Theorem lidrididd 13595

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 13594) 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 6058	. . . . 5 |- ( x = y -> ( L .+ x ) = ( L .+ y ) )
7		id 19	. . . . 5 |- ( x = y -> x = y )
8	6, 7	eqeq12d 2247	. . . 4 |- ( x = y -> ( ( L .+ x ) = x <-> ( L .+ y ) = y ) )
9	8	rspcv 2917	. . 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 13594	. . . 4 |- ( ph -> L = R )
14		oveq1 6057	. . . . . . . 8 |- ( x = y -> ( x .+ R ) = ( y .+ R ) )
15	14, 7	eqeq12d 2247	. . . . . . 7 |- ( x = y -> ( ( x .+ R ) = x <-> ( y .+ R ) = y ) )
16	15	rspcv 2917	. . . . . 6 |- ( y e. B -> ( A. x e. B ( x .+ R ) = x -> ( y .+ R ) = y ) )
17		oveq2 6058	. . . . . . . . 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 2265	. . . . . . 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 13593	1 |- ( ph -> L = .0. )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   A.wral 2520   ` cfv 5352  (class class class)co 6050   Basecbs 13212   +g cplusg 13290   0gc0g 13469

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360  df-riota 6003  df-ov 6053  df-inn 9238  df-ndx 13215  df-slot 13216  df-base 13218  df-0g 13471

This theorem is referenced by: (None)