Theorem gsumvallem2 17372

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 17269	. 2 |- ( G e. Mnd -> O C_ { .0. } )
6	1, 2	mndidcl 17308	. . . 4 |- ( G e. Mnd -> .0. e. B )
7	1, 3, 2	mndlrid 17310	. . . . 5 |- ( ( G e. Mnd /\ y e. B ) -> ( ( .0. .+ y ) = y /\ ( y .+ .0. ) = y ) )
8	7	ralrimiva 2966	. . . 4 |- ( G e. Mnd -> A. y e. B ( ( .0. .+ y ) = y /\ ( y .+ .0. ) = y ) )
9		oveq1 6657	. . . . . . . 8 |- ( x = .0. -> ( x .+ y ) = ( .0. .+ y ) )
10	9	eqeq1d 2624	. . . . . . 7 |- ( x = .0. -> ( ( x .+ y ) = y <-> ( .0. .+ y ) = y ) )
11		oveq2 6658	. . . . . . . 8 |- ( x = .0. -> ( y .+ x ) = ( y .+ .0. ) )
12	11	eqeq1d 2624	. . . . . . 7 |- ( x = .0. -> ( ( y .+ x ) = y <-> ( y .+ .0. ) = y ) )
13	10, 12	anbi12d 747	. . . . . 6 |- ( x = .0. -> ( ( ( x .+ y ) = y /\ ( y .+ x ) = y ) <-> ( ( .0. .+ y ) = y /\ ( y .+ .0. ) = y ) ) )
14	13	ralbidv 2986	. . . . 5 |- ( x = .0. -> ( A. y e. B ( ( x .+ y ) = y /\ ( y .+ x ) = y ) <-> A. y e. B ( ( .0. .+ y ) = y /\ ( y .+ .0. ) = y ) ) )
15	14, 4	elrab2 3366	. . . 4 |- ( .0. e. O <-> ( .0. e. B /\ A. y e. B ( ( .0. .+ y ) = y /\ ( y .+ .0. ) = y ) ) )
16	6, 8, 15	sylanbrc 698	. . 3 |- ( G e. Mnd -> .0. e. O )
17	16	snssd 4340	. 2 |- ( G e. Mnd -> { .0. } C_ O )
18	5, 17	eqssd 3620	1 |- ( G e. Mnd -> O = { .0. } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   {csn 4177   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   0gc0g 16100   Mndcmnd 17294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-riota 6611  df-ov 6653  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295

This theorem is referenced by:  gsumz  17374  gsumval3a  18304