Theorem gsumvallem2 13512

Description: Lemma for properties of the set of identities of G. The set of identities of a monoid is exactly the unique identity element. (Contributed by Mario Carneiro, 7-Dec-2014.)

Hypotheses

Ref	Expression
gsumvallem2.b	|- B = ( Base ` G )
gsumvallem2.z	|- .0. = ( 0g ` G )
gsumvallem2.p	|- .+ = ( +g ` G )
gsumvallem2.o	|- O = { x e. B | A. y e. B ( ( x .+ y ) = y /\ ( y .+ x ) = y ) }

Assertion

Ref	Expression
gsumvallem2	|- ( G e. Mnd -> O = { .0. } )

Distinct variable groups:   x, y, B   x, G, y   x, .+ , y   x, .0. , y

Allowed substitution hints:   O( x, y)

Proof of Theorem gsumvallem2

Step	Hyp	Ref	Expression
1		gsumvallem2.b	. . 3 |- B = ( Base ` G )
2		gsumvallem2.z	. . 3 |- .0. = ( 0g ` G )
3		gsumvallem2.p	. . 3 |- .+ = ( +g ` G )
4		gsumvallem2.o	. . 3 |- O = { x e. B | A. y e. B ( ( x .+ y ) = y /\ ( y .+ x ) = y ) }
5	1, 2, 3, 4	mgmidsssn0 13403	. 2 |- ( G e. Mnd -> O C_ { .0. } )
6	1, 2	mndidcl 13449	. . . 4 |- ( G e. Mnd -> .0. e. B )
7	1, 3, 2	mndlrid 13453	. . . . 5 |- ( ( G e. Mnd /\ y e. B ) -> ( ( .0. .+ y ) = y /\ ( y .+ .0. ) = y ) )
8	7	ralrimiva 2603	. . . 4 |- ( G e. Mnd -> A. y e. B ( ( .0. .+ y ) = y /\ ( y .+ .0. ) = y ) )
9		oveq1 6001	. . . . . . 7 |- ( x = .0. -> ( x .+ y ) = ( .0. .+ y ) )
10	9	eqeq1d 2238	. . . . . 6 |- ( x = .0. -> ( ( x .+ y ) = y <-> ( .0. .+ y ) = y ) )
11	10	ovanraleqv 6018	. . . . 5 |- ( x = .0. -> ( A. y e. B ( ( x .+ y ) = y /\ ( y .+ x ) = y ) <-> A. y e. B ( ( .0. .+ y ) = y /\ ( y .+ .0. ) = y ) ) )
12	11, 4	elrab2 2962	. . . 4 |- ( .0. e. O <-> ( .0. e. B /\ A. y e. B ( ( .0. .+ y ) = y /\ ( y .+ .0. ) = y ) ) )
13	6, 8, 12	sylanbrc 417	. . 3 |- ( G e. Mnd -> .0. e. O )
14	13	snssd 3812	. 2 |- ( G e. Mnd -> { .0. } C_ O )
15	5, 14	eqssd 3241	1 |- ( G e. Mnd -> O = { .0. } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   A.wral 2508   {crab 2512   {csn 3666   ` cfv 5314  (class class class)co 5994   Basecbs 13018   +g cplusg 13096   0gc0g 13275   Mndcmnd 13435

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-cnex 8078  ax-resscn 8079  ax-1re 8081  ax-addrcl 8084

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-iota 5274  df-fun 5316  df-fn 5317  df-fv 5322  df-riota 5947  df-ov 5997  df-inn 9099  df-2 9157  df-ndx 13021  df-slot 13022  df-base 13024  df-plusg 13109  df-0g 13277  df-mgm 13375  df-sgrp 13421  df-mnd 13436

This theorem is referenced by: (None)