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

Theorem grpsubpropd2 16463
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 997 . . . . . 6  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  ph )
2 simp2 998 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  a  e.  (
Base `  G )
)
3 grpsubpropd2.1 . . . . . . . 8  |-  ( ph  ->  B  =  ( Base `  G ) )
433ad2ant1 1018 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  B  =  (
Base `  G )
)
52, 4eleqtrrd 2493 . . . . . 6  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  a  e.  B
)
6 grpsubpropd2.3 . . . . . . . . 9  |-  ( ph  ->  G  e.  Grp )
763ad2ant1 1018 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  G  e.  Grp )
8 simp3 999 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  b  e.  (
Base `  G )
)
9 eqid 2402 . . . . . . . . 9  |-  ( Base `  G )  =  (
Base `  G )
10 eqid 2402 . . . . . . . . 9  |-  ( invg `  G )  =  ( invg `  G )
119, 10grpinvcl 16417 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  b  e.  ( Base `  G ) )  -> 
( ( invg `  G ) `  b
)  e.  ( Base `  G ) )
127, 8, 11syl2anc 659 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  ( ( invg `  G ) `
 b )  e.  ( Base `  G
) )
1312, 4eleqtrrd 2493 . . . . . 6  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  ( ( invg `  G ) `
 b )  e.  B )
14 grpsubpropd2.4 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  G ) y )  =  ( x ( +g  `  H ) y ) )
1514oveqrspc2v 6300 . . . . . 6  |-  ( (
ph  /\  ( a  e.  B  /\  (
( invg `  G ) `  b
)  e.  B ) )  ->  ( a
( +g  `  G ) ( ( invg `  G ) `  b
) )  =  ( a ( +g  `  H
) ( ( invg `  G ) `
 b ) ) )
161, 5, 13, 15syl12anc 1228 . . . . 5  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  ( a ( +g  `  G ) ( ( invg `  G ) `  b
) )  =  ( a ( +g  `  H
) ( ( invg `  G ) `
 b ) ) )
17 grpsubpropd2.2 . . . . . . . . 9  |-  ( ph  ->  B  =  ( Base `  H ) )
183, 17, 14grpinvpropd 16435 . . . . . . . 8  |-  ( ph  ->  ( invg `  G )  =  ( invg `  H
) )
1918fveq1d 5850 . . . . . . 7  |-  ( ph  ->  ( ( invg `  G ) `  b
)  =  ( ( invg `  H
) `  b )
)
2019oveq2d 6293 . . . . . 6  |-  ( ph  ->  ( a ( +g  `  H ) ( ( invg `  G
) `  b )
)  =  ( a ( +g  `  H
) ( ( invg `  H ) `
 b ) ) )
21203ad2ant1 1018 . . . . 5  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  ( a ( +g  `  H ) ( ( invg `  G ) `  b
) )  =  ( a ( +g  `  H
) ( ( invg `  H ) `
 b ) ) )
2216, 21eqtrd 2443 . . . 4  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  ( a ( +g  `  G ) ( ( invg `  G ) `  b
) )  =  ( a ( +g  `  H
) ( ( invg `  H ) `
 b ) ) )
2322mpt2eq3dva 6341 . . 3  |-  ( ph  ->  ( a  e.  (
Base `  G ) ,  b  e.  ( Base `  G )  |->  ( a ( +g  `  G
) ( ( invg `  G ) `
 b ) ) )  =  ( a  e.  ( Base `  G
) ,  b  e.  ( Base `  G
)  |->  ( a ( +g  `  H ) ( ( invg `  H ) `  b
) ) ) )
243, 17eqtr3d 2445 . . . 4  |-  ( ph  ->  ( Base `  G
)  =  ( Base `  H ) )
25 mpt2eq12 6337 . . . 4  |-  ( ( ( Base `  G
)  =  ( Base `  H )  /\  ( Base `  G )  =  ( Base `  H
) )  ->  (
a  e.  ( Base `  G ) ,  b  e.  ( Base `  G
)  |->  ( a ( +g  `  H ) ( ( invg `  H ) `  b
) ) )  =  ( a  e.  (
Base `  H ) ,  b  e.  ( Base `  H )  |->  ( a ( +g  `  H
) ( ( invg `  H ) `
 b ) ) ) )
2624, 24, 25syl2anc 659 . . 3  |-  ( ph  ->  ( a  e.  (
Base `  G ) ,  b  e.  ( Base `  G )  |->  ( a ( +g  `  H
) ( ( invg `  H ) `
 b ) ) )  =  ( a  e.  ( Base `  H
) ,  b  e.  ( Base `  H
)  |->  ( a ( +g  `  H ) ( ( invg `  H ) `  b
) ) ) )
2723, 26eqtrd 2443 . 2  |-  ( ph  ->  ( a  e.  (
Base `  G ) ,  b  e.  ( Base `  G )  |->  ( a ( +g  `  G
) ( ( invg `  G ) `
 b ) ) )  =  ( a  e.  ( Base `  H
) ,  b  e.  ( Base `  H
)  |->  ( a ( +g  `  H ) ( ( invg `  H ) `  b
) ) ) )
28 eqid 2402 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
29 eqid 2402 . . 3  |-  ( -g `  G )  =  (
-g `  G )
309, 28, 10, 29grpsubfval 16414 . 2  |-  ( -g `  G )  =  ( a  e.  ( Base `  G ) ,  b  e.  ( Base `  G
)  |->  ( a ( +g  `  G ) ( ( invg `  G ) `  b
) ) )
31 eqid 2402 . . 3  |-  ( Base `  H )  =  (
Base `  H )
32 eqid 2402 . . 3  |-  ( +g  `  H )  =  ( +g  `  H )
33 eqid 2402 . . 3  |-  ( invg `  H )  =  ( invg `  H )
34 eqid 2402 . . 3  |-  ( -g `  H )  =  (
-g `  H )
3531, 32, 33, 34grpsubfval 16414 . 2  |-  ( -g `  H )  =  ( a  e.  ( Base `  H ) ,  b  e.  ( Base `  H
)  |->  ( a ( +g  `  H ) ( ( invg `  H ) `  b
) ) )
3627, 30, 353eqtr4g 2468 1  |-  ( ph  ->  ( -g `  G
)  =  ( -g `  H ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   ` cfv 5568  (class class class)co 6277    |-> cmpt2 6279   Basecbs 14839   +g cplusg 14907   Grpcgrp 16375   invgcminusg 16376   -gcsg 16377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-0g 15054  df-mgm 16194  df-sgrp 16233  df-mnd 16243  df-grp 16379  df-minusg 16380  df-sbg 16381
This theorem is referenced by:  ngppropd  21441
  Copyright terms: Public domain W3C validator