Theorem mndprop 12773

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 2178	. . 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 5884	. . . 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 12772	. 2 |- ( T. -> ( K e. Mnd <-> L e. Mnd ) )
8	7	mptru 1362	1 |- ( K e. Mnd <-> L e. Mnd )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1353   T. wtru 1354   e. wcel 2148   ` cfv 5214  (class class class)co 5871   Basecbs 12453   +g cplusg 12527   Mndcmnd 12748

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-cnex 7898  ax-resscn 7899  ax-1re 7901  ax-addrcl 7904

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-iota 5176  df-fun 5216  df-fn 5217  df-fv 5222  df-ov 5874  df-inn 8915  df-2 8973  df-ndx 12456  df-slot 12457  df-base 12459  df-plusg 12540  df-mgm 12706  df-sgrp 12739  df-mnd 12749

This theorem is referenced by: (None)