Theorem mndissubm 13841

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 1034	. . 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 1035	. . 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 13794	. . . . . . 7 |- ( G e. Mnd -> G e. Mgm )
4		mndmgm 13794	. . . . . . 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 492	. . . . 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 1026	. . . . . 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 493	. . . . 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 13740	. . . . 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 1278	. . . 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 2632	. . 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 2238	. . . . 5 |- ( +g ` G ) = ( +g ` G )
17	10, 15, 16	issubm 13838	. . . 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 492	. . 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 1211	. 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 ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   C_ wss 3220   X. cxp 4774   |` cres 4778   ` cfv 5380  (class class class)co 6088   Basecbs 13410   +g cplusg 13490   0gc0g 13669  Mgmcmgm 13733   Mndcmnd 13788  SubMndcsubmnd 13824

This proof depends on 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 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-cnex 8276  ax-resscn 8277  ax-1re 8279  ax-addrcl 8282

This proof 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-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-iota 5340  df-fun 5382  df-fn 5383  df-fv 5388  df-ov 6091  df-inn 9314  df-2 9372  df-ndx 13413  df-slot 13414  df-base 13416  df-plusg 13503  df-mgm 13735  df-sgrp 13776  df-mnd 13789  df-submnd 13826

This theorem is used by: (None)