Theorem mndprop 12861

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 2188	. . 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 5901	. . . 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 12860	. 2 |- ( T. -> ( K e. Mnd <-> L e. Mnd ) )
8	7	mptru 1372	1 |- ( K e. Mnd <-> L e. Mnd )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1363   T. wtru 1364   e. wcel 2158   ` cfv 5228  (class class class)co 5888   Basecbs 12475   +g cplusg 12550   Mndcmnd 12836

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-cnex 7915  ax-resscn 7916  ax-1re 7918  ax-addrcl 7921

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-iota 5190  df-fun 5230  df-fn 5231  df-fv 5236  df-ov 5891  df-inn 8933  df-2 8991  df-ndx 12478  df-slot 12479  df-base 12481  df-plusg 12563  df-mgm 12793  df-sgrp 12826  df-mnd 12837

This theorem is referenced by: (None)