Theorem dalem15 34349

Description: Lemma for dath 34407. 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 2460	. . . 4 |- ( LVols ` K ) = ( LVols ` K )
8		dalem15.y	. . . 4 |- Y = ( ( P .\/ Q ) .\/ R )
9		dalem15.z	. . . 4 |- Z = ( ( S .\/ T ) .\/ U )
10		eqid 2460	. . . 4 |- ( Y .\/ C ) = ( Y .\/ C )
11	2, 3, 4, 5, 6, 7, 8, 9, 10	dalem14 34348	. . 3 |- ( ( ph /\ Y =/= Z ) -> ( Y .\/ Z ) e. ( LVols ` K ) )
12	2	dalemkehl 34294	. . . . 5 |- ( ph -> K e. HL )
13	2	dalemyeo 34303	. . . . 5 |- ( ph -> Y e. O )
14	2	dalemzeo 34304	. . . . 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 34293	. . . . 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 1223	. . . 4 |- ( ph -> ( ( Y ./\ Z ) e. N <-> ( Y .\/ Z ) e. ( LVols ` K ) ) )
19	18	adantr 465	. . 3 |- ( ( ph /\ Y =/= Z ) -> ( ( Y ./\ Z ) e. N <-> ( Y .\/ Z ) e. ( LVols ` K ) ) )
20	11, 19	mpbird 232	. 2 |- ( ( ph /\ Y =/= Z ) -> ( Y ./\ Z ) e. N )
21	1, 20	syl5eqel 2552	1 |- ( ( ph /\ Y =/= Z ) -> X e. N )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 968   = wceq 1374   e. wcel 1762   =/= wne 2655   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   Basecbs 14479   lecple 14551   joincjn 15420   meetcmee 15421   Atomscatm 33935   HLchlt 34022   LLinesclln 34162   LPlanesclpl 34163   LVolsclvol 34164

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-poset 15422  df-plt 15434  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-p0 15515  df-lat 15522  df-clat 15584  df-oposet 33848  df-ol 33850  df-oml 33851  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023  df-llines 34169  df-lplanes 34170  df-lvols 34171

This theorem is referenced by:  dalem16  34350  dalem53  34396