Theorem grpidd 13529

Description: Deduce the identity element of a magma from its properties. (Contributed by Mario Carneiro, 6-Jan-2015.)

Hypotheses

Ref	Expression
grpidd.b	|- ( ph -> B = ( Base ` G ) )
grpidd.p	|- ( ph -> .+ = ( +g ` G ) )
grpidd.z	|- ( ph -> .0. e. B )
grpidd.i	|- ( ( ph /\ x e. B ) -> ( .0. .+ x ) = x )
grpidd.j	|- ( ( ph /\ x e. B ) -> ( x .+ .0. ) = x )

Assertion

Ref	Expression
grpidd	|- ( ph -> .0. = ( 0g ` G ) )

Distinct variable groups:   x, G   ph, x   x, .0.

Allowed substitution hints:   B( x)   .+ ( x)

Proof of Theorem grpidd

Step	Hyp	Ref	Expression
1		eqid 2231	. 2 |- ( Base ` G ) = ( Base ` G )
2		eqid 2231	. 2 |- ( 0g ` G ) = ( 0g ` G )
3		eqid 2231	. 2 |- ( +g ` G ) = ( +g ` G )
4		grpidd.z	. . 3 |- ( ph -> .0. e. B )
5		grpidd.b	. . 3 |- ( ph -> B = ( Base ` G ) )
6	4, 5	eleqtrd 2310	. 2 |- ( ph -> .0. e. ( Base ` G ) )
7	5	eleq2d 2301	. . . 4 |- ( ph -> ( x e. B <-> x e. ( Base ` G ) ) )
8	7	biimpar 297	. . 3 |- ( ( ph /\ x e. ( Base ` G ) ) -> x e. B )
9		grpidd.p	. . . . . 6 |- ( ph -> .+ = ( +g ` G ) )
10	9	adantr 276	. . . . 5 |- ( ( ph /\ x e. B ) -> .+ = ( +g ` G ) )
11	10	oveqd 6045	. . . 4 |- ( ( ph /\ x e. B ) -> ( .0. .+ x ) = ( .0. ( +g ` G ) x ) )
12		grpidd.i	. . . 4 |- ( ( ph /\ x e. B ) -> ( .0. .+ x ) = x )
13	11, 12	eqtr3d 2266	. . 3 |- ( ( ph /\ x e. B ) -> ( .0. ( +g ` G ) x ) = x )
14	8, 13	syldan 282	. 2 |- ( ( ph /\ x e. ( Base ` G ) ) -> ( .0. ( +g ` G ) x ) = x )
15	10	oveqd 6045	. . . 4 |- ( ( ph /\ x e. B ) -> ( x .+ .0. ) = ( x ( +g ` G ) .0. ) )
16		grpidd.j	. . . 4 |- ( ( ph /\ x e. B ) -> ( x .+ .0. ) = x )
17	15, 16	eqtr3d 2266	. . 3 |- ( ( ph /\ x e. B ) -> ( x ( +g ` G ) .0. ) = x )
18	8, 17	syldan 282	. 2 |- ( ( ph /\ x e. ( Base ` G ) ) -> ( x ( +g ` G ) .0. ) = x )
19	1, 2, 3, 6, 14, 18	ismgmid2 13526	1 |- ( ph -> .0. = ( 0g ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   ` cfv 5333  (class class class)co 6028   Basecbs 13145   +g cplusg 13223   0gc0g 13402

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-cnex 8166  ax-resscn 8167  ax-1re 8169  ax-addrcl 8172

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-riota 5981  df-ov 6031  df-inn 9186  df-ndx 13148  df-slot 13149  df-base 13151  df-0g 13404

This theorem is referenced by:  ress0g  13589  imasmnd2  13598  mnd1id  13602  isgrpde  13668