Theorem 0subm 13116

Description: The zero submonoid of an arbitrary monoid. (Contributed by AV, 17-Feb-2024.)

Hypothesis

Ref	Expression
0subm.z	|- .0. = ( 0g ` G )

Assertion

Ref	Expression
0subm	|- ( G e. Mnd -> { .0. } e. (SubMnd ` G ) )

Proof of Theorem 0subm

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2196	. . . 4 |- ( Base ` G ) = ( Base ` G )
2		0subm.z	. . . 4 |- .0. = ( 0g ` G )
3	1, 2	mndidcl 13071	. . 3 |- ( G e. Mnd -> .0. e. ( Base ` G ) )
4	3	snssd 3767	. 2 |- ( G e. Mnd -> { .0. } C_ ( Base ` G ) )
5		snidg 3651	. . 3 |- ( .0. e. ( Base ` G ) -> .0. e. { .0. } )
6	3, 5	syl 14	. 2 |- ( G e. Mnd -> .0. e. { .0. } )
7		velsn 3639	. . . . 5 |- ( a e. { .0. } <-> a = .0. )
8		velsn 3639	. . . . 5 |- ( b e. { .0. } <-> b = .0. )
9	7, 8	anbi12i 460	. . . 4 |- ( ( a e. { .0. } /\ b e. { .0. } ) <-> ( a = .0. /\ b = .0. ) )
10		eqid 2196	. . . . . . . 8 |- ( +g ` G ) = ( +g ` G )
11	1, 10, 2	mndlid 13076	. . . . . . 7 |- ( ( G e. Mnd /\ .0. e. ( Base ` G ) ) -> ( .0. ( +g ` G ) .0. ) = .0. )
12	3, 11	mpdan 421	. . . . . 6 |- ( G e. Mnd -> ( .0. ( +g ` G ) .0. ) = .0. )
13	12, 3	eqeltrd 2273	. . . . . . 7 |- ( G e. Mnd -> ( .0. ( +g ` G ) .0. ) e. ( Base ` G ) )
14		elsng 3637	. . . . . . 7 |- ( ( .0. ( +g ` G ) .0. ) e. ( Base ` G ) -> ( ( .0. ( +g ` G ) .0. ) e. { .0. } <-> ( .0. ( +g ` G ) .0. ) = .0. ) )
15	13, 14	syl 14	. . . . . 6 |- ( G e. Mnd -> ( ( .0. ( +g ` G ) .0. ) e. { .0. } <-> ( .0. ( +g ` G ) .0. ) = .0. ) )
16	12, 15	mpbird 167	. . . . 5 |- ( G e. Mnd -> ( .0. ( +g ` G ) .0. ) e. { .0. } )
17		oveq12 5931	. . . . . 6 |- ( ( a = .0. /\ b = .0. ) -> ( a ( +g ` G ) b ) = ( .0. ( +g ` G ) .0. ) )
18	17	eleq1d 2265	. . . . 5 |- ( ( a = .0. /\ b = .0. ) -> ( ( a ( +g ` G ) b ) e. { .0. } <-> ( .0. ( +g ` G ) .0. ) e. { .0. } ) )
19	16, 18	syl5ibrcom 157	. . . 4 |- ( G e. Mnd -> ( ( a = .0. /\ b = .0. ) -> ( a ( +g ` G ) b ) e. { .0. } ) )
20	9, 19	biimtrid 152	. . 3 |- ( G e. Mnd -> ( ( a e. { .0. } /\ b e. { .0. } ) -> ( a ( +g ` G ) b ) e. { .0. } ) )
21	20	ralrimivv 2578	. 2 |- ( G e. Mnd -> A. a e. { .0. } A. b e. { .0. } ( a ( +g ` G ) b ) e. { .0. } )
22	1, 2, 10	issubm 13104	. 2 |- ( G e. Mnd -> ( { .0. } e. (SubMnd ` G ) <-> ( { .0. } C_ ( Base ` G ) /\ .0. e. { .0. } /\ A. a e. { .0. } A. b e. { .0. } ( a ( +g ` G ) b ) e. { .0. } ) ) )
23	4, 6, 21, 22	mpbir3and 1182	1 |- ( G e. Mnd -> { .0. } e. (SubMnd ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   A.wral 2475   C_ wss 3157   {csn 3622   ` cfv 5258  (class class class)co 5922   Basecbs 12678   +g cplusg 12755   0gc0g 12927   Mndcmnd 13057  SubMndcsubmnd 13090

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-riota 5877  df-ov 5925  df-inn 8991  df-2 9049  df-ndx 12681  df-slot 12682  df-base 12684  df-plusg 12768  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-submnd 13092

This theorem is referenced by:  0subg  13329