MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpsubpropd2 Unicode version

Theorem grpsubpropd2 14567
Description: Strong property deduction for the group subtraction operation. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
grpsubpropd2.1  |-  ( ph  ->  B  =  ( Base `  G ) )
grpsubpropd2.2  |-  ( ph  ->  B  =  ( Base `  H ) )
grpsubpropd2.3  |-  ( ph  ->  G  e.  Grp )
grpsubpropd2.4  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  G ) y )  =  ( x ( +g  `  H ) y ) )
Assertion
Ref Expression
grpsubpropd2  |-  ( ph  ->  ( -g `  G
)  =  ( -g `  H ) )
Distinct variable groups:    x, y, B    x, G, y    x, H, y    ph, x, y

Proof of Theorem grpsubpropd2
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 955 . . . . . 6  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  ph )
2 simp2 956 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  a  e.  (
Base `  G )
)
3 grpsubpropd2.1 . . . . . . . 8  |-  ( ph  ->  B  =  ( Base `  G ) )
433ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  B  =  (
Base `  G )
)
52, 4eleqtrrd 2360 . . . . . 6  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  a  e.  B
)
6 grpsubpropd2.3 . . . . . . . . 9  |-  ( ph  ->  G  e.  Grp )
763ad2ant1 976 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  G  e.  Grp )
8 simp3 957 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  b  e.  (
Base `  G )
)
9 eqid 2283 . . . . . . . . 9  |-  ( Base `  G )  =  (
Base `  G )
10 eqid 2283 . . . . . . . . 9  |-  ( inv g `  G )  =  ( inv g `  G )
119, 10grpinvcl 14527 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  b  e.  ( Base `  G ) )  -> 
( ( inv g `  G ) `  b
)  e.  ( Base `  G ) )
127, 8, 11syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  ( ( inv g `  G ) `
 b )  e.  ( Base `  G
) )
1312, 4eleqtrrd 2360 . . . . . 6  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  ( ( inv g `  G ) `
 b )  e.  B )
14 grpsubpropd2.4 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  G ) y )  =  ( x ( +g  `  H ) y ) )
1514proplem 13592 . . . . . 6  |-  ( (
ph  /\  ( a  e.  B  /\  (
( inv g `  G ) `  b
)  e.  B ) )  ->  ( a
( +g  `  G ) ( ( inv g `  G ) `  b
) )  =  ( a ( +g  `  H
) ( ( inv g `  G ) `
 b ) ) )
161, 5, 13, 15syl12anc 1180 . . . . 5  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  ( a ( +g  `  G ) ( ( inv g `  G ) `  b
) )  =  ( a ( +g  `  H
) ( ( inv g `  G ) `
 b ) ) )
17 grpsubpropd2.2 . . . . . . . . 9  |-  ( ph  ->  B  =  ( Base `  H ) )
183, 17, 14grpinvpropd 14543 . . . . . . . 8  |-  ( ph  ->  ( inv g `  G )  =  ( inv g `  H
) )
1918fveq1d 5527 . . . . . . 7  |-  ( ph  ->  ( ( inv g `  G ) `  b
)  =  ( ( inv g `  H
) `  b )
)
2019oveq2d 5874 . . . . . 6  |-  ( ph  ->  ( a ( +g  `  H ) ( ( inv g `  G
) `  b )
)  =  ( a ( +g  `  H
) ( ( inv g `  H ) `
 b ) ) )
21203ad2ant1 976 . . . . 5  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  ( a ( +g  `  H ) ( ( inv g `  G ) `  b
) )  =  ( a ( +g  `  H
) ( ( inv g `  H ) `
 b ) ) )
2216, 21eqtrd 2315 . . . 4  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  ( a ( +g  `  G ) ( ( inv g `  G ) `  b
) )  =  ( a ( +g  `  H
) ( ( inv g `  H ) `
 b ) ) )
2322mpt2eq3dva 5912 . . 3  |-  ( ph  ->  ( a  e.  (
Base `  G ) ,  b  e.  ( Base `  G )  |->  ( a ( +g  `  G
) ( ( inv g `  G ) `
 b ) ) )  =  ( a  e.  ( Base `  G
) ,  b  e.  ( Base `  G
)  |->  ( a ( +g  `  H ) ( ( inv g `  H ) `  b
) ) ) )
243, 17eqtr3d 2317 . . . 4  |-  ( ph  ->  ( Base `  G
)  =  ( Base `  H ) )
25 mpt2eq12 5908 . . . 4  |-  ( ( ( Base `  G
)  =  ( Base `  H )  /\  ( Base `  G )  =  ( Base `  H
) )  ->  (
a  e.  ( Base `  G ) ,  b  e.  ( Base `  G
)  |->  ( a ( +g  `  H ) ( ( inv g `  H ) `  b
) ) )  =  ( a  e.  (
Base `  H ) ,  b  e.  ( Base `  H )  |->  ( a ( +g  `  H
) ( ( inv g `  H ) `
 b ) ) ) )
2624, 24, 25syl2anc 642 . . 3  |-  ( ph  ->  ( a  e.  (
Base `  G ) ,  b  e.  ( Base `  G )  |->  ( a ( +g  `  H
) ( ( inv g `  H ) `
 b ) ) )  =  ( a  e.  ( Base `  H
) ,  b  e.  ( Base `  H
)  |->  ( a ( +g  `  H ) ( ( inv g `  H ) `  b
) ) ) )
2723, 26eqtrd 2315 . 2  |-  ( ph  ->  ( a  e.  (
Base `  G ) ,  b  e.  ( Base `  G )  |->  ( a ( +g  `  G
) ( ( inv g `  G ) `
 b ) ) )  =  ( a  e.  ( Base `  H
) ,  b  e.  ( Base `  H
)  |->  ( a ( +g  `  H ) ( ( inv g `  H ) `  b
) ) ) )
28 eqid 2283 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
29 eqid 2283 . . 3  |-  ( -g `  G )  =  (
-g `  G )
309, 28, 10, 29grpsubfval 14524 . 2  |-  ( -g `  G )  =  ( a  e.  ( Base `  G ) ,  b  e.  ( Base `  G
)  |->  ( a ( +g  `  G ) ( ( inv g `  G ) `  b
) ) )
31 eqid 2283 . . 3  |-  ( Base `  H )  =  (
Base `  H )
32 eqid 2283 . . 3  |-  ( +g  `  H )  =  ( +g  `  H )
33 eqid 2283 . . 3  |-  ( inv g `  H )  =  ( inv g `  H )
34 eqid 2283 . . 3  |-  ( -g `  H )  =  (
-g `  H )
3531, 32, 33, 34grpsubfval 14524 . 2  |-  ( -g `  H )  =  ( a  e.  ( Base `  H ) ,  b  e.  ( Base `  H
)  |->  ( a ( +g  `  H ) ( ( inv g `  H ) `  b
) ) )
3627, 30, 353eqtr4g 2340 1  |-  ( ph  ->  ( -g `  G
)  =  ( -g `  H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   Basecbs 13148   +g cplusg 13208   Grpcgrp 14362   inv gcminusg 14363   -gcsg 14365
This theorem is referenced by:  ngppropd  18153
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-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-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  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-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491
  Copyright terms: Public domain W3C validator