Theorem 0subm 13714

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 2234	. . . 4 |- ( Base ` G ) = ( Base ` G )
2		0subm.z	. . . 4 |- .0. = ( 0g ` G )
3	1, 2	mndidcl 13660	. . 3 |- ( G e. Mnd -> .0. e. ( Base ` G ) )
4	3	snssd 3841	. 2 |- ( G e. Mnd -> { .0. } C_ ( Base ` G ) )
5		snidg 3720	. . 3 |- ( .0. e. ( Base ` G ) -> .0. e. { .0. } )
6	3, 5	syl 14	. 2 |- ( G e. Mnd -> .0. e. { .0. } )
7		velsn 3708	. . . . 5 |- ( a e. { .0. } <-> a = .0. )
8		velsn 3708	. . . . 5 |- ( b e. { .0. } <-> b = .0. )
9	7, 8	anbi12i 460	. . . 4 |- ( ( a e. { .0. } /\ b e. { .0. } ) <-> ( a = .0. /\ b = .0. ) )
10		eqid 2234	. . . . . . . 8 |- ( +g ` G ) = ( +g ` G )
11	1, 10, 2	mndlid 13665	. . . . . . 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 2311	. . . . . . 7 |- ( G e. Mnd -> ( .0. ( +g ` G ) .0. ) e. ( Base ` G ) )
14		elsng 3706	. . . . . . 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 6061	. . . . . 6 |- ( ( a = .0. /\ b = .0. ) -> ( a ( +g ` G ) b ) = ( .0. ( +g ` G ) .0. ) )
18	17	eleq1d 2303	. . . . 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 2625	. 2 |- ( G e. Mnd -> A. a e. { .0. } A. b e. { .0. } ( a ( +g ` G ) b ) e. { .0. } )
22	1, 2, 10	issubm 13702	. 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 1207	1 |- ( G e. Mnd -> { .0. } e. (SubMnd ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   A.wral 2522   C_ wss 3213   {csn 3691   ` cfv 5354  (class class class)co 6052   Basecbs 13229   +g cplusg 13307   0gc0g 13486   Mndcmnd 13646  SubMndcsubmnd 13688

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-cnex 8220  ax-resscn 8221  ax-1re 8223  ax-addrcl 8226

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-iota 5314  df-fun 5356  df-fn 5357  df-fv 5362  df-riota 6005  df-ov 6055  df-inn 9240  df-2 9298  df-ndx 13232  df-slot 13233  df-base 13235  df-plusg 13320  df-0g 13488  df-mgm 13586  df-sgrp 13632  df-mnd 13647  df-submnd 13690

This theorem is referenced by:  0subg  13933