Theorem ltrniotaidvalN

Description: Value of the unique translation specified by identity value. (Contributed by NM, 25-Aug-2014) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ltrniotaidval.b	|- B = ( Base ` K )
	ltrniotaidval.l	|- .<_ = ( le ` K )
	ltrniotaidval.a	|- A = ( Atoms ` K )
	ltrniotaidval.h	|- H = ( LHyp ` K )
	ltrniotaidval.t	|- T = ( ( LTrn ` K ) ` W )
	ltrniotaidval.f	|- F = ( iota_ f e. T ( f ` P ) = P )
Assertion	ltrniotaidvalN	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> F = ( _I |` B ) )

Proof

Step	Hyp	Ref	Expression
1		ltrniotaidval.b	|- B = ( Base ` K )
2		ltrniotaidval.l	|- .<_ = ( le ` K )
3		ltrniotaidval.a	|- A = ( Atoms ` K )
4		ltrniotaidval.h	|- H = ( LHyp ` K )
5		ltrniotaidval.t	|- T = ( ( LTrn ` K ) ` W )
6		ltrniotaidval.f	|- F = ( iota_ f e. T ( f ` P ) = P )
7	2 3 4 5 6	ltrniotaval	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( F ` P ) = P )
8	7	3anidm23	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( F ` P ) = P )
9		simpl	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( K e. HL /\ W e. H ) )
10	2 3 4 5 6	ltrniotacl	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( P e. A /\ -. P .<_ W ) ) -> F e. T )
11	10	3anidm23	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> F e. T )
12		simpr	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( P e. A /\ -. P .<_ W ) )
13	1 2 3 4 5	ltrnideq	|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( F = ( _I |` B ) <-> ( F ` P ) = P ) )
14	9 11 12 13	syl3anc	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( F = ( _I |` B ) <-> ( F ` P ) = P ) )
15	8 14	mpbird	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> F = ( _I |` B ) )