Theorem dihjatcc

Description: Isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 29-Sep-2014)

	Ref	Expression
Hypotheses	dihjatcc.l	|- .<_ = ( le ` K )
	dihjatcc.h	|- H = ( LHyp ` K )
	dihjatcc.j	|- .\/ = ( join ` K )
	dihjatcc.a	|- A = ( Atoms ` K )
	dihjatcc.u	|- U = ( ( DVecH ` K ) ` W )
	dihjatcc.s	|- .(+) = ( LSSum ` U )
	dihjatcc.i	|- I = ( ( DIsoH ` K ) ` W )
	dihjatcc.k	|- ( ph -> ( K e. HL /\ W e. H ) )
	dihjatcc.p	|- ( ph -> ( P e. A /\ -. P .<_ W ) )
	dihjatcc.q	|- ( ph -> ( Q e. A /\ -. Q .<_ W ) )
Assertion	dihjatcc	|- ( ph -> ( I ` ( P .\/ Q ) ) = ( ( I ` P ) .(+) ( I ` Q ) ) )

Proof

Step	Hyp	Ref	Expression
1		dihjatcc.l	|- .<_ = ( le ` K )
2		dihjatcc.h	|- H = ( LHyp ` K )
3		dihjatcc.j	|- .\/ = ( join ` K )
4		dihjatcc.a	|- A = ( Atoms ` K )
5		dihjatcc.u	|- U = ( ( DVecH ` K ) ` W )
6		dihjatcc.s	|- .(+) = ( LSSum ` U )
7		dihjatcc.i	|- I = ( ( DIsoH ` K ) ` W )
8		dihjatcc.k	|- ( ph -> ( K e. HL /\ W e. H ) )
9		dihjatcc.p	|- ( ph -> ( P e. A /\ -. P .<_ W ) )
10		dihjatcc.q	|- ( ph -> ( Q e. A /\ -. Q .<_ W ) )
11		eqid	|- ( Base ` K ) = ( Base ` K )
12		eqid	|- ( meet ` K ) = ( meet ` K )
13		eqid	|- ( ( P .\/ Q ) ( meet ` K ) W ) = ( ( P .\/ Q ) ( meet ` K ) W )
14		eqid	|- ( ( oc ` K ) ` W ) = ( ( oc ` K ) ` W )
15		eqid	|- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
16		eqid	|- ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W )
17		eqid	|- ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W )
18		eqid	|- ( iota_ d e. ( ( LTrn ` K ) ` W ) ( d ` ( ( oc ` K ) ` W ) ) = P ) = ( iota_ d e. ( ( LTrn ` K ) ` W ) ( d ` ( ( oc ` K ) ` W ) ) = P )
19		eqid	|- ( iota_ d e. ( ( LTrn ` K ) ` W ) ( d ` ( ( oc ` K ) ` W ) ) = Q ) = ( iota_ d e. ( ( LTrn ` K ) ` W ) ( d ` ( ( oc ` K ) ` W ) ) = Q )
20		eqid	|- ( a e. ( ( TEndo ` K ) ` W ) |-> ( d e. ( ( LTrn ` K ) ` W ) |-> `' ( a ` d ) ) ) = ( a e. ( ( TEndo ` K ) ` W ) |-> ( d e. ( ( LTrn ` K ) ` W ) |-> `' ( a ` d ) ) )
21		eqid	|- ( d e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) = ( d e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) )
22		eqid	|- ( a e. ( ( TEndo ` K ) ` W ) , b e. ( ( TEndo ` K ) ` W ) |-> ( d e. ( ( LTrn ` K ) ` W ) |-> ( ( a ` d ) o. ( b ` d ) ) ) ) = ( a e. ( ( TEndo ` K ) ` W ) , b e. ( ( TEndo ` K ) ` W ) |-> ( d e. ( ( LTrn ` K ) ` W ) |-> ( ( a ` d ) o. ( b ` d ) ) ) )
23	11 1 2 3 12 4 5 6 7 13 8 9 10 14 15 16 17 18 19 20 21 22	dihjatcclem4	|- ( ph -> ( I ` ( ( P .\/ Q ) ( meet ` K ) W ) ) C_ ( ( I ` P ) .(+) ( I ` Q ) ) )
24	11 1 2 3 12 4 5 6 7 13 8 9 10 23	dihjatcclem2	|- ( ph -> ( I ` ( P .\/ Q ) ) = ( ( I ` P ) .(+) ( I ` Q ) ) )