Theorem ismgmid2 13212

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 2579	. . 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 5951	. . . . . . . 8 |- ( e = U -> ( e .+ x ) = ( U .+ x ) )
10	9	eqeq1d 2214	. . . . . . 7 |- ( e = U -> ( ( e .+ x ) = x <-> ( U .+ x ) = x ) )
11	10	ovanraleqv 5968	. . . . . 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 2877	. . . . 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 13209	. . 3 |- ( ph -> ( ( U e. B /\ A. x e. B ( ( U .+ x ) = x /\ ( x .+ U ) = x ) ) <-> .0. = U ) )
15	1, 5, 14	mpbi2and 946	. 2 |- ( ph -> .0. = U )
16	15	eqcomd 2211	1 |- ( ph -> U = .0. )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   A.wral 2484   E.wrex 2485   ` cfv 5271  (class class class)co 5944   Basecbs 12832   +g cplusg 12909   0gc0g 13088

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-cnex 8016  ax-resscn 8017  ax-1re 8019  ax-addrcl 8022

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-iota 5232  df-fun 5273  df-fn 5274  df-fv 5279  df-riota 5899  df-ov 5947  df-inn 9037  df-ndx 12835  df-slot 12836  df-base 12838  df-0g 13090

This theorem is referenced by:  lidrididd  13214  grpidd  13215  mhmid  13451  ringidss  13791