Theorem mndissubm 13340

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 1006	. . 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 1007	. . 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 13287	. . . . . . 7 |- ( G e. Mnd -> G e. Mgm )
4		mndmgm 13287	. . . . . . 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 998	. . . . . 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 13226	. . . . 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 1250	. . . 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 2588	. . 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 2205	. . . . 5 |- ( +g ` G ) = ( +g ` G )
17	10, 15, 16	issubm 13337	. . . 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 1183	. 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 981   = wceq 1373   e. wcel 2176   A.wral 2484   C_ wss 3166   X. cxp 4674   |` cres 4678   ` cfv 5272  (class class class)co 5946   Basecbs 12865   +g cplusg 12942   0gc0g 13121  Mgmcmgm 13219   Mndcmnd 13281  SubMndcsubmnd 13323

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 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-cnex 8018  ax-resscn 8019  ax-1re 8021  ax-addrcl 8024

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-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 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-iota 5233  df-fun 5274  df-fn 5275  df-fv 5280  df-ov 5949  df-inn 9039  df-2 9097  df-ndx 12868  df-slot 12869  df-base 12871  df-plusg 12955  df-mgm 13221  df-sgrp 13267  df-mnd 13282  df-submnd 13325

This theorem is referenced by: (None)