Theorem lidrididd 12869

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 12868) 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 5908	. . . . 5 |- ( x = y -> ( L .+ x ) = ( L .+ y ) )
7		id 19	. . . . 5 |- ( x = y -> x = y )
8	6, 7	eqeq12d 2204	. . . 4 |- ( x = y -> ( ( L .+ x ) = x <-> ( L .+ y ) = y ) )
9	8	rspcv 2852	. . 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 12868	. . . 4 |- ( ph -> L = R )
14		oveq1 5907	. . . . . . . 8 |- ( x = y -> ( x .+ R ) = ( y .+ R ) )
15	14, 7	eqeq12d 2204	. . . . . . 7 |- ( x = y -> ( ( x .+ R ) = x <-> ( y .+ R ) = y ) )
16	15	rspcv 2852	. . . . . 6 |- ( y e. B -> ( A. x e. B ( x .+ R ) = x -> ( y .+ R ) = y ) )
17		oveq2 5908	. . . . . . . . 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 2222	. . . . . . 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 12867	1 |- ( ph -> L = .0. )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2160   A.wral 2468   ` cfv 5238  (class class class)co 5900   Basecbs 12523   +g cplusg 12600   0gc0g 12772

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-cnex 7937  ax-resscn 7938  ax-1re 7940  ax-addrcl 7943

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-br 4022  df-opab 4083  df-mpt 4084  df-id 4314  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-iota 5199  df-fun 5240  df-fn 5241  df-fv 5246  df-riota 5855  df-ov 5903  df-inn 8955  df-ndx 12526  df-slot 12527  df-base 12529  df-0g 12774

This theorem is referenced by: (None)