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Metamath Proof Explorer

Theorem ablsub2inv

Description: Abelian group subtraction of two inverses. (Contributed by Stefan O'Rear, 24-May-2015)

Ref Expression
Hypotheses ablsub2inv.b 
|- B = ( Base ` G )
ablsub2inv.m 
|- .- = ( -g ` G )
ablsub2inv.n 
|- N = ( invg ` G )
ablsub2inv.g 
|- ( ph -> G e. Abel )
ablsub2inv.x 
|- ( ph -> X e. B )
ablsub2inv.y 
|- ( ph -> Y e. B )
Assertion ablsub2inv 
|- ( ph -> ( ( N ` X ) .- ( N ` Y ) ) = ( Y .- X ) )

Proof

Step Hyp Ref Expression
1 ablsub2inv.b 
 |-  B = ( Base ` G )
2 ablsub2inv.m 
 |-  .- = ( -g ` G )
3 ablsub2inv.n 
 |-  N = ( invg ` G )
4 ablsub2inv.g 
 |-  ( ph -> G e. Abel )
5 ablsub2inv.x 
 |-  ( ph -> X e. B )
6 ablsub2inv.y 
 |-  ( ph -> Y e. B )
7 eqid 
 |-  ( +g ` G ) = ( +g ` G )
8 ablgrp 
 |-  ( G e. Abel -> G e. Grp )
9 4 8 syl 
 |-  ( ph -> G e. Grp )
10 1 3 grpinvcl 
 |-  ( ( G e. Grp /\ X e. B ) -> ( N ` X ) e. B )
11 9 5 10 syl2anc 
 |-  ( ph -> ( N ` X ) e. B )
12 1 7 2 3 9 11 6 grpsubinv 
 |-  ( ph -> ( ( N ` X ) .- ( N ` Y ) ) = ( ( N ` X ) ( +g ` G ) Y ) )
13 1 7 ablcom 
 |-  ( ( G e. Abel /\ ( N ` X ) e. B /\ Y e. B ) -> ( ( N ` X ) ( +g ` G ) Y ) = ( Y ( +g ` G ) ( N ` X ) ) )
14 4 11 6 13 syl3anc 
 |-  ( ph -> ( ( N ` X ) ( +g ` G ) Y ) = ( Y ( +g ` G ) ( N ` X ) ) )
15 1 3 grpinvinv 
 |-  ( ( G e. Grp /\ Y e. B ) -> ( N ` ( N ` Y ) ) = Y )
16 9 6 15 syl2anc 
 |-  ( ph -> ( N ` ( N ` Y ) ) = Y )
17 16 oveq1d 
 |-  ( ph -> ( ( N ` ( N ` Y ) ) ( +g ` G ) ( N ` X ) ) = ( Y ( +g ` G ) ( N ` X ) ) )
18 14 17 eqtr4d 
 |-  ( ph -> ( ( N ` X ) ( +g ` G ) Y ) = ( ( N ` ( N ` Y ) ) ( +g ` G ) ( N ` X ) ) )
19 1 3 grpinvcl 
 |-  ( ( G e. Grp /\ Y e. B ) -> ( N ` Y ) e. B )
20 9 6 19 syl2anc 
 |-  ( ph -> ( N ` Y ) e. B )
21 1 7 3 grpinvadd 
 |-  ( ( G e. Grp /\ X e. B /\ ( N ` Y ) e. B ) -> ( N ` ( X ( +g ` G ) ( N ` Y ) ) ) = ( ( N ` ( N ` Y ) ) ( +g ` G ) ( N ` X ) ) )
22 9 5 20 21 syl3anc 
 |-  ( ph -> ( N ` ( X ( +g ` G ) ( N ` Y ) ) ) = ( ( N ` ( N ` Y ) ) ( +g ` G ) ( N ` X ) ) )
23 18 22 eqtr4d 
 |-  ( ph -> ( ( N ` X ) ( +g ` G ) Y ) = ( N ` ( X ( +g ` G ) ( N ` Y ) ) ) )
24 1 7 3 2 grpsubval 
 |-  ( ( X e. B /\ Y e. B ) -> ( X .- Y ) = ( X ( +g ` G ) ( N ` Y ) ) )
25 5 6 24 syl2anc 
 |-  ( ph -> ( X .- Y ) = ( X ( +g ` G ) ( N ` Y ) ) )
26 25 fveq2d 
 |-  ( ph -> ( N ` ( X .- Y ) ) = ( N ` ( X ( +g ` G ) ( N ` Y ) ) ) )
27 23 26 eqtr4d 
 |-  ( ph -> ( ( N ` X ) ( +g ` G ) Y ) = ( N ` ( X .- Y ) ) )
28 1 2 3 grpinvsub 
 |-  ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( N ` ( X .- Y ) ) = ( Y .- X ) )
29 9 5 6 28 syl3anc 
 |-  ( ph -> ( N ` ( X .- Y ) ) = ( Y .- X ) )
30 12 27 29 3eqtrd 
 |-  ( ph -> ( ( N ` X ) .- ( N ` Y ) ) = ( Y .- X ) )