Theorem grpinvnz 13674

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 5639	. . . . . 6 |- ( ( N ` X ) = .0. -> ( N ` ( N ` X ) ) = ( N ` .0. ) )
2	1	adantl 277	. . . . 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 13670	. . . . . 6 |- ( ( G e. Grp /\ X e. B ) -> ( N ` ( N ` X ) ) = X )
6	5	adantr 276	. . . . 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 13663	. . . . . 6 |- ( G e. Grp -> ( N ` .0. ) = .0. )
9	8	ad2antrr 488	. . . . 5 |- ( ( ( G e. Grp /\ X e. B ) /\ ( N ` X ) = .0. ) -> ( N ` .0. ) = .0. )
10	2, 6, 9	3eqtr3d 2271	. . . 4 |- ( ( ( G e. Grp /\ X e. B ) /\ ( N ` X ) = .0. ) -> X = .0. )
11	10	ex 115	. . 3 |- ( ( G e. Grp /\ X e. B ) -> ( ( N ` X ) = .0. -> X = .0. ) )
12	11	necon3d 2445	. 2 |- ( ( G e. Grp /\ X e. B ) -> ( X =/= .0. -> ( N ` X ) =/= .0. ) )
13	12	3impia 1226	1 |- ( ( G e. Grp /\ X e. B /\ X =/= .0. ) -> ( N ` X ) =/= .0. )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1004   = wceq 1397   e. wcel 2201   =/= wne 2401   ` cfv 5325   Basecbs 13102   0gc0g 13359   Grpcgrp 13603   invgcminusg 13604

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4203  ax-sep 4206  ax-pow 4263  ax-pr 4298  ax-un 4529  ax-cnex 8125  ax-resscn 8126  ax-1re 8128  ax-addrcl 8131

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-reu 2516  df-rmo 2517  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-un 3203  df-in 3205  df-ss 3212  df-pw 3653  df-sn 3674  df-pr 3675  df-op 3677  df-uni 3893  df-int 3928  df-iun 3971  df-br 4088  df-opab 4150  df-mpt 4151  df-id 4389  df-xp 4730  df-rel 4731  df-cnv 4732  df-co 4733  df-dm 4734  df-rn 4735  df-res 4736  df-ima 4737  df-iota 5285  df-fun 5327  df-fn 5328  df-f 5329  df-f1 5330  df-fo 5331  df-f1o 5332  df-fv 5333  df-riota 5973  df-ov 6023  df-inn 9146  df-2 9204  df-ndx 13105  df-slot 13106  df-base 13108  df-plusg 13193  df-0g 13361  df-mgm 13459  df-sgrp 13505  df-mnd 13520  df-grp 13606  df-minusg 13607

This theorem is referenced by:  grpinvnzcl  13675