Theorem grpolidinv 27355

Description: A group has a left identity element, and every member has a left inverse. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)

Hypothesis

Ref	Expression
grpfo.1	|- X = ran G

Assertion

Ref	Expression
grpolidinv	|- ( G e. GrpOp -> E. u e. X A. x e. X ( ( u G x ) = x /\ E. y e. X ( y G x ) = u ) )

Distinct variable groups:   x, y, u, G   u, X, x, y

Proof of Theorem grpolidinv

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpfo.1	. . . 4 |- X = ran G
2	1	isgrpo 27351	. . 3 |- ( G e. GrpOp -> ( G e. GrpOp <-> ( G : ( X X. X ) --> X /\ A. x e. X A. y e. X A. z e. X ( ( x G y ) G z ) = ( x G ( y G z ) ) /\ E. u e. X A. x e. X ( ( u G x ) = x /\ E. y e. X ( y G x ) = u ) ) ) )
3	2	ibi 256	. 2 |- ( G e. GrpOp -> ( G : ( X X. X ) --> X /\ A. x e. X A. y e. X A. z e. X ( ( x G y ) G z ) = ( x G ( y G z ) ) /\ E. u e. X A. x e. X ( ( u G x ) = x /\ E. y e. X ( y G x ) = u ) ) )
4	3	simp3d 1075	1 |- ( G e. GrpOp -> E. u e. X A. x e. X ( ( u G x ) = x /\ E. y e. X ( y G x ) = u ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   X. cxp 5112   ran crn 5115   -->wf 5884  (class class class)co 6650   GrpOpcgr 27343

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-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-ral 2917  df-rex 2918  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-ov 6653  df-grpo 27347

This theorem is referenced by:  grpon0  27356  grpoidinv  27362