Theorem gsumvallem2 13637

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 13528	. 2 |- ( G e. Mnd -> O C_ { .0. } )
6	1, 2	mndidcl 13574	. . . 4 |- ( G e. Mnd -> .0. e. B )
7	1, 3, 2	mndlrid 13578	. . . . 5 |- ( ( G e. Mnd /\ y e. B ) -> ( ( .0. .+ y ) = y /\ ( y .+ .0. ) = y ) )
8	7	ralrimiva 2606	. . . 4 |- ( G e. Mnd -> A. y e. B ( ( .0. .+ y ) = y /\ ( y .+ .0. ) = y ) )
9		oveq1 6035	. . . . . . 7 |- ( x = .0. -> ( x .+ y ) = ( .0. .+ y ) )
10	9	eqeq1d 2240	. . . . . 6 |- ( x = .0. -> ( ( x .+ y ) = y <-> ( .0. .+ y ) = y ) )
11	10	ovanraleqv 6052	. . . . 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 2966	. . . 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 3823	. 2 |- ( G e. Mnd -> { .0. } C_ O )
15	5, 14	eqssd 3245	1 |- ( G e. Mnd -> O = { .0. } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   A.wral 2511   {crab 2515   {csn 3673   ` cfv 5333  (class class class)co 6028   Basecbs 13143   +g cplusg 13221   0gc0g 13400   Mndcmnd 13560

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-cnex 8166  ax-resscn 8167  ax-1re 8169  ax-addrcl 8172

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-riota 5981  df-ov 6031  df-inn 9187  df-2 9245  df-ndx 13146  df-slot 13147  df-base 13149  df-plusg 13234  df-0g 13402  df-mgm 13500  df-sgrp 13546  df-mnd 13561

This theorem is referenced by: (None)