dia2dimlem6

Description: Lemma for dia2dim . Eliminate auxiliary translations G and D . (Contributed by NM, 8-Sep-2014)

	Ref	Expression
Hypotheses	dia2dimlem6.l	\|- .<_ = ( le ` K )
	dia2dimlem6.j	\|- .\/ = ( join ` K )
	dia2dimlem6.m	\|- ./\ = ( meet ` K )
	dia2dimlem6.a	\|- A = ( Atoms ` K )
	dia2dimlem6.h	\|- H = ( LHyp ` K )
	dia2dimlem6.t	\|- T = ( ( LTrn ` K ) ` W )
	dia2dimlem6.r	\|- R = ( ( trL ` K ) ` W )
	dia2dimlem6.y	\|- Y = ( ( DVecA ` K ) ` W )
	dia2dimlem6.s	\|- S = ( LSubSp ` Y )
	dia2dimlem6.pl	\|- .(+) = ( LSSum ` Y )
	dia2dimlem6.n	\|- N = ( LSpan ` Y )
	dia2dimlem6.i	\|- I = ( ( DIsoA ` K ) ` W )
	dia2dimlem6.q	\|- Q = ( ( P .\/ U ) ./\ ( ( F ` P ) .\/ V ) )
	dia2dimlem6.k	\|- ( ph -> ( K e. HL /\ W e. H ) )
	dia2dimlem6.u	\|- ( ph -> ( U e. A /\ U .<_ W ) )
	dia2dimlem6.v	\|- ( ph -> ( V e. A /\ V .<_ W ) )
	dia2dimlem6.p	\|- ( ph -> ( P e. A /\ -. P .<_ W ) )
	dia2dimlem6.f	\|- ( ph -> ( F e. T /\ ( F ` P ) =/= P ) )
	dia2dimlem6.rf	\|- ( ph -> ( R ` F ) .<_ ( U .\/ V ) )
	dia2dimlem6.uv	\|- ( ph -> U =/= V )
	dia2dimlem6.ru	\|- ( ph -> ( R ` F ) =/= U )
	dia2dimlem6.rv	\|- ( ph -> ( R ` F ) =/= V )
Assertion	dia2dimlem6	\|- ( ph -> F e. ( ( I ` U ) .(+) ( I ` V ) ) )

Proof

Step	Hyp	Ref	Expression
1		dia2dimlem6.l	\|- .<_ = ( le ` K )
2		dia2dimlem6.j	\|- .\/ = ( join ` K )
3		dia2dimlem6.m	\|- ./\ = ( meet ` K )
4		dia2dimlem6.a	\|- A = ( Atoms ` K )
5		dia2dimlem6.h	\|- H = ( LHyp ` K )
6		dia2dimlem6.t	\|- T = ( ( LTrn ` K ) ` W )
7		dia2dimlem6.r	\|- R = ( ( trL ` K ) ` W )
8		dia2dimlem6.y	\|- Y = ( ( DVecA ` K ) ` W )
9		dia2dimlem6.s	\|- S = ( LSubSp ` Y )
10		dia2dimlem6.pl	\|- .(+) = ( LSSum ` Y )
11		dia2dimlem6.n	\|- N = ( LSpan ` Y )
12		dia2dimlem6.i	\|- I = ( ( DIsoA ` K ) ` W )
13		dia2dimlem6.q	\|- Q = ( ( P .\/ U ) ./\ ( ( F ` P ) .\/ V ) )
14		dia2dimlem6.k	\|- ( ph -> ( K e. HL /\ W e. H ) )
15		dia2dimlem6.u	\|- ( ph -> ( U e. A /\ U .<_ W ) )
16		dia2dimlem6.v	\|- ( ph -> ( V e. A /\ V .<_ W ) )
17		dia2dimlem6.p	\|- ( ph -> ( P e. A /\ -. P .<_ W ) )
18		dia2dimlem6.f	\|- ( ph -> ( F e. T /\ ( F ` P ) =/= P ) )
19		dia2dimlem6.rf	\|- ( ph -> ( R ` F ) .<_ ( U .\/ V ) )
20		dia2dimlem6.uv	\|- ( ph -> U =/= V )
21		dia2dimlem6.ru	\|- ( ph -> ( R ` F ) =/= U )
22		dia2dimlem6.rv	\|- ( ph -> ( R ` F ) =/= V )
23	1 2 3 4 5 6 7 13 14 15 16 17 18 19 20 21	dia2dimlem1	\|- ( ph -> ( Q e. A /\ -. Q .<_ W ) )
24	18	simpld	\|- ( ph -> F e. T )
25	1 4 5 6	ltrnel	\|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( F ` P ) e. A /\ -. ( F ` P ) .<_ W ) )
26	14 24 17 25	syl3anc	\|- ( ph -> ( ( F ` P ) e. A /\ -. ( F ` P ) .<_ W ) )
27	1 4 5 6	cdleme50ex	\|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( ( F ` P ) e. A /\ -. ( F ` P ) .<_ W ) ) -> E. d e. T ( d ` Q ) = ( F ` P ) )
28	14 23 26 27	syl3anc	\|- ( ph -> E. d e. T ( d ` Q ) = ( F ` P ) )
29	1 4 5 6	cdleme50ex	\|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> E. g e. T ( g ` P ) = Q )
30	14 17 23 29	syl3anc	\|- ( ph -> E. g e. T ( g ` P ) = Q )
31	14	3ad2ant1	\|- ( ( ph /\ ( g e. T /\ ( g ` P ) = Q /\ d e. T ) /\ ( d ` Q ) = ( F ` P ) ) -> ( K e. HL /\ W e. H ) )
32	15	3ad2ant1	\|- ( ( ph /\ ( g e. T /\ ( g ` P ) = Q /\ d e. T ) /\ ( d ` Q ) = ( F ` P ) ) -> ( U e. A /\ U .<_ W ) )
33	16	3ad2ant1	\|- ( ( ph /\ ( g e. T /\ ( g ` P ) = Q /\ d e. T ) /\ ( d ` Q ) = ( F ` P ) ) -> ( V e. A /\ V .<_ W ) )
34	17	3ad2ant1	\|- ( ( ph /\ ( g e. T /\ ( g ` P ) = Q /\ d e. T ) /\ ( d ` Q ) = ( F ` P ) ) -> ( P e. A /\ -. P .<_ W ) )
35	18	3ad2ant1	\|- ( ( ph /\ ( g e. T /\ ( g ` P ) = Q /\ d e. T ) /\ ( d ` Q ) = ( F ` P ) ) -> ( F e. T /\ ( F ` P ) =/= P ) )
36	19	3ad2ant1	\|- ( ( ph /\ ( g e. T /\ ( g ` P ) = Q /\ d e. T ) /\ ( d ` Q ) = ( F ` P ) ) -> ( R ` F ) .<_ ( U .\/ V ) )
37	20	3ad2ant1	\|- ( ( ph /\ ( g e. T /\ ( g ` P ) = Q /\ d e. T ) /\ ( d ` Q ) = ( F ` P ) ) -> U =/= V )
38	21	3ad2ant1	\|- ( ( ph /\ ( g e. T /\ ( g ` P ) = Q /\ d e. T ) /\ ( d ` Q ) = ( F ` P ) ) -> ( R ` F ) =/= U )
39	22	3ad2ant1	\|- ( ( ph /\ ( g e. T /\ ( g ` P ) = Q /\ d e. T ) /\ ( d ` Q ) = ( F ` P ) ) -> ( R ` F ) =/= V )
40		simp21	\|- ( ( ph /\ ( g e. T /\ ( g ` P ) = Q /\ d e. T ) /\ ( d ` Q ) = ( F ` P ) ) -> g e. T )
41		simp22	\|- ( ( ph /\ ( g e. T /\ ( g ` P ) = Q /\ d e. T ) /\ ( d ` Q ) = ( F ` P ) ) -> ( g ` P ) = Q )
42		simp23	\|- ( ( ph /\ ( g e. T /\ ( g ` P ) = Q /\ d e. T ) /\ ( d ` Q ) = ( F ` P ) ) -> d e. T )
43		simp3	\|- ( ( ph /\ ( g e. T /\ ( g ` P ) = Q /\ d e. T ) /\ ( d ` Q ) = ( F ` P ) ) -> ( d ` Q ) = ( F ` P ) )
44	1 2 3 4 5 6 7 8 9 10 11 12 13 31 32 33 34 35 36 37 38 39 40 41 42 43	dia2dimlem5	\|- ( ( ph /\ ( g e. T /\ ( g ` P ) = Q /\ d e. T ) /\ ( d ` Q ) = ( F ` P ) ) -> F e. ( ( I ` U ) .(+) ( I ` V ) ) )
45	44	3exp	\|- ( ph -> ( ( g e. T /\ ( g ` P ) = Q /\ d e. T ) -> ( ( d ` Q ) = ( F ` P ) -> F e. ( ( I ` U ) .(+) ( I ` V ) ) ) ) )
46	45	3expd	\|- ( ph -> ( g e. T -> ( ( g ` P ) = Q -> ( d e. T -> ( ( d ` Q ) = ( F ` P ) -> F e. ( ( I ` U ) .(+) ( I ` V ) ) ) ) ) ) )
47	46	rexlimdv	\|- ( ph -> ( E. g e. T ( g ` P ) = Q -> ( d e. T -> ( ( d ` Q ) = ( F ` P ) -> F e. ( ( I ` U ) .(+) ( I ` V ) ) ) ) ) )
48	30 47	mpd	\|- ( ph -> ( d e. T -> ( ( d ` Q ) = ( F ` P ) -> F e. ( ( I ` U ) .(+) ( I ` V ) ) ) ) )
49	48	rexlimdv	\|- ( ph -> ( E. d e. T ( d ` Q ) = ( F ` P ) -> F e. ( ( I ` U ) .(+) ( I ` V ) ) ) )
50	28 49	mpd	\|- ( ph -> F e. ( ( I ` U ) .(+) ( I ` V ) ) )

Metamath Proof Explorer

Theorem dia2dimlem6

Proof