Theorem grpinvnz 13478

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 5589	. . . . . 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 13474	. . . . . 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 13467	. . . . . 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 2247	. . . 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 2421	. 2 |- ( ( G e. Grp /\ X e. B ) -> ( X =/= .0. -> ( N ` X ) =/= .0. ) )
13	12	3impia 1203	1 |- ( ( G e. Grp /\ X e. B /\ X =/= .0. ) -> ( N ` X ) =/= .0. )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 981   = wceq 1373   e. wcel 2177   =/= wne 2377   ` cfv 5280   Basecbs 12907   0gc0g 13163   Grpcgrp 13407   invgcminusg 13408

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-cnex 8036  ax-resscn 8037  ax-1re 8039  ax-addrcl 8042

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-riota 5912  df-ov 5960  df-inn 9057  df-2 9115  df-ndx 12910  df-slot 12911  df-base 12913  df-plusg 12997  df-0g 13165  df-mgm 13263  df-sgrp 13309  df-mnd 13324  df-grp 13410  df-minusg 13411

This theorem is referenced by:  grpinvnzcl  13479