Theorem dalem9 34958

Description: Lemma for dath 35022. Since -. C .<_ Y, the join Y .\/ C forms a 3-dimensional space. (Contributed by NM, 20-Jul-2012.)

Hypotheses

Ref	Expression
dalema.ph	|- ( ph <-> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( Y e. O /\ Z e. O ) /\ ( ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( C .<_ ( P .\/ S ) /\ C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) ) ) ) )
dalemc.l	|- .<_ = ( le ` K )
dalemc.j	|- .\/ = ( join ` K )
dalemc.a	|- A = ( Atoms ` K )
dalem9.o	|- O = ( LPlanes ` K )
dalem9.v	|- V = ( LVols ` K )
dalem9.y	|- Y = ( ( P .\/ Q ) .\/ R )
dalem9.z	|- Z = ( ( S .\/ T ) .\/ U )
dalem9.w	|- W = ( Y .\/ C )

Assertion

Ref	Expression
dalem9	|- ( ( ph /\ Y =/= Z ) -> W e. V )

Proof of Theorem dalem9

Step	Hyp	Ref	Expression
1		dalem9.w	. 2 |- W = ( Y .\/ C )
2		dalema.ph	. . . . 5 |- ( ph <-> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( Y e. O /\ Z e. O ) /\ ( ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( C .<_ ( P .\/ S ) /\ C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) ) ) ) )
3	2	dalemkehl 34909	. . . 4 |- ( ph -> K e. HL )
4	3	adantr 481	. . 3 |- ( ( ph /\ Y =/= Z ) -> K e. HL )
5	2	dalemyeo 34918	. . . 4 |- ( ph -> Y e. O )
6	5	adantr 481	. . 3 |- ( ( ph /\ Y =/= Z ) -> Y e. O )
7		dalemc.l	. . . . 5 |- .<_ = ( le ` K )
8		dalemc.j	. . . . 5 |- .\/ = ( join ` K )
9		dalemc.a	. . . . 5 |- A = ( Atoms ` K )
10		dalem9.o	. . . . 5 |- O = ( LPlanes ` K )
11		dalem9.y	. . . . 5 |- Y = ( ( P .\/ Q ) .\/ R )
12	2, 7, 8, 9, 10, 11	dalemcea 34946	. . . 4 |- ( ph -> C e. A )
13	12	adantr 481	. . 3 |- ( ( ph /\ Y =/= Z ) -> C e. A )
14		dalem9.z	. . . 4 |- Z = ( ( S .\/ T ) .\/ U )
15	2, 7, 8, 9, 10, 11, 14	dalem-cly 34957	. . 3 |- ( ( ph /\ Y =/= Z ) -> -. C .<_ Y )
16		dalem9.v	. . . 4 |- V = ( LVols ` K )
17	7, 8, 9, 10, 16	lvoli3 34863	. . 3 |- ( ( ( K e. HL /\ Y e. O /\ C e. A ) /\ -. C .<_ Y ) -> ( Y .\/ C ) e. V )
18	4, 6, 13, 15, 17	syl31anc 1329	. 2 |- ( ( ph /\ Y =/= Z ) -> ( Y .\/ C ) e. V )
19	1, 18	syl5eqel 2705	1 |- ( ( ph /\ Y =/= Z ) -> W e. V )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   Atomscatm 34550   HLchlt 34637   LPlanesclpl 34778   LVolsclvol 34779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-llines 34784  df-lplanes 34785  df-lvols 34786

This theorem is referenced by:  dalem13  34962  dalem14  34963