MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  divsgrp2 Structured version   Unicode version

Theorem divsgrp2 15775
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 2451 . . . 4  |-  ( u  e.  V  |->  [ u ]  .~  )  =  ( u  e.  V  |->  [ u ]  .~  )
4 divsgrp2.r . . . . 5  |-  ( ph  ->  .~  Er  V )
5 fvex 5799 . . . . . 6  |-  ( Base `  R )  e.  _V
62, 5syl6eqel 2547 . . . . 5  |-  ( ph  ->  V  e.  _V )
7 erex 7225 . . . . 5  |-  (  .~  Er  V  ->  ( V  e.  _V  ->  .~  e.  _V ) )
84, 6, 7sylc 60 . . . 4  |-  ( ph  ->  .~  e.  _V )
9 divsgrp2.x . . . 4  |-  ( ph  ->  R  e.  X )
101, 2, 3, 8, 9divsval 14582 . . 3  |-  ( ph  ->  U  =  ( ( u  e.  V  |->  [ u ]  .~  )  "s  R ) )
11 divsgrp2.p . . 3  |-  ( ph  ->  .+  =  ( +g  `  R ) )
121, 2, 3, 8, 9divslem 14583 . . 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 1189 . . . 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 14589 . . 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 465 . . . . 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 7247 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  [ ( ( x 
.+  y )  .+  z ) ]  .~  =  [ ( x  .+  ( y  .+  z
) ) ]  .~  )
206adantr 465 . . . . 5  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  V  e.  _V )
2117, 20, 3divsfval 14587 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( ( u  e.  V  |->  [ u ]  .~  ) `  ( ( x  .+  y ) 
.+  z ) )  =  [ ( ( x  .+  y ) 
.+  z ) ]  .~  )
2217, 20, 3divsfval 14587 . . . 4  |-  ( (
ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  -> 
( ( u  e.  V  |->  [ u ]  .~  ) `  ( x 
.+  ( y  .+  z ) ) )  =  [ ( x 
.+  ( y  .+  z ) ) ]  .~  )
2319, 21, 223eqtr4d 2502 . . 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 465 . . . . 5  |-  ( (
ph  /\  x  e.  V )  ->  .~  Er  V )
26 divsgrp2.4 . . . . 5  |-  ( (
ph  /\  x  e.  V )  ->  (  .0.  .+  x )  .~  x )
2725, 26erthi 7247 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  [ (  .0.  .+  x ) ]  .~  =  [ x ]  .~  )
286adantr 465 . . . . 5  |-  ( (
ph  /\  x  e.  V )  ->  V  e.  _V )
2925, 28, 3divsfval 14587 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (
( u  e.  V  |->  [ u ]  .~  ) `  (  .0.  .+  x ) )  =  [ (  .0.  .+  x ) ]  .~  )
3025, 28, 3divsfval 14587 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (
( u  e.  V  |->  [ u ]  .~  ) `  x )  =  [ x ]  .~  )
3127, 29, 303eqtr4d 2502 . . 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 7213 . . . . 5  |-  ( (
ph  /\  x  e.  V )  ->  .0.  .~  ( N  .+  x
) )
3525, 34erthi 7247 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  [  .0.  ]  .~  =  [ ( N  .+  x ) ]  .~  )
3625, 28, 3divsfval 14587 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (
( u  e.  V  |->  [ u ]  .~  ) `  .0.  )  =  [  .0.  ]  .~  )
3725, 28, 3divsfval 14587 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  (
( u  e.  V  |->  [ u ]  .~  ) `  ( N  .+  x ) )  =  [ ( N  .+  x ) ]  .~  )
3835, 36, 373eqtr4rd 2503 . . 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 15772 . 2  |-  ( ph  ->  ( U  e.  Grp  /\  ( ( u  e.  V  |->  [ u ]  .~  ) `  .0.  )  =  ( 0g `  U ) ) )
404, 6, 3divsfval 14587 . . . . 5  |-  ( ph  ->  ( ( u  e.  V  |->  [ u ]  .~  ) `  .0.  )  =  [  .0.  ]  .~  )
4140eqcomd 2459 . . . 4  |-  ( ph  ->  [  .0.  ]  .~  =  ( ( u  e.  V  |->  [ u ]  .~  ) `  .0.  ) )
4241eqeq1d 2453 . . 3  |-  ( ph  ->  ( [  .0.  ]  .~  =  ( 0g `  U )  <->  ( (
u  e.  V  |->  [ u ]  .~  ) `  .0.  )  =  ( 0g `  U ) ) )
4342anbi2d 703 . 2  |-  ( ph  ->  ( ( U  e. 
Grp  /\  [  .0.  ]  .~  =  ( 0g
`  U ) )  <-> 
( U  e.  Grp  /\  ( ( u  e.  V  |->  [ u ]  .~  ) `  .0.  )  =  ( 0g `  U ) ) ) )
4439, 43mpbird 232 1  |-  ( ph  ->  ( U  e.  Grp  /\ 
[  .0.  ]  .~  =  ( 0g `  U ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   _Vcvv 3068   class class class wbr 4390    |-> cmpt 4448   ` cfv 5516  (class class class)co 6190    Er wer 7198   [cec 7199   /.cqs 7200   Basecbs 14276   +g cplusg 14340   0gc0g 14480    /.s cqus 14545   Grpcgrp 15512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-1o 7020  df-oadd 7024  df-er 7201  df-ec 7203  df-qs 7207  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-sup 7792  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-2 10481  df-3 10482  df-4 10483  df-5 10484  df-6 10485  df-7 10486  df-8 10487  df-9 10488  df-10 10489  df-n0 10681  df-z 10748  df-dec 10857  df-uz 10963  df-fz 11539  df-struct 14278  df-ndx 14279  df-slot 14280  df-base 14281  df-plusg 14353  df-mulr 14354  df-sca 14356  df-vsca 14357  df-ip 14358  df-tset 14359  df-ple 14360  df-ds 14362  df-0g 14482  df-imas 14548  df-divs 14549  df-mnd 15517  df-grp 15647
This theorem is referenced by:  divsgrp  15838  frgp0  16361  pi1grplem  20737
  Copyright terms: Public domain W3C validator