Theorem sgrpidmndm 13223

Description: A semigroup with an identity element which is inhabited is a monoid. Of course there could be monoids with the empty set as identity element, but these cannot be proven to be monoids with this theorem. (Contributed by AV, 29-Jan-2024.)

Hypotheses

Ref	Expression
sgrpidmnd.b	|- B = ( Base ` G )
sgrpidmnd.0	|- .0. = ( 0g ` G )

Assertion

Ref	Expression
sgrpidmndm	|- ( ( G e. Smgrp /\ E. e e. B ( E. w w e. e /\ e = .0. ) ) -> G e. Mnd )

Distinct variable groups:   B, e, w   e, G, w   w, .0.   w, e

Allowed substitution hint:   .0. ( e)

Proof of Theorem sgrpidmndm

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp-4r 542	. . . . . . . . . 10 |- ( ( ( ( ( G e. Smgrp /\ e e. B ) /\ w e. e ) /\ e = .0. ) /\ x e. B ) -> e e. B )
2		simpllr 534	. . . . . . . . . . 11 |- ( ( ( ( ( G e. Smgrp /\ e e. B ) /\ w e. e ) /\ e = .0. ) /\ x e. B ) -> w e. e )
3	2	19.8ad 1613	. . . . . . . . . 10 |- ( ( ( ( ( G e. Smgrp /\ e e. B ) /\ w e. e ) /\ e = .0. ) /\ x e. B ) -> E. w w e. e )
4		simplr 528	. . . . . . . . . . 11 |- ( ( ( ( ( G e. Smgrp /\ e e. B ) /\ w e. e ) /\ e = .0. ) /\ x e. B ) -> e = .0. )
5		sgrpidmnd.b	. . . . . . . . . . . . . 14 |- B = ( Base ` G )
6		eqid 2204	. . . . . . . . . . . . . 14 |- ( +g ` G ) = ( +g ` G )
7		sgrpidmnd.0	. . . . . . . . . . . . . 14 |- .0. = ( 0g ` G )
8	5, 6, 7	grpidvalg 13176	. . . . . . . . . . . . 13 |- ( G e. Smgrp -> .0. = ( iota y ( y e. B /\ A. x e. B ( ( y ( +g ` G ) x ) = x /\ ( x ( +g ` G ) y ) = x ) ) ) )
9	8	eqeq2d 2216	. . . . . . . . . . . 12 |- ( G e. Smgrp -> ( e = .0. <-> e = ( iota y ( y e. B /\ A. x e. B ( ( y ( +g ` G ) x ) = x /\ ( x ( +g ` G ) y ) = x ) ) ) ) )
10	9	ad4antr 494	. . . . . . . . . . 11 |- ( ( ( ( ( G e. Smgrp /\ e e. B ) /\ w e. e ) /\ e = .0. ) /\ x e. B ) -> ( e = .0. <-> e = ( iota y ( y e. B /\ A. x e. B ( ( y ( +g ` G ) x ) = x /\ ( x ( +g ` G ) y ) = x ) ) ) ) )
11	4, 10	mpbid 147	. . . . . . . . . 10 |- ( ( ( ( ( G e. Smgrp /\ e e. B ) /\ w e. e ) /\ e = .0. ) /\ x e. B ) -> e = ( iota y ( y e. B /\ A. x e. B ( ( y ( +g ` G ) x ) = x /\ ( x ( +g ` G ) y ) = x ) ) ) )
12	1, 3, 11	3jca 1179	. . . . . . . . 9 |- ( ( ( ( ( G e. Smgrp /\ e e. B ) /\ w e. e ) /\ e = .0. ) /\ x e. B ) -> ( e e. B /\ E. w w e. e /\ e = ( iota y ( y e. B /\ A. x e. B ( ( y ( +g ` G ) x ) = x /\ ( x ( +g ` G ) y ) = x ) ) ) ) )
13		simpr 110	. . . . . . . . 9 |- ( ( ( ( ( G e. Smgrp /\ e e. B ) /\ w e. e ) /\ e = .0. ) /\ x e. B ) -> x e. B )
14		eleq1w 2265	. . . . . . . . . . . 12 |- ( y = e -> ( y e. B <-> e e. B ) )
15		oveq1 5950	. . . . . . . . . . . . . 14 |- ( y = e -> ( y ( +g ` G ) x ) = ( e ( +g ` G ) x ) )
16	15	eqeq1d 2213	. . . . . . . . . . . . 13 |- ( y = e -> ( ( y ( +g ` G ) x ) = x <-> ( e ( +g ` G ) x ) = x ) )
17	16	ovanraleqv 5967	. . . . . . . . . . . 12 |- ( y = e -> ( A. x e. B ( ( y ( +g ` G ) x ) = x /\ ( x ( +g ` G ) y ) = x ) <-> A. x e. B ( ( e ( +g ` G ) x ) = x /\ ( x ( +g ` G ) e ) = x ) ) )
18	14, 17	anbi12d 473	. . . . . . . . . . 11 |- ( y = e -> ( ( y e. B /\ A. x e. B ( ( y ( +g ` G ) x ) = x /\ ( x ( +g ` G ) y ) = x ) ) <-> ( e e. B /\ A. x e. B ( ( e ( +g ` G ) x ) = x /\ ( x ( +g ` G ) e ) = x ) ) ) )
19	18	iotam 5262	. . . . . . . . . 10 |- ( ( e e. B /\ E. w w e. e /\ e = ( iota y ( y e. B /\ A. x e. B ( ( y ( +g ` G ) x ) = x /\ ( x ( +g ` G ) y ) = x ) ) ) ) -> ( e e. B /\ A. x e. B ( ( e ( +g ` G ) x ) = x /\ ( x ( +g ` G ) e ) = x ) ) )
20		rsp 2552	. . . . . . . . . 10 |- ( A. x e. B ( ( e ( +g ` G ) x ) = x /\ ( x ( +g ` G ) e ) = x ) -> ( x e. B -> ( ( e ( +g ` G ) x ) = x /\ ( x ( +g ` G ) e ) = x ) ) )
21	19, 20	simpl2im 386	. . . . . . . . 9 |- ( ( e e. B /\ E. w w e. e /\ e = ( iota y ( y e. B /\ A. x e. B ( ( y ( +g ` G ) x ) = x /\ ( x ( +g ` G ) y ) = x ) ) ) ) -> ( x e. B -> ( ( e ( +g ` G ) x ) = x /\ ( x ( +g ` G ) e ) = x ) ) )
22	12, 13, 21	sylc 62	. . . . . . . 8 |- ( ( ( ( ( G e. Smgrp /\ e e. B ) /\ w e. e ) /\ e = .0. ) /\ x e. B ) -> ( ( e ( +g ` G ) x ) = x /\ ( x ( +g ` G ) e ) = x ) )
23	22	ralrimiva 2578	. . . . . . 7 |- ( ( ( ( G e. Smgrp /\ e e. B ) /\ w e. e ) /\ e = .0. ) -> A. x e. B ( ( e ( +g ` G ) x ) = x /\ ( x ( +g ` G ) e ) = x ) )
24	23	exp31 364	. . . . . 6 |- ( ( G e. Smgrp /\ e e. B ) -> ( w e. e -> ( e = .0. -> A. x e. B ( ( e ( +g ` G ) x ) = x /\ ( x ( +g ` G ) e ) = x ) ) ) )
25	24	exlimdv 1841	. . . . 5 |- ( ( G e. Smgrp /\ e e. B ) -> ( E. w w e. e -> ( e = .0. -> A. x e. B ( ( e ( +g ` G ) x ) = x /\ ( x ( +g ` G ) e ) = x ) ) ) )
26	25	impd 254	. . . 4 |- ( ( G e. Smgrp /\ e e. B ) -> ( ( E. w w e. e /\ e = .0. ) -> A. x e. B ( ( e ( +g ` G ) x ) = x /\ ( x ( +g ` G ) e ) = x ) ) )
27	26	reximdva 2607	. . 3 |- ( G e. Smgrp -> ( E. e e. B ( E. w w e. e /\ e = .0. ) -> E. e e. B A. x e. B ( ( e ( +g ` G ) x ) = x /\ ( x ( +g ` G ) e ) = x ) ) )
28	27	imdistani 445	. 2 |- ( ( G e. Smgrp /\ E. e e. B ( E. w w e. e /\ e = .0. ) ) -> ( G e. Smgrp /\ E. e e. B A. x e. B ( ( e ( +g ` G ) x ) = x /\ ( x ( +g ` G ) e ) = x ) ) )
29	5, 6	ismnddef 13221	. 2 |- ( G e. Mnd <-> ( G e. Smgrp /\ E. e e. B A. x e. B ( ( e ( +g ` G ) x ) = x /\ ( x ( +g ` G ) e ) = x ) ) )
30	28, 29	sylibr 134	1 |- ( ( G e. Smgrp /\ E. e e. B ( E. w w e. e /\ e = .0. ) ) -> G e. Mnd )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1372   E.wex 1514   e. wcel 2175   A.wral 2483   E.wrex 2484   iotacio 5229   ` cfv 5270  (class class class)co 5943   Basecbs 12803   +g cplusg 12880   0gc0g 13059  Smgrpcsgrp 13204   Mndcmnd 13219

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-cnex 8015  ax-resscn 8016  ax-1re 8018  ax-addrcl 8021

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-iota 5231  df-fun 5272  df-fn 5273  df-fv 5278  df-riota 5898  df-ov 5946  df-inn 9036  df-2 9094  df-ndx 12806  df-slot 12807  df-base 12809  df-plusg 12893  df-0g 13061  df-mnd 13220

This theorem is referenced by: (None)