Theorem 0subm 13768

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 2238	. . . 4 |- ( Base ` G ) = ( Base ` G )
2		0subm.z	. . . 4 |- .0. = ( 0g ` G )
3	1, 2	mndidcl 13720	. . 3 |- ( G e. Mnd -> .0. e. ( Base ` G ) )
4	3	snssd 3855	. 2 |- ( G e. Mnd -> { .0. } C_ ( Base ` G ) )
5		snidg 3734	. . 3 |- ( .0. e. ( Base ` G ) -> .0. e. { .0. } )
6	3, 5	syl 14	. 2 |- ( G e. Mnd -> .0. e. { .0. } )
7		velsn 3722	. . . . 5 |- ( a e. { .0. } <-> a = .0. )
8		velsn 3722	. . . . 5 |- ( b e. { .0. } <-> b = .0. )
9	7, 8	anbi12i 464	. . . 4 |- ( ( a e. { .0. } /\ b e. { .0. } ) <-> ( a = .0. /\ b = .0. ) )
10		eqid 2238	. . . . . . . 8 |- ( +g ` G ) = ( +g ` G )
11	1, 10, 2	mndlid 13725	. . . . . . 7 |- ( ( G e. Mnd /\ .0. e. ( Base ` G ) ) -> ( .0. ( +g ` G ) .0. ) = .0. )
12	3, 11	mpdan 425	. . . . . 6 |- ( G e. Mnd -> ( .0. ( +g ` G ) .0. ) = .0. )
13	12, 3	eqeltrd 2315	. . . . . . 7 |- ( G e. Mnd -> ( .0. ( +g ` G ) .0. ) e. ( Base ` G ) )
14		elsng 3720	. . . . . . 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 6084	. . . . . 6 |- ( ( a = .0. /\ b = .0. ) -> ( a ( +g ` G ) b ) = ( .0. ( +g ` G ) .0. ) )
18	17	eleq1d 2307	. . . . 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 2631	. 2 |- ( G e. Mnd -> A. a e. { .0. } A. b e. { .0. } ( a ( +g ` G ) b ) e. { .0. } )
22	1, 2, 10	issubm 13756	. 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 1211	1 |- ( G e. Mnd -> { .0. } e. (SubMnd ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   C_ wss 3220   {csn 3705   ` cfv 5372  (class class class)co 6075   Basecbs 13330   +g cplusg 13408   0gc0g 13587   Mndcmnd 13706  SubMndcsubmnd 13742

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-riota 6028  df-ov 6078  df-inn 9284  df-2 9342  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-submnd 13744

This theorem is referenced by:  0subg  13979