Theorem ismgmid2 12663

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 2548	. . 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 5872	. . . . . . . 8 |- ( e = U -> ( e .+ x ) = ( U .+ x ) )
10	9	eqeq1d 2184	. . . . . . 7 |- ( e = U -> ( ( e .+ x ) = x <-> ( U .+ x ) = x ) )
11	10	ovanraleqv 5889	. . . . . 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 2839	. . . . 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 12660	. . 3 |- ( ph -> ( ( U e. B /\ A. x e. B ( ( U .+ x ) = x /\ ( x .+ U ) = x ) ) <-> .0. = U ) )
15	1, 5, 14	mpbi2and 943	. 2 |- ( ph -> .0. = U )
16	15	eqcomd 2181	1 |- ( ph -> U = .0. )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2146   A.wral 2453   E.wrex 2454   ` cfv 5208  (class class class)co 5865   Basecbs 12427   +g cplusg 12491   0gc0g 12625

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-cnex 7877  ax-resscn 7878  ax-1re 7880  ax-addrcl 7883

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-iota 5170  df-fun 5210  df-fn 5211  df-fv 5216  df-riota 5821  df-ov 5868  df-inn 8891  df-ndx 12430  df-slot 12431  df-base 12433  df-0g 12627

This theorem is referenced by:  lidrididd  12665  grpidd  12666  mhmid  12838