Theorem lidrididd 13702

Description: If there is a left and right identity element for any binary operation (group operation) .+, the left identity element (and therefore also the right identity element according to lidrideqd 13701) is equal to the two-sided identity element. (Contributed by AV, 26-Dec-2023.)

Hypotheses

Ref	Expression
lidrideqd.l	|- ( ph -> L e. B )
lidrideqd.r	|- ( ph -> R e. B )
lidrideqd.li	|- ( ph -> A. x e. B ( L .+ x ) = x )
lidrideqd.ri	|- ( ph -> A. x e. B ( x .+ R ) = x )
lidrideqd.b	|- B = ( Base ` G )
lidrideqd.p	|- .+ = ( +g ` G )
lidrididd.o	|- .0. = ( 0g ` G )

Assertion

Ref	Expression
lidrididd	|- ( ph -> L = .0. )

Distinct variable groups:   x, B   x, L   x, R   x, .+

Allowed substitution hints:   ph( x)   G( x)   .0. ( x)

Proof of Theorem lidrididd

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lidrideqd.b	. 2 |- B = ( Base ` G )
2		lidrididd.o	. 2 |- .0. = ( 0g ` G )
3		lidrideqd.p	. 2 |- .+ = ( +g ` G )
4		lidrideqd.l	. 2 |- ( ph -> L e. B )
5		lidrideqd.li	. . 3 |- ( ph -> A. x e. B ( L .+ x ) = x )
6		oveq2 6093	. . . . 5 |- ( x = y -> ( L .+ x ) = ( L .+ y ) )
7		id 19	. . . . 5 |- ( x = y -> x = y )
8	6, 7	eqeq12d 2253	. . . 4 |- ( x = y -> ( ( L .+ x ) = x <-> ( L .+ y ) = y ) )
9	8	rspcv 2925	. . 3 |- ( y e. B -> ( A. x e. B ( L .+ x ) = x -> ( L .+ y ) = y ) )
10	5, 9	mpan9 281	. 2 |- ( ( ph /\ y e. B ) -> ( L .+ y ) = y )
11		lidrideqd.ri	. . . 4 |- ( ph -> A. x e. B ( x .+ R ) = x )
12		lidrideqd.r	. . . . 5 |- ( ph -> R e. B )
13	4, 12, 5, 11	lidrideqd 13701	. . . 4 |- ( ph -> L = R )
14		oveq1 6092	. . . . . . . 8 |- ( x = y -> ( x .+ R ) = ( y .+ R ) )
15	14, 7	eqeq12d 2253	. . . . . . 7 |- ( x = y -> ( ( x .+ R ) = x <-> ( y .+ R ) = y ) )
16	15	rspcv 2925	. . . . . 6 |- ( y e. B -> ( A. x e. B ( x .+ R ) = x -> ( y .+ R ) = y ) )
17		oveq2 6093	. . . . . . . . 9 |- ( L = R -> ( y .+ L ) = ( y .+ R ) )
18	17	adantl 277	. . . . . . . 8 |- ( ( ( y .+ R ) = y /\ L = R ) -> ( y .+ L ) = ( y .+ R ) )
19		simpl 109	. . . . . . . 8 |- ( ( ( y .+ R ) = y /\ L = R ) -> ( y .+ R ) = y )
20	18, 19	eqtrd 2271	. . . . . . 7 |- ( ( ( y .+ R ) = y /\ L = R ) -> ( y .+ L ) = y )
21	20	ex 115	. . . . . 6 |- ( ( y .+ R ) = y -> ( L = R -> ( y .+ L ) = y ) )
22	16, 21	syl6com 35	. . . . 5 |- ( A. x e. B ( x .+ R ) = x -> ( y e. B -> ( L = R -> ( y .+ L ) = y ) ) )
23	22	com23 78	. . . 4 |- ( A. x e. B ( x .+ R ) = x -> ( L = R -> ( y e. B -> ( y .+ L ) = y ) ) )
24	11, 13, 23	sylc 62	. . 3 |- ( ph -> ( y e. B -> ( y .+ L ) = y ) )
25	24	imp 124	. 2 |- ( ( ph /\ y e. B ) -> ( y .+ L ) = y )
26	1, 2, 3, 4, 10, 25	ismgmid2 13700	1 |- ( ph -> L = .0. )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   ` cfv 5377  (class class class)co 6085   Basecbs 13352   +g cplusg 13431   0gc0g 13610

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385  df-riota 6038  df-ov 6088  df-inn 9305  df-ndx 13355  df-slot 13356  df-base 13358  df-0g 13612

This theorem is used by: (None)