Theorem grpinvnz 17486

Description: The inverse of a nonzero group element is not zero. (Contributed by Stefan O'Rear, 27-Feb-2015.)

Hypotheses

Ref	Expression
grpinvnzcl.b	|- B = ( Base ` G )
grpinvnzcl.z	|- .0. = ( 0g ` G )
grpinvnzcl.n	|- N = ( invg ` G )

Assertion

Ref	Expression
grpinvnz	|- ( ( G e. Grp /\ X e. B /\ X =/= .0. ) -> ( N ` X ) =/= .0. )

Proof of Theorem grpinvnz

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . 6 |- ( ( N ` X ) = .0. -> ( N ` ( N ` X ) ) = ( N ` .0. ) )
2	1	adantl 482	. . . . 5 |- ( ( ( G e. Grp /\ X e. B ) /\ ( N ` X ) = .0. ) -> ( N ` ( N ` X ) ) = ( N ` .0. ) )
3		grpinvnzcl.b	. . . . . . 7 |- B = ( Base ` G )
4		grpinvnzcl.n	. . . . . . 7 |- N = ( invg ` G )
5	3, 4	grpinvinv 17482	. . . . . 6 |- ( ( G e. Grp /\ X e. B ) -> ( N ` ( N ` X ) ) = X )
6	5	adantr 481	. . . . 5 |- ( ( ( G e. Grp /\ X e. B ) /\ ( N ` X ) = .0. ) -> ( N ` ( N ` X ) ) = X )
7		grpinvnzcl.z	. . . . . . 7 |- .0. = ( 0g ` G )
8	7, 4	grpinvid 17476	. . . . . 6 |- ( G e. Grp -> ( N ` .0. ) = .0. )
9	8	ad2antrr 762	. . . . 5 |- ( ( ( G e. Grp /\ X e. B ) /\ ( N ` X ) = .0. ) -> ( N ` .0. ) = .0. )
10	2, 6, 9	3eqtr3d 2664	. . . 4 |- ( ( ( G e. Grp /\ X e. B ) /\ ( N ` X ) = .0. ) -> X = .0. )
11	10	ex 450	. . 3 |- ( ( G e. Grp /\ X e. B ) -> ( ( N ` X ) = .0. -> X = .0. ) )
12	11	necon3d 2815	. 2 |- ( ( G e. Grp /\ X e. B ) -> ( X =/= .0. -> ( N ` X ) =/= .0. ) )
13	12	3impia 1261	1 |- ( ( G e. Grp /\ X e. B /\ X =/= .0. ) -> ( N ` X ) =/= .0. )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888   Basecbs 15857   0gc0g 16100   Grpcgrp 17422   invgcminusg 17423

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426

This theorem is referenced by:  grpinvnzcl  17487