Theorem dalem15 33287

Description: Lemma for dath 33345. The axis of perspectivity X is a line. (Contributed by NM, 21-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 )
dalem15.m	|- ./\ = ( meet ` K )
dalem15.n	|- N = ( LLines ` K )
dalem15.o	|- O = ( LPlanes ` K )
dalem15.y	|- Y = ( ( P .\/ Q ) .\/ R )
dalem15.z	|- Z = ( ( S .\/ T ) .\/ U )
dalem15.x	|- X = ( Y ./\ Z )

Assertion

Ref	Expression
dalem15	|- ( ( ph /\ Y =/= Z ) -> X e. N )

Proof of Theorem dalem15

Step	Hyp	Ref	Expression
1		dalem15.x	. 2 |- X = ( Y ./\ Z )
2		dalema.ph	. . . 4 |- ( 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		dalemc.l	. . . 4 |- .<_ = ( le ` K )
4		dalemc.j	. . . 4 |- .\/ = ( join ` K )
5		dalemc.a	. . . 4 |- A = ( Atoms ` K )
6		dalem15.o	. . . 4 |- O = ( LPlanes ` K )
7		eqid 2461	. . . 4 |- ( LVols ` K ) = ( LVols ` K )
8		dalem15.y	. . . 4 |- Y = ( ( P .\/ Q ) .\/ R )
9		dalem15.z	. . . 4 |- Z = ( ( S .\/ T ) .\/ U )
10		eqid 2461	. . . 4 |- ( Y .\/ C ) = ( Y .\/ C )
11	2, 3, 4, 5, 6, 7, 8, 9, 10	dalem14 33286	. . 3 |- ( ( ph /\ Y =/= Z ) -> ( Y .\/ Z ) e. ( LVols ` K ) )
12	2	dalemkehl 33232	. . . . 5 |- ( ph -> K e. HL )
13	2	dalemyeo 33241	. . . . 5 |- ( ph -> Y e. O )
14	2	dalemzeo 33242	. . . . 5 |- ( ph -> Z e. O )
15		dalem15.m	. . . . . 6 |- ./\ = ( meet ` K )
16		dalem15.n	. . . . . 6 |- N = ( LLines ` K )
17	4, 15, 16, 6, 7	2lplnmj 33231	. . . . 5 |- ( ( K e. HL /\ Y e. O /\ Z e. O ) -> ( ( Y ./\ Z ) e. N <-> ( Y .\/ Z ) e. ( LVols ` K ) ) )
18	12, 13, 14, 17	syl3anc 1276	. . . 4 |- ( ph -> ( ( Y ./\ Z ) e. N <-> ( Y .\/ Z ) e. ( LVols ` K ) ) )
19	18	adantr 471	. . 3 |- ( ( ph /\ Y =/= Z ) -> ( ( Y ./\ Z ) e. N <-> ( Y .\/ Z ) e. ( LVols ` K ) ) )
20	11, 19	mpbird 240	. 2 |- ( ( ph /\ Y =/= Z ) -> ( Y ./\ Z ) e. N )
21	1, 20	syl5eqel 2543	1 |- ( ( ph /\ Y =/= Z ) -> X e. N )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 375   /\ w3a 991   = wceq 1454   e. wcel 1897   =/= wne 2632   class class class wbr 4415   ` cfv 5600  (class class class)co 6314   Basecbs 15169   lecple 15245   joincjn 16237   meetcmee 16238   Atomscatm 32873   HLchlt 32960   LLinesclln 33100   LPlanesclpl 33101   LVolsclvol 33102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-reu 2755  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6276  df-ov 6317  df-oprab 6318  df-preset 16221  df-poset 16239  df-plt 16252  df-lub 16268  df-glb 16269  df-join 16270  df-meet 16271  df-p0 16333  df-lat 16340  df-clat 16402  df-oposet 32786  df-ol 32788  df-oml 32789  df-covers 32876  df-ats 32877  df-atl 32908  df-cvlat 32932  df-hlat 32961  df-llines 33107  df-lplanes 33108  df-lvols 33109

This theorem is referenced by:  dalem16  33288  dalem53  33334