Theorem ismgmid2 13327

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 2581	. . 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 5974	. . . . . . . 8 |- ( e = U -> ( e .+ x ) = ( U .+ x ) )
10	9	eqeq1d 2216	. . . . . . 7 |- ( e = U -> ( ( e .+ x ) = x <-> ( U .+ x ) = x ) )
11	10	ovanraleqv 5991	. . . . . 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 2884	. . . . 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 13324	. . 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 2213	1 |- ( ph -> U = .0. )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   A.wral 2486   E.wrex 2487   ` cfv 5290  (class class class)co 5967   Basecbs 12947   +g cplusg 13024   0gc0g 13203

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-cnex 8051  ax-resscn 8052  ax-1re 8054  ax-addrcl 8057

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298  df-riota 5922  df-ov 5970  df-inn 9072  df-ndx 12950  df-slot 12951  df-base 12953  df-0g 13205

This theorem is referenced by:  lidrididd  13329  grpidd  13330  mhmid  13566  ringidss  13906