Theorem ismgmid2 13413

Description: Show that a given element is the identity element of a magma. (Contributed by Mario Carneiro, 27-Dec-2014.)

Hypotheses

Ref	Expression
ismgmid.b	|- B = ( Base ` G )
ismgmid.o	|- .0. = ( 0g ` G )
ismgmid.p	|- .+ = ( +g ` G )
ismgmid2.u	|- ( ph -> U e. B )
ismgmid2.l	|- ( ( ph /\ x e. B ) -> ( U .+ x ) = x )
ismgmid2.r	|- ( ( ph /\ x e. B ) -> ( x .+ U ) = x )

Assertion

Ref	Expression
ismgmid2	|- ( ph -> U = .0. )

Distinct variable groups:   x, .+   x, .0.   x, B   x, G   x, U   ph, x

Proof of Theorem ismgmid2

Dummy variable e is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ismgmid2.u	. . 3 |- ( ph -> U e. B )
2		ismgmid2.l	. . . . 5 |- ( ( ph /\ x e. B ) -> ( U .+ x ) = x )
3		ismgmid2.r	. . . . 5 |- ( ( ph /\ x e. B ) -> ( x .+ U ) = x )
4	2, 3	jca 306	. . . 4 |- ( ( ph /\ x e. B ) -> ( ( U .+ x ) = x /\ ( x .+ U ) = x ) )
5	4	ralrimiva 2603	. . 3 |- ( ph -> A. x e. B ( ( U .+ x ) = x /\ ( x .+ U ) = x ) )
6		ismgmid.b	. . . 4 |- B = ( Base ` G )
7		ismgmid.o	. . . 4 |- .0. = ( 0g ` G )
8		ismgmid.p	. . . 4 |- .+ = ( +g ` G )
9		oveq1 6008	. . . . . . . 8 |- ( e = U -> ( e .+ x ) = ( U .+ x ) )
10	9	eqeq1d 2238	. . . . . . 7 |- ( e = U -> ( ( e .+ x ) = x <-> ( U .+ x ) = x ) )
11	10	ovanraleqv 6025	. . . . . 6 |- ( e = U -> ( A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) <-> A. x e. B ( ( U .+ x ) = x /\ ( x .+ U ) = x ) ) )
12	11	rspcev 2907	. . . . 5 |- ( ( U e. B /\ A. x e. B ( ( U .+ x ) = x /\ ( x .+ U ) = x ) ) -> E. e e. B A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) )
13	1, 5, 12	syl2anc 411	. . . 4 |- ( ph -> E. e e. B A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) )
14	6, 7, 8, 13	ismgmid 13410	. . 3 |- ( ph -> ( ( U e. B /\ A. x e. B ( ( U .+ x ) = x /\ ( x .+ U ) = x ) ) <-> .0. = U ) )
15	1, 5, 14	mpbi2and 949	. 2 |- ( ph -> .0. = U )
16	15	eqcomd 2235	1 |- ( ph -> U = .0. )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   A.wral 2508   E.wrex 2509   ` cfv 5318  (class class class)co 6001   Basecbs 13032   +g cplusg 13110   0gc0g 13289

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 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-cnex 8090  ax-resscn 8091  ax-1re 8093  ax-addrcl 8096

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 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326  df-riota 5954  df-ov 6004  df-inn 9111  df-ndx 13035  df-slot 13036  df-base 13038  df-0g 13291

This theorem is referenced by:  lidrididd  13415  grpidd  13416  mhmid  13652  ringidss  13992