Theorem sgrpidmndm 13502

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 544	. . . . . . . . . 10 |- ( ( ( ( ( G e. Smgrp /\ e e. B ) /\ w e. e ) /\ e = .0. ) /\ x e. B ) -> e e. B )
2		simpllr 536	. . . . . . . . . . 11 |- ( ( ( ( ( G e. Smgrp /\ e e. B ) /\ w e. e ) /\ e = .0. ) /\ x e. B ) -> w e. e )
3	2	19.8ad 1639	. . . . . . . . . 10 |- ( ( ( ( ( G e. Smgrp /\ e e. B ) /\ w e. e ) /\ e = .0. ) /\ x e. B ) -> E. w w e. e )
4		simplr 529	. . . . . . . . . . 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 2231	. . . . . . . . . . . . . 14 |- ( +g ` G ) = ( +g ` G )
7		sgrpidmnd.0	. . . . . . . . . . . . . 14 |- .0. = ( 0g ` G )
8	5, 6, 7	grpidvalg 13455	. . . . . . . . . . . . 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 2243	. . . . . . . . . . . 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 1203	. . . . . . . . 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 2292	. . . . . . . . . . . 12 |- ( y = e -> ( y e. B <-> e e. B ) )
15		oveq1 6024	. . . . . . . . . . . . . 14 |- ( y = e -> ( y ( +g ` G ) x ) = ( e ( +g ` G ) x ) )
16	15	eqeq1d 2240	. . . . . . . . . . . . 13 |- ( y = e -> ( ( y ( +g ` G ) x ) = x <-> ( e ( +g ` G ) x ) = x ) )
17	16	ovanraleqv 6041	. . . . . . . . . . . 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 5318	. . . . . . . . . 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 2579	. . . . . . . . . 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 2605	. . . . . . 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 1867	. . . . 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 2634	. . 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 13500	. 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 1004   = wceq 1397   E.wex 1540   e. wcel 2202   A.wral 2510   E.wrex 2511   iotacio 5284   ` cfv 5326  (class class class)co 6017   Basecbs 13081   +g cplusg 13159   0gc0g 13338  Smgrpcsgrp 13483   Mndcmnd 13498

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-riota 5970  df-ov 6020  df-inn 9143  df-2 9201  df-ndx 13084  df-slot 13085  df-base 13087  df-plusg 13172  df-0g 13340  df-mnd 13499

This theorem is referenced by: (None)