Theorem 0subm 13056

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 2193	. . . 4 |- ( Base ` G ) = ( Base ` G )
2		0subm.z	. . . 4 |- .0. = ( 0g ` G )
3	1, 2	mndidcl 13011	. . 3 |- ( G e. Mnd -> .0. e. ( Base ` G ) )
4	3	snssd 3763	. 2 |- ( G e. Mnd -> { .0. } C_ ( Base ` G ) )
5		snidg 3647	. . 3 |- ( .0. e. ( Base ` G ) -> .0. e. { .0. } )
6	3, 5	syl 14	. 2 |- ( G e. Mnd -> .0. e. { .0. } )
7		velsn 3635	. . . . 5 |- ( a e. { .0. } <-> a = .0. )
8		velsn 3635	. . . . 5 |- ( b e. { .0. } <-> b = .0. )
9	7, 8	anbi12i 460	. . . 4 |- ( ( a e. { .0. } /\ b e. { .0. } ) <-> ( a = .0. /\ b = .0. ) )
10		eqid 2193	. . . . . . . 8 |- ( +g ` G ) = ( +g ` G )
11	1, 10, 2	mndlid 13016	. . . . . . 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 2270	. . . . . . 7 |- ( G e. Mnd -> ( .0. ( +g ` G ) .0. ) e. ( Base ` G ) )
14		elsng 3633	. . . . . . 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 5927	. . . . . 6 |- ( ( a = .0. /\ b = .0. ) -> ( a ( +g ` G ) b ) = ( .0. ( +g ` G ) .0. ) )
18	17	eleq1d 2262	. . . . 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 2575	. 2 |- ( G e. Mnd -> A. a e. { .0. } A. b e. { .0. } ( a ( +g ` G ) b ) e. { .0. } )
22	1, 2, 10	issubm 13044	. 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 2164   A.wral 2472   C_ wss 3153   {csn 3618   ` cfv 5254  (class class class)co 5918   Basecbs 12618   +g cplusg 12695   0gc0g 12867   Mndcmnd 12997  SubMndcsubmnd 13030

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262  df-riota 5873  df-ov 5921  df-inn 8983  df-2 9041  df-ndx 12621  df-slot 12622  df-base 12624  df-plusg 12708  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-submnd 13032

This theorem is referenced by:  0subg  13269