Theorem mndissubm 12942

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 12898	. . . . . . 7 |- ( G e. Mnd -> G e. Mgm )
4		mndmgm 12898	. . . . . . 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 12840	. . . . 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 2572	. . 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 2189	. . . . 5 |- ( +g ` G ) = ( +g ` G )
17	10, 15, 16	issubm 12939	. . . 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 2160   A.wral 2468   C_ wss 3144   X. cxp 4642   |` cres 4646   ` cfv 5235  (class class class)co 5897   Basecbs 12515   +g cplusg 12592   0gc0g 12764  Mgmcmgm 12833   Mndcmnd 12892  SubMndcsubmnd 12925

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 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-cnex 7933  ax-resscn 7934  ax-1re 7936  ax-addrcl 7939

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243  df-ov 5900  df-inn 8951  df-2 9009  df-ndx 12518  df-slot 12519  df-base 12521  df-plusg 12605  df-mgm 12835  df-sgrp 12880  df-mnd 12893  df-submnd 12927

This theorem is referenced by: (None)