Theorem dfgrp2e 13783

Description: Alternate definition of a group as a set with a closed, associative operation, a left identity and a left inverse for each element. Alternate definition in [Lang] p. 7. (Contributed by NM, 10-Oct-2006.) (Revised by AV, 28-Aug-2021.)

Hypotheses

Ref	Expression
dfgrp2.b	|- B = ( Base ` G )
dfgrp2.p	|- .+ = ( +g ` G )

Assertion

Ref	Expression
dfgrp2e	|- ( G e. Grp <-> ( A. x e. B A. y e. B ( ( x .+ y ) e. B /\ A. z e. B ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) ) /\ E. n e. B A. x e. B ( ( n .+ x ) = x /\ E. i e. B ( i .+ x ) = n ) ) )

Distinct variable groups:   B, i, n, x   i, G, n, x   .+ , i, n, x   y, B, z, x   y, G, z   y, .+ , z

Proof of Theorem dfgrp2e

Step	Hyp	Ref	Expression
1		dfgrp2.b	. . 3 |- B = ( Base ` G )
2		dfgrp2.p	. . 3 |- .+ = ( +g ` G )
3	1, 2	dfgrp2 13782	. 2 |- ( G e. Grp <-> ( G e. Smgrp /\ E. n e. B A. x e. B ( ( n .+ x ) = x /\ E. i e. B ( i .+ x ) = n ) ) )
4		rexm 3613	. . . 4 |- ( E. n e. B A. x e. B ( ( n .+ x ) = x /\ E. i e. B ( i .+ x ) = n ) -> E. n n e. B )
5	1	basmex 13356	. . . . 5 |- ( n e. B -> G e. _V )
6	5	exlimiv 1647	. . . 4 |- ( E. n n e. B -> G e. _V )
7	1, 2	issgrpv 13667	. . . 4 |- ( G e. _V -> ( G e. Smgrp <-> A. x e. B A. y e. B ( ( x .+ y ) e. B /\ A. z e. B ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) ) ) )
8	4, 6, 7	3syl 17	. . 3 |- ( E. n e. B A. x e. B ( ( n .+ x ) = x /\ E. i e. B ( i .+ x ) = n ) -> ( G e. Smgrp <-> A. x e. B A. y e. B ( ( x .+ y ) e. B /\ A. z e. B ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) ) ) )
9	8	pm5.32ri 455	. 2 |- ( ( G e. Smgrp /\ E. n e. B A. x e. B ( ( n .+ x ) = x /\ E. i e. B ( i .+ x ) = n ) ) <-> ( A. x e. B A. y e. B ( ( x .+ y ) e. B /\ A. z e. B ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) ) /\ E. n e. B A. x e. B ( ( n .+ x ) = x /\ E. i e. B ( i .+ x ) = n ) ) )
10	3, 9	bitri 184	1 |- ( G e. Grp <-> ( A. x e. B A. y e. B ( ( x .+ y ) e. B /\ A. z e. B ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) ) /\ E. n e. B A. x e. B ( ( n .+ x ) = x /\ E. i e. B ( i .+ x ) = n ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2205   A.wral 2522   E.wrex 2523   _Vcvv 2815   ` cfv 5357  (class class class)co 6058   Basecbs 13296   +g cplusg 13374  Smgrpcsgrp 13664   Grpcgrp 13755

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-riota 6011  df-ov 6061  df-inn 9255  df-2 9313  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-grp 13758

This theorem is referenced by: (None)