Theorem mndprop 13474

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

Hypotheses

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

Assertion

Ref	Expression
mndprop	|- ( K e. Mnd <-> L e. Mnd )

Proof of Theorem mndprop

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

Step	Hyp	Ref	Expression
1		eqidd 2230	. . 3 |- ( T. -> ( Base ` K ) = ( Base ` K ) )
2		mndprop.b	. . . 4 |- ( Base ` K ) = ( Base ` L )
3	2	a1i 9	. . 3 |- ( T. -> ( Base ` K ) = ( Base ` L ) )
4		mndprop.p	. . . . 5 |- ( +g ` K ) = ( +g ` L )
5	4	oveqi 6014	. . . 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	mndpropd 13473	. 2 |- ( T. -> ( K e. Mnd <-> L e. Mnd ) )
8	7	mptru 1404	1 |- ( K e. Mnd <-> L e. Mnd )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1395   T. wtru 1396   e. wcel 2200   ` cfv 5318  (class class class)co 6001   Basecbs 13032   +g cplusg 13110   Mndcmnd 13449

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-cnex 8090  ax-resscn 8091  ax-1re 8093  ax-addrcl 8096

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326  df-ov 6004  df-inn 9111  df-2 9169  df-ndx 13035  df-slot 13036  df-base 13038  df-plusg 13123  df-mgm 13389  df-sgrp 13435  df-mnd 13450

This theorem is referenced by: (None)