Metamath Proof Explorer

Theorem dvh4dimat

Description: There is an atom that is outside the subspace sum of 3 others. (Contributed by NM, 25-Apr-2015)

Ref Expression
Hypotheses dvh4dimat.h 
|- H = ( LHyp ` K )
dvh4dimat.u 
|- U = ( ( DVecH ` K ) ` W )
dvh4dimat.s 
|- .(+) = ( LSSum ` U )
dvh4dimat.a 
|- A = ( LSAtoms ` U )
dvh4dimat.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
dvh4dimat.p 
|- ( ph -> P e. A )
dvh4dimat.q 
|- ( ph -> Q e. A )
dvh4dimat.r 
|- ( ph -> R e. A )
Assertion dvh4dimat 
|- ( ph -> E. s e. A -. s C_ ( ( P .(+) Q ) .(+) R ) )

Proof

Step Hyp Ref Expression
1 dvh4dimat.h 
 |-  H = ( LHyp ` K )
2 dvh4dimat.u 
 |-  U = ( ( DVecH ` K ) ` W )
3 dvh4dimat.s 
 |-  .(+) = ( LSSum ` U )
4 dvh4dimat.a 
 |-  A = ( LSAtoms ` U )
5 dvh4dimat.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
6 dvh4dimat.p 
 |-  ( ph -> P e. A )
7 dvh4dimat.q 
 |-  ( ph -> Q e. A )
8 dvh4dimat.r 
 |-  ( ph -> R e. A )
9 5 simpld 
 |-  ( ph -> K e. HL )
10 eqid 
 |-  ( Atoms ` K ) = ( Atoms ` K )
11 eqid 
 |-  ( ( DIsoH ` K ) ` W ) = ( ( DIsoH ` K ) ` W )
12 10 1 2 11 4 dihlatat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ P e. A ) -> ( `' ( ( DIsoH ` K ) ` W ) ` P ) e. ( Atoms ` K ) )
13 5 6 12 syl2anc 
 |-  ( ph -> ( `' ( ( DIsoH ` K ) ` W ) ` P ) e. ( Atoms ` K ) )
14 10 1 2 11 4 dihlatat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ Q e. A ) -> ( `' ( ( DIsoH ` K ) ` W ) ` Q ) e. ( Atoms ` K ) )
15 5 7 14 syl2anc 
 |-  ( ph -> ( `' ( ( DIsoH ` K ) ` W ) ` Q ) e. ( Atoms ` K ) )
16 10 1 2 11 4 dihlatat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ R e. A ) -> ( `' ( ( DIsoH ` K ) ` W ) ` R ) e. ( Atoms ` K ) )
17 5 8 16 syl2anc 
 |-  ( ph -> ( `' ( ( DIsoH ` K ) ` W ) ` R ) e. ( Atoms ` K ) )
18 eqid 
 |-  ( join ` K ) = ( join ` K )
19 eqid 
 |-  ( le ` K ) = ( le ` K )
20 18 19 10 3dim3 
 |-  ( ( K e. HL /\ ( ( `' ( ( DIsoH ` K ) ` W ) ` P ) e. ( Atoms ` K ) /\ ( `' ( ( DIsoH ` K ) ` W ) ` Q ) e. ( Atoms ` K ) /\ ( `' ( ( DIsoH ` K ) ` W ) ` R ) e. ( Atoms ` K ) ) ) -> E. r e. ( Atoms ` K ) -. r ( le ` K ) ( ( ( `' ( ( DIsoH ` K ) ` W ) ` P ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` Q ) ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` R ) ) )
21 9 13 15 17 20 syl13anc 
 |-  ( ph -> E. r e. ( Atoms ` K ) -. r ( le ` K ) ( ( ( `' ( ( DIsoH ` K ) ` W ) ` P ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` Q ) ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` R ) ) )
22 5 adantr 
 |-  ( ( ph /\ r e. ( Atoms ` K ) ) -> ( K e. HL /\ W e. H ) )
23 1 2 11 4 dih1dimat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ P e. A ) -> P e. ran ( ( DIsoH ` K ) ` W ) )
24 5 6 23 syl2anc 
 |-  ( ph -> P e. ran ( ( DIsoH ` K ) ` W ) )
25 1 11 2 3 4 5 24 7 dihsmatrn 
 |-  ( ph -> ( P .(+) Q ) e. ran ( ( DIsoH ` K ) ` W ) )
26 25 adantr 
 |-  ( ( ph /\ r e. ( Atoms ` K ) ) -> ( P .(+) Q ) e. ran ( ( DIsoH ` K ) ` W ) )
27 8 adantr 
 |-  ( ( ph /\ r e. ( Atoms ` K ) ) -> R e. A )
28 18 1 11 2 3 4 22 26 27 dihjat4 
 |-  ( ( ph /\ r e. ( Atoms ` K ) ) -> ( ( P .(+) Q ) .(+) R ) = ( ( ( DIsoH ` K ) ` W ) ` ( ( `' ( ( DIsoH ` K ) ` W ) ` ( P .(+) Q ) ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` R ) ) ) )
29 24 adantr 
 |-  ( ( ph /\ r e. ( Atoms ` K ) ) -> P e. ran ( ( DIsoH ` K ) ` W ) )
30 7 adantr 
 |-  ( ( ph /\ r e. ( Atoms ` K ) ) -> Q e. A )
31 18 1 11 2 3 4 22 29 30 dihjat6 
 |-  ( ( ph /\ r e. ( Atoms ` K ) ) -> ( `' ( ( DIsoH ` K ) ` W ) ` ( P .(+) Q ) ) = ( ( `' ( ( DIsoH ` K ) ` W ) ` P ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` Q ) ) )
32 31 fvoveq1d 
 |-  ( ( ph /\ r e. ( Atoms ` K ) ) -> ( ( ( DIsoH ` K ) ` W ) ` ( ( `' ( ( DIsoH ` K ) ` W ) ` ( P .(+) Q ) ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` R ) ) ) = ( ( ( DIsoH ` K ) ` W ) ` ( ( ( `' ( ( DIsoH ` K ) ` W ) ` P ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` Q ) ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` R ) ) ) )
33 28 32 eqtrd 
 |-  ( ( ph /\ r e. ( Atoms ` K ) ) -> ( ( P .(+) Q ) .(+) R ) = ( ( ( DIsoH ` K ) ` W ) ` ( ( ( `' ( ( DIsoH ` K ) ` W ) ` P ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` Q ) ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` R ) ) ) )
34 33 sseq2d 
 |-  ( ( ph /\ r e. ( Atoms ` K ) ) -> ( ( ( ( DIsoH ` K ) ` W ) ` r ) C_ ( ( P .(+) Q ) .(+) R ) <-> ( ( ( DIsoH ` K ) ` W ) ` r ) C_ ( ( ( DIsoH ` K ) ` W ) ` ( ( ( `' ( ( DIsoH ` K ) ` W ) ` P ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` Q ) ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` R ) ) ) ) )
35 eqid 
 |-  ( Base ` K ) = ( Base ` K )
36 35 10 atbase 
 |-  ( r e. ( Atoms ` K ) -> r e. ( Base ` K ) )
37 36 adantl 
 |-  ( ( ph /\ r e. ( Atoms ` K ) ) -> r e. ( Base ` K ) )
38 9 hllatd 
 |-  ( ph -> K e. Lat )
39 35 18 10 hlatjcl 
 |-  ( ( K e. HL /\ ( `' ( ( DIsoH ` K ) ` W ) ` P ) e. ( Atoms ` K ) /\ ( `' ( ( DIsoH ` K ) ` W ) ` Q ) e. ( Atoms ` K ) ) -> ( ( `' ( ( DIsoH ` K ) ` W ) ` P ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` Q ) ) e. ( Base ` K ) )
40 9 13 15 39 syl3anc 
 |-  ( ph -> ( ( `' ( ( DIsoH ` K ) ` W ) ` P ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` Q ) ) e. ( Base ` K ) )
41 35 10 atbase 
 |-  ( ( `' ( ( DIsoH ` K ) ` W ) ` R ) e. ( Atoms ` K ) -> ( `' ( ( DIsoH ` K ) ` W ) ` R ) e. ( Base ` K ) )
42 17 41 syl 
 |-  ( ph -> ( `' ( ( DIsoH ` K ) ` W ) ` R ) e. ( Base ` K ) )
43 35 18 latjcl 
 |-  ( ( K e. Lat /\ ( ( `' ( ( DIsoH ` K ) ` W ) ` P ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` Q ) ) e. ( Base ` K ) /\ ( `' ( ( DIsoH ` K ) ` W ) ` R ) e. ( Base ` K ) ) -> ( ( ( `' ( ( DIsoH ` K ) ` W ) ` P ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` Q ) ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` R ) ) e. ( Base ` K ) )
44 38 40 42 43 syl3anc 
 |-  ( ph -> ( ( ( `' ( ( DIsoH ` K ) ` W ) ` P ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` Q ) ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` R ) ) e. ( Base ` K ) )
45 44 adantr 
 |-  ( ( ph /\ r e. ( Atoms ` K ) ) -> ( ( ( `' ( ( DIsoH ` K ) ` W ) ` P ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` Q ) ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` R ) ) e. ( Base ` K ) )
46 35 19 1 11 dihord 
 |-  ( ( ( K e. HL /\ W e. H ) /\ r e. ( Base ` K ) /\ ( ( ( `' ( ( DIsoH ` K ) ` W ) ` P ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` Q ) ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` R ) ) e. ( Base ` K ) ) -> ( ( ( ( DIsoH ` K ) ` W ) ` r ) C_ ( ( ( DIsoH ` K ) ` W ) ` ( ( ( `' ( ( DIsoH ` K ) ` W ) ` P ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` Q ) ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` R ) ) ) <-> r ( le ` K ) ( ( ( `' ( ( DIsoH ` K ) ` W ) ` P ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` Q ) ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` R ) ) ) )
47 22 37 45 46 syl3anc 
 |-  ( ( ph /\ r e. ( Atoms ` K ) ) -> ( ( ( ( DIsoH ` K ) ` W ) ` r ) C_ ( ( ( DIsoH ` K ) ` W ) ` ( ( ( `' ( ( DIsoH ` K ) ` W ) ` P ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` Q ) ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` R ) ) ) <-> r ( le ` K ) ( ( ( `' ( ( DIsoH ` K ) ` W ) ` P ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` Q ) ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` R ) ) ) )
48 34 47 bitr2d 
 |-  ( ( ph /\ r e. ( Atoms ` K ) ) -> ( r ( le ` K ) ( ( ( `' ( ( DIsoH ` K ) ` W ) ` P ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` Q ) ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` R ) ) <-> ( ( ( DIsoH ` K ) ` W ) ` r ) C_ ( ( P .(+) Q ) .(+) R ) ) )
49 48 notbid 
 |-  ( ( ph /\ r e. ( Atoms ` K ) ) -> ( -. r ( le ` K ) ( ( ( `' ( ( DIsoH ` K ) ` W ) ` P ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` Q ) ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` R ) ) <-> -. ( ( ( DIsoH ` K ) ` W ) ` r ) C_ ( ( P .(+) Q ) .(+) R ) ) )
50 49 rexbidva 
 |-  ( ph -> ( E. r e. ( Atoms ` K ) -. r ( le ` K ) ( ( ( `' ( ( DIsoH ` K ) ` W ) ` P ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` Q ) ) ( join ` K ) ( `' ( ( DIsoH ` K ) ` W ) ` R ) ) <-> E. r e. ( Atoms ` K ) -. ( ( ( DIsoH ` K ) ` W ) ` r ) C_ ( ( P .(+) Q ) .(+) R ) ) )
51 21 50 mpbid 
 |-  ( ph -> E. r e. ( Atoms ` K ) -. ( ( ( DIsoH ` K ) ` W ) ` r ) C_ ( ( P .(+) Q ) .(+) R ) )
52 10 1 2 11 4 dihatlat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ r e. ( Atoms ` K ) ) -> ( ( ( DIsoH ` K ) ` W ) ` r ) e. A )
53 5 52 sylan 
 |-  ( ( ph /\ r e. ( Atoms ` K ) ) -> ( ( ( DIsoH ` K ) ` W ) ` r ) e. A )
54 10 1 2 11 4 dihlatat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ s e. A ) -> ( `' ( ( DIsoH ` K ) ` W ) ` s ) e. ( Atoms ` K ) )
55 5 54 sylan 
 |-  ( ( ph /\ s e. A ) -> ( `' ( ( DIsoH ` K ) ` W ) ` s ) e. ( Atoms ` K ) )
56 5 adantr 
 |-  ( ( ph /\ s e. A ) -> ( K e. HL /\ W e. H ) )
57 1 2 11 4 dih1dimat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ s e. A ) -> s e. ran ( ( DIsoH ` K ) ` W ) )
58 5 57 sylan 
 |-  ( ( ph /\ s e. A ) -> s e. ran ( ( DIsoH ` K ) ` W ) )
59 1 11 dihcnvid2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ s e. ran ( ( DIsoH ` K ) ` W ) ) -> ( ( ( DIsoH ` K ) ` W ) ` ( `' ( ( DIsoH ` K ) ` W ) ` s ) ) = s )
60 56 58 59 syl2anc 
 |-  ( ( ph /\ s e. A ) -> ( ( ( DIsoH ` K ) ` W ) ` ( `' ( ( DIsoH ` K ) ` W ) ` s ) ) = s )
61 60 eqcomd 
 |-  ( ( ph /\ s e. A ) -> s = ( ( ( DIsoH ` K ) ` W ) ` ( `' ( ( DIsoH ` K ) ` W ) ` s ) ) )
62 fveq2 
 |-  ( r = ( `' ( ( DIsoH ` K ) ` W ) ` s ) -> ( ( ( DIsoH ` K ) ` W ) ` r ) = ( ( ( DIsoH ` K ) ` W ) ` ( `' ( ( DIsoH ` K ) ` W ) ` s ) ) )
63 62 rspceeqv 
 |-  ( ( ( `' ( ( DIsoH ` K ) ` W ) ` s ) e. ( Atoms ` K ) /\ s = ( ( ( DIsoH ` K ) ` W ) ` ( `' ( ( DIsoH ` K ) ` W ) ` s ) ) ) -> E. r e. ( Atoms ` K ) s = ( ( ( DIsoH ` K ) ` W ) ` r ) )
64 55 61 63 syl2anc 
 |-  ( ( ph /\ s e. A ) -> E. r e. ( Atoms ` K ) s = ( ( ( DIsoH ` K ) ` W ) ` r ) )
65 sseq1 
 |-  ( s = ( ( ( DIsoH ` K ) ` W ) ` r ) -> ( s C_ ( ( P .(+) Q ) .(+) R ) <-> ( ( ( DIsoH ` K ) ` W ) ` r ) C_ ( ( P .(+) Q ) .(+) R ) ) )
66 65 notbid 
 |-  ( s = ( ( ( DIsoH ` K ) ` W ) ` r ) -> ( -. s C_ ( ( P .(+) Q ) .(+) R ) <-> -. ( ( ( DIsoH ` K ) ` W ) ` r ) C_ ( ( P .(+) Q ) .(+) R ) ) )
67 66 adantl 
 |-  ( ( ph /\ s = ( ( ( DIsoH ` K ) ` W ) ` r ) ) -> ( -. s C_ ( ( P .(+) Q ) .(+) R ) <-> -. ( ( ( DIsoH ` K ) ` W ) ` r ) C_ ( ( P .(+) Q ) .(+) R ) ) )
68 53 64 67 rexxfrd 
 |-  ( ph -> ( E. s e. A -. s C_ ( ( P .(+) Q ) .(+) R ) <-> E. r e. ( Atoms ` K ) -. ( ( ( DIsoH ` K ) ` W ) ` r ) C_ ( ( P .(+) Q ) .(+) R ) ) )
69 51 68 mpbird 
 |-  ( ph -> E. s e. A -. s C_ ( ( P .(+) Q ) .(+) R ) )