Theorem grpolinv 27380

Description: The left inverse of a group element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
grpinv.1	|- X = ran G
grpinv.2	|- U = (GId ` G )
grpinv.3	|- N = ( inv ` G )

Assertion

Ref	Expression
grpolinv	|- ( ( G e. GrpOp /\ A e. X ) -> ( ( N ` A ) G A ) = U )

Proof of Theorem grpolinv

Step	Hyp	Ref	Expression
1		grpinv.1	. . 3 |- X = ran G
2		grpinv.2	. . 3 |- U = (GId ` G )
3		grpinv.3	. . 3 |- N = ( inv ` G )
4	1, 2, 3	grpoinv 27379	. 2 |- ( ( G e. GrpOp /\ A e. X ) -> ( ( ( N ` A ) G A ) = U /\ ( A G ( N ` A ) ) = U ) )
5	4	simpld 475	1 |- ( ( G e. GrpOp /\ A e. X ) -> ( ( N ` A ) G A ) = U )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ran crn 5115   ` cfv 5888  (class class class)co 6650   GrpOpcgr 27343  GIdcgi 27344   invcgn 27345

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-pr 4906  ax-un 6949

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-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-grpo 27347  df-gid 27348  df-ginv 27349

This theorem is referenced by:  grpoinvid1  27382  grpoinvid2  27383  grpolcan  27384  grpo2inv  27385  grponpcan  27397  nvlinv  27507  hhssabloilem  28118  rngoaddneg2  33728  isdrngo2  33757