Theorem grpprop 13773

Description: If two structures have the same group components (properties), one is a group iff the other one is. (Contributed by NM, 11-Oct-2013.)

Hypotheses

Ref	Expression
grpprop.b	|- ( Base ` K ) = ( Base ` L )
grpprop.p	|- ( +g ` K ) = ( +g ` L )

Assertion

Ref	Expression
grpprop	|- ( K e. Grp <-> L e. Grp )

Proof of Theorem grpprop

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2235	. . 3 |- ( T. -> ( Base ` K ) = ( Base ` K ) )
2		grpprop.b	. . . 4 |- ( Base ` K ) = ( Base ` L )
3	2	a1i 9	. . 3 |- ( T. -> ( Base ` K ) = ( Base ` L ) )
4		grpprop.p	. . . . 5 |- ( +g ` K ) = ( +g ` L )
5	4	oveqi 6071	. . . 4 |- ( x ( +g ` K ) y ) = ( x ( +g ` L ) y )
6	5	a1i 9	. . 3 |- ( ( T. /\ ( x e. ( Base ` K ) /\ y e. ( Base ` K ) ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )
7	1, 3, 6	grppropd 13772	. 2 |- ( T. -> ( K e. Grp <-> L e. Grp ) )
8	7	mptru 1407	1 |- ( K e. Grp <-> L e. Grp )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1398   T. wtru 1399   e. wcel 2205   ` cfv 5357  (class class class)co 6058   Basecbs 13296   +g cplusg 13374   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-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:  rmodislmod  14625