Theorem grpinvnz 13853

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 5690	. . . . . 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 13849	. . . . . 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 13842	. . . . . 6 |- ( G e. Grp -> ( N ` .0. ) = .0. )
9	8	ad2antrr 492	. . . . 5 |- ( ( ( G e. Grp /\ X e. B ) /\ ( N ` X ) = .0. ) -> ( N ` .0. ) = .0. )
10	2, 6, 9	3eqtr3d 2279	. . . 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 2464	. 2 |- ( ( G e. Grp /\ X e. B ) -> ( X =/= .0. -> ( N ` X ) =/= .0. ) )
13	12	3impia 1231	1 |- ( ( G e. Grp /\ X e. B /\ X =/= .0. ) -> ( N ` X ) =/= .0. )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   ` cfv 5372   Basecbs 13330   0gc0g 13587   Grpcgrp 13782   invgcminusg 13783

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-inn 9284  df-2 9342  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-minusg 13786

This theorem is referenced by:  grpinvnzcl  13854