Theorem isnmnd 17298

Description: A condition for a structure not to be a monoid: every element of the base set is not a left identity for at least one element of the base set. (Contributed by AV, 4-Feb-2020.)

Hypotheses

Ref	Expression
isnmnd.b	|- B = ( Base ` M )
isnmnd.o	|- .o. = ( +g ` M )

Assertion

Ref	Expression
isnmnd	|- ( A. z e. B E. x e. B ( z .o. x ) =/= x -> M e/ Mnd )

Distinct variable groups:   x, B, z   x, M, z   x, .o. , z

Proof of Theorem isnmnd

Step	Hyp	Ref	Expression
1		neneq 2800	. . . . . . . 8 |- ( ( z .o. x ) =/= x -> -. ( z .o. x ) = x )
2	1	intnanrd 963	. . . . . . 7 |- ( ( z .o. x ) =/= x -> -. ( ( z .o. x ) = x /\ ( x .o. z ) = x ) )
3	2	reximi 3011	. . . . . 6 |- ( E. x e. B ( z .o. x ) =/= x -> E. x e. B -. ( ( z .o. x ) = x /\ ( x .o. z ) = x ) )
4	3	ralimi 2952	. . . . 5 |- ( A. z e. B E. x e. B ( z .o. x ) =/= x -> A. z e. B E. x e. B -. ( ( z .o. x ) = x /\ ( x .o. z ) = x ) )
5		rexnal 2995	. . . . . . 7 |- ( E. x e. B -. ( ( z .o. x ) = x /\ ( x .o. z ) = x ) <-> -. A. x e. B ( ( z .o. x ) = x /\ ( x .o. z ) = x ) )
6	5	ralbii 2980	. . . . . 6 |- ( A. z e. B E. x e. B -. ( ( z .o. x ) = x /\ ( x .o. z ) = x ) <-> A. z e. B -. A. x e. B ( ( z .o. x ) = x /\ ( x .o. z ) = x ) )
7		ralnex 2992	. . . . . 6 |- ( A. z e. B -. A. x e. B ( ( z .o. x ) = x /\ ( x .o. z ) = x ) <-> -. E. z e. B A. x e. B ( ( z .o. x ) = x /\ ( x .o. z ) = x ) )
8	6, 7	bitri 264	. . . . 5 |- ( A. z e. B E. x e. B -. ( ( z .o. x ) = x /\ ( x .o. z ) = x ) <-> -. E. z e. B A. x e. B ( ( z .o. x ) = x /\ ( x .o. z ) = x ) )
9	4, 8	sylib 208	. . . 4 |- ( A. z e. B E. x e. B ( z .o. x ) =/= x -> -. E. z e. B A. x e. B ( ( z .o. x ) = x /\ ( x .o. z ) = x ) )
10	9	intnand 962	. . 3 |- ( A. z e. B E. x e. B ( z .o. x ) =/= x -> -. ( M e. SGrp /\ E. z e. B A. x e. B ( ( z .o. x ) = x /\ ( x .o. z ) = x ) ) )
11		isnmnd.b	. . . 4 |- B = ( Base ` M )
12		isnmnd.o	. . . 4 |- .o. = ( +g ` M )
13	11, 12	ismnddef 17296	. . 3 |- ( M e. Mnd <-> ( M e. SGrp /\ E. z e. B A. x e. B ( ( z .o. x ) = x /\ ( x .o. z ) = x ) ) )
14	10, 13	sylnibr 319	. 2 |- ( A. z e. B E. x e. B ( z .o. x ) =/= x -> -. M e. Mnd )
15		df-nel 2898	. 2 |- ( M e/ Mnd <-> -. M e. Mnd )
16	14, 15	sylibr 224	1 |- ( A. z e. B E. x e. B ( z .o. x ) =/= x -> M e/ Mnd )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   e/ wnel 2897   A.wral 2912   E.wrex 2913   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941  SGrpcsgrp 17283   Mndcmnd 17294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-mnd 17295

This theorem is referenced by:  sgrp2nmndlem5  17416  copisnmnd  41809  nnsgrpnmnd  41818  2zrngnring  41952