .+  y ) 
.+  z ) ]  .~  )
2217, 20, 3divsfval 13449 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( ( u  e.  V  |->  [ u ]  .~  ) `  ( x 
.+  ( y  .+  z ) ) )  =  [ ( x 
.+  ( y  .+  z ) ) ]  .~  )
2319, 21, 223eqtr4d 2325 . . 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 6706 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  [ (  .0.  .+  x ) ]  .~  =  [ x ]  .~  )
286adantr 451 . . . . 5  |-  ( (
ph  /\  x  e.  V )  ->  V  e.  _V )
2925, 28, 3divsfval 13449 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (
( u  e.  V  |->  [ u ]  .~  ) `  (  .0.  .+  x ) )  =  [ (  .0.  .+  x ) ]  .~  )
3025, 28, 3divsfval 13449 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (
( u  e.  V  |->  [ u ]  .~  ) `  x )  =  [ x ]  .~  )
3127, 29, 303eqtr4d 2325 . . 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 6672 . . . . 5  |-  ( (
ph  /\  x  e.  V )  ->  .0.  .~  ( N  .+  x
) )
3525, 34erthi 6706 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  [  .0.  ]  .~  =  [ ( N  .+  x ) ]  .~  )
3625, 28, 3divsfval 13449 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (
( u  e.  V  |->  [ u ]  .~  ) `  .0.  )  =  [  .0.  ]  .~  )
3725, 28, 3divsfval 13449 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (
( u  e.  V  |->  [ u ]  .~  ) `  ( N  .+  x ) )  =  [ ( N  .+  x ) ]  .~  )
3835, 36, 373eqtr4rd 2326 . . 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 14610 . 2  |-  ( ph  ->  ( U  e.  Grp  /\  ( ( u  e.  V  |->  [ u ]  .~  ) `  .0.  )  =  ( 0g `  U ) ) )
404, 6, 3divsfval 13449 . . . . 5  |-  ( ph  ->  ( ( u  e.  V  |->  [ u ]  .~  ) `  .0.  )  =  [  .0.  ]  .~  )
4140eqcomd 2288 . . . 4  |-  ( ph  ->  [  .0.  ]  .~  =  ( ( u  e.  V  |->  [ u ]  .~  ) `  .0.  ) )
4241eqeq1d 2291 . . 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 934    = wceq 1623    e. wcel 1684   _Vcvv 2788   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858    Er wer 6657   [cec 6658   /.cqs 6659   Basecbs 13148   +g cplusg 13208   0gc0g 13400    /.s cqus 13408   Grpcgrp 14362
This theorem is referenced by:  divsgrp  14672  frgp0  15069  pi1grplem  18547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-ec 6662  df-qs 6666  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-imas 13411  df-divs 13412  df-mnd 14367  df-grp 14489
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