Theorem mndissubm 13047

Description: If the base set of a monoid is contained in the base set of another monoid, and the group operation of the monoid is the restriction of the group operation of the other monoid to its base set, and the identity element of the the other monoid is contained in the base set of the monoid, then the (base set of the) monoid is a submonoid of the other monoid. (Contributed by AV, 17-Feb-2024.)

Hypotheses

Ref	Expression
mndissubm.b	|- B = ( Base ` G )
mndissubm.s	|- S = ( Base ` H )
mndissubm.z	|- .0. = ( 0g ` G )

Assertion

Ref	Expression
mndissubm	|- ( ( G e. Mnd /\ H e. Mnd ) -> ( ( S C_ B /\ .0. e. S /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) -> S e. (SubMnd ` G ) ) )

Proof of Theorem mndissubm

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

Step	Hyp	Ref	Expression
1		simpr1 1005	. . 3 |- ( ( ( G e. Mnd /\ H e. Mnd ) /\ ( S C_ B /\ .0. e. S /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) -> S C_ B )
2		simpr2 1006	. . 3 |- ( ( ( G e. Mnd /\ H e. Mnd ) /\ ( S C_ B /\ .0. e. S /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) -> .0. e. S )
3		mndmgm 13003	. . . . . . 7 |- ( G e. Mnd -> G e. Mgm )
4		mndmgm 13003	. . . . . . 7 |- ( H e. Mnd -> H e. Mgm )
5	3, 4	anim12i 338	. . . . . 6 |- ( ( G e. Mnd /\ H e. Mnd ) -> ( G e. Mgm /\ H e. Mgm ) )
6	5	ad2antrr 488	. . . . 5 |- ( ( ( ( G e. Mnd /\ H e. Mnd ) /\ ( S C_ B /\ .0. e. S /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) /\ ( a e. S /\ b e. S ) ) -> ( G e. Mgm /\ H e. Mgm ) )
7		3simpb 997	. . . . . 6 |- ( ( S C_ B /\ .0. e. S /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) -> ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) )
8	7	ad2antlr 489	. . . . 5 |- ( ( ( ( G e. Mnd /\ H e. Mnd ) /\ ( S C_ B /\ .0. e. S /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) /\ ( a e. S /\ b e. S ) ) -> ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) )
9		simpr 110	. . . . 5 |- ( ( ( ( G e. Mnd /\ H e. Mnd ) /\ ( S C_ B /\ .0. e. S /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) /\ ( a e. S /\ b e. S ) ) -> ( a e. S /\ b e. S ) )
10		mndissubm.b	. . . . . 6 |- B = ( Base ` G )
11		mndissubm.s	. . . . . 6 |- S = ( Base ` H )
12	10, 11	mgmsscl 12944	. . . . 5 |- ( ( ( G e. Mgm /\ H e. Mgm ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) /\ ( a e. S /\ b e. S ) ) -> ( a ( +g ` G ) b ) e. S )
13	6, 8, 9, 12	syl3anc 1249	. . . 4 |- ( ( ( ( G e. Mnd /\ H e. Mnd ) /\ ( S C_ B /\ .0. e. S /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) /\ ( a e. S /\ b e. S ) ) -> ( a ( +g ` G ) b ) e. S )
14	13	ralrimivva 2576	. . 3 |- ( ( ( G e. Mnd /\ H e. Mnd ) /\ ( S C_ B /\ .0. e. S /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) -> A. a e. S A. b e. S ( a ( +g ` G ) b ) e. S )
15		mndissubm.z	. . . . 5 |- .0. = ( 0g ` G )
16		eqid 2193	. . . . 5 |- ( +g ` G ) = ( +g ` G )
17	10, 15, 16	issubm 13044	. . . 4 |- ( G e. Mnd -> ( S e. (SubMnd ` G ) <-> ( S C_ B /\ .0. e. S /\ A. a e. S A. b e. S ( a ( +g ` G ) b ) e. S ) ) )
18	17	ad2antrr 488	. . 3 |- ( ( ( G e. Mnd /\ H e. Mnd ) /\ ( S C_ B /\ .0. e. S /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) -> ( S e. (SubMnd ` G ) <-> ( S C_ B /\ .0. e. S /\ A. a e. S A. b e. S ( a ( +g ` G ) b ) e. S ) ) )
19	1, 2, 14, 18	mpbir3and 1182	. 2 |- ( ( ( G e. Mnd /\ H e. Mnd ) /\ ( S C_ B /\ .0. e. S /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) -> S e. (SubMnd ` G ) )
20	19	ex 115	1 |- ( ( G e. Mnd /\ H e. Mnd ) -> ( ( S C_ B /\ .0. e. S /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) -> S e. (SubMnd ` G ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   A.wral 2472   C_ wss 3153   X. cxp 4657   |` cres 4661   ` cfv 5254  (class class class)co 5918   Basecbs 12618   +g cplusg 12695   0gc0g 12867  Mgmcmgm 12937   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-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-ov 5921  df-inn 8983  df-2 9041  df-ndx 12621  df-slot 12622  df-base 12624  df-plusg 12708  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-submnd 13032

This theorem is referenced by: (None)