Theorem gsumvallem2 13783

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 13687	. 2 |- ( G e. Mnd -> O C_ { .0. } )
6	1, 2	mndidcl 13726	. . . 4 |- ( G e. Mnd -> .0. e. B )
7	1, 3, 2	mndlrid 13730	. . . . 5 |- ( ( G e. Mnd /\ y e. B ) -> ( ( .0. .+ y ) = y /\ ( y .+ .0. ) = y ) )
8	7	ralrimiva 2623	. . . 4 |- ( G e. Mnd -> A. y e. B ( ( .0. .+ y ) = y /\ ( y .+ .0. ) = y ) )
9		oveq1 6086	. . . . . . 7 |- ( x = .0. -> ( x .+ y ) = ( .0. .+ y ) )
10	9	eqeq1d 2247	. . . . . 6 |- ( x = .0. -> ( ( x .+ y ) = y <-> ( .0. .+ y ) = y ) )
11	10	ovanraleqv 6103	. . . . 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 2985	. . . 4 |- ( .0. e. O <-> ( .0. e. B /\ A. y e. B ( ( .0. .+ y ) = y /\ ( y .+ .0. ) = y ) ) )
13	6, 8, 12	sylanbrc 421	. . 3 |- ( G e. Mnd -> .0. e. O )
14	13	snssd 3858	. 2 |- ( G e. Mnd -> { .0. } C_ O )
15	5, 14	eqssd 3265	1 |- ( G e. Mnd -> O = { .0. } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   {crab 2532   {csn 3708   ` cfv 5375  (class class class)co 6079   Basecbs 13335   +g cplusg 13414   0gc0g 13593   Mndcmnd 13712

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-cnex 8264  ax-resscn 8265  ax-1re 8267  ax-addrcl 8270

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-iota 5335  df-fun 5377  df-fn 5378  df-fv 5383  df-riota 6032  df-ov 6082  df-inn 9288  df-2 9346  df-ndx 13338  df-slot 13339  df-base 13341  df-plusg 13427  df-0g 13595  df-mgm 13659  df-sgrp 13700  df-mnd 13713

This theorem is referenced by: (None)