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Theorem divsgrp2 14891
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 2404 . . . 4  |-  ( u  e.  V  |->  [ u ]  .~  )  =  ( u  e.  V  |->  [ u ]  .~  )
4 divsgrp2.r . . . . 5  |-  ( ph  ->  .~  Er  V )
5 fvex 5701 . . . . . 6  |-  ( Base `  R )  e.  _V
62, 5syl6eqel 2492 . . . . 5  |-  ( ph  ->  V  e.  _V )
7 erex 6888 . . . . 5  |-  (  .~  Er  V  ->  ( V  e.  _V  ->  .~  e.  _V ) )
84, 6, 7sylc 58 . . . 4  |-  ( ph  ->  .~  e.  _V )
9 divsgrp2.x . . . 4  |-  ( ph  ->  R  e.  X )
101, 2, 3, 8, 9divsval 13722 . . 3  |-  ( ph  ->  U  =  ( ( u  e.  V  |->  [ u ]  .~  )  "s  R ) )
11 divsgrp2.p . . 3  |-  ( ph  ->  .+  =  ( +g  `  R ) )
121, 2, 3, 8, 9divslem 13723 . . 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 1154 . . . 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 13729 . . 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 452 . . . . 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 6910 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  [ ( ( x 
.+  y )  .+  z ) ]  .~  =  [ ( x  .+  ( y  .+  z
) ) ]  .~  )
206adantr 452 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  V  e.  _V )
2117, 20, 3divsfval 13727 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( ( u  e.  V  |->  [ u ]  .~  ) `  ( ( x  .+  y ) 
.+  z ) )  =  [ ( ( x  .+  y ) 
.+  z ) ]  .~  )
2217, 20, 3divsfval 13727 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( ( u  e.  V  |->  [ u ]  .~  ) `  ( x 
.+  ( y  .+  z ) ) )  =  [ ( x 
.+  ( y  .+  z ) ) ]  .~  )
2319, 21, 223eqtr4d 2446 . . 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 452 . . . . 5  |-  ( (
ph  /\  x  e.  V )  ->  .~  Er  V )
26 divsgrp2.4 . . . . 5  |-  ( (
ph  /\  x  e.  V )  ->  (  .0.  .+  x )  .~  x )
2725, 26erthi 6910 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  [ (  .0.  .+  x ) ]  .~  =  [ x ]  .~  )
286adantr 452 . . . . 5  |-  ( (
ph  /\  x  e.  V )  ->  V  e.  _V )
2925, 28, 3divsfval 13727 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (
( u  e.  V  |->  [ u ]  .~  ) `  (  .0.  .+  x ) )  =  [ (  .0.  .+  x ) ]  .~  )
3025, 28, 3divsfval 13727 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (
( u  e.  V  |->  [ u ]  .~  ) `  x )  =  [ x ]  .~  )
3127, 29, 303eqtr4d 2446 . . 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 6876 . . . . 5  |-  ( (
ph  /\  x  e.  V )  ->  .0.  .~  ( N  .+  x
) )
3525, 34erthi 6910 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  [  .0.  ]  .~  =  [ ( N  .+  x ) ]  .~  )
3625, 28, 3divsfval 13727 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (
( u  e.  V  |->  [ u ]  .~  ) `  .0.  )  =  [  .0.  ]  .~  )
3725, 28, 3divsfval 13727 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (
( u  e.  V  |->  [ u ]  .~  ) `  ( N  .+  x ) )  =  [ ( N  .+  x ) ]  .~  )
3835, 36, 373eqtr4rd 2447 . . 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 14888 . 2  |-  ( ph  ->  ( U  e.  Grp  /\  ( ( u  e.  V  |->  [ u ]  .~  ) `  .0.  )  =  ( 0g `  U ) ) )
404, 6, 3divsfval 13727 . . . . 5  |-  ( ph  ->  ( ( u  e.  V  |->  [ u ]  .~  ) `  .0.  )  =  [  .0.  ]  .~  )
4140eqcomd 2409 . . . 4  |-  ( ph  ->  [  .0.  ]  .~  =  ( ( u  e.  V  |->  [ u ]  .~  ) `  .0.  ) )
4241eqeq1d 2412 . . 3  |-  ( ph  ->  ( [  .0.  ]  .~  =  ( 0g `  U )  <->  ( (
u  e.  V  |->  [ u ]  .~  ) `  .0.  )  =  ( 0g `  U ) ) )
4342anbi2d 685 . 2  |-  ( ph  ->  ( ( U  e. 
Grp  /\  [  .0.  ]  .~  =  ( 0g
`  U ) )  <-> 
( U  e.  Grp  /\  ( ( u  e.  V  |->  [ u ]  .~  ) `  .0.  )  =  ( 0g `  U ) ) ) )
4439, 43mpbird 224 1  |-  ( ph  ->  ( U  e.  Grp  /\ 
[  .0.  ]  .~  =  ( 0g `  U ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2916   class class class wbr 4172    e. cmpt 4226   ` cfv 5413  (class class class)co 6040    Er wer 6861   [cec 6862   /.cqs 6863   Basecbs 13424   +g cplusg 13484   0gc0g 13678    /.s cqus 13686   Grpcgrp 14640
This theorem is referenced by:  divsgrp  14950  frgp0  15347  pi1grplem  19027
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-ec 6866  df-qs 6870  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-plusg 13497  df-mulr 13498  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-0g 13682  df-imas 13689  df-divs 13690  df-mnd 14645  df-grp 14767
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