Theorem 0subm 13316

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 2205	. . . 4 |- ( Base ` G ) = ( Base ` G )
2		0subm.z	. . . 4 |- .0. = ( 0g ` G )
3	1, 2	mndidcl 13262	. . 3 |- ( G e. Mnd -> .0. e. ( Base ` G ) )
4	3	snssd 3778	. 2 |- ( G e. Mnd -> { .0. } C_ ( Base ` G ) )
5		snidg 3662	. . 3 |- ( .0. e. ( Base ` G ) -> .0. e. { .0. } )
6	3, 5	syl 14	. 2 |- ( G e. Mnd -> .0. e. { .0. } )
7		velsn 3650	. . . . 5 |- ( a e. { .0. } <-> a = .0. )
8		velsn 3650	. . . . 5 |- ( b e. { .0. } <-> b = .0. )
9	7, 8	anbi12i 460	. . . 4 |- ( ( a e. { .0. } /\ b e. { .0. } ) <-> ( a = .0. /\ b = .0. ) )
10		eqid 2205	. . . . . . . 8 |- ( +g ` G ) = ( +g ` G )
11	1, 10, 2	mndlid 13267	. . . . . . 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 2282	. . . . . . 7 |- ( G e. Mnd -> ( .0. ( +g ` G ) .0. ) e. ( Base ` G ) )
14		elsng 3648	. . . . . . 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 5953	. . . . . 6 |- ( ( a = .0. /\ b = .0. ) -> ( a ( +g ` G ) b ) = ( .0. ( +g ` G ) .0. ) )
18	17	eleq1d 2274	. . . . 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 2587	. 2 |- ( G e. Mnd -> A. a e. { .0. } A. b e. { .0. } ( a ( +g ` G ) b ) e. { .0. } )
22	1, 2, 10	issubm 13304	. 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 1183	1 |- ( G e. Mnd -> { .0. } e. (SubMnd ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2176   A.wral 2484   C_ wss 3166   {csn 3633   ` cfv 5271  (class class class)co 5944   Basecbs 12832   +g cplusg 12909   0gc0g 13088   Mndcmnd 13248  SubMndcsubmnd 13290

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-cnex 8016  ax-resscn 8017  ax-1re 8019  ax-addrcl 8022

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-iota 5232  df-fun 5273  df-fn 5274  df-fv 5279  df-riota 5899  df-ov 5947  df-inn 9037  df-2 9095  df-ndx 12835  df-slot 12836  df-base 12838  df-plusg 12922  df-0g 13090  df-mgm 13188  df-sgrp 13234  df-mnd 13249  df-submnd 13292

This theorem is referenced by:  0subg  13535