Theorem dalem15 35142

Description: Lemma for dath 35200. 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 2443	. . . 4 |- ( LVols ` K ) = ( LVols ` K )
8		dalem15.y	. . . 4 |- Y = ( ( P .\/ Q ) .\/ R )
9		dalem15.z	. . . 4 |- Z = ( ( S .\/ T ) .\/ U )
10		eqid 2443	. . . 4 |- ( Y .\/ C ) = ( Y .\/ C )
11	2, 3, 4, 5, 6, 7, 8, 9, 10	dalem14 35141	. . 3 |- ( ( ph /\ Y =/= Z ) -> ( Y .\/ Z ) e. ( LVols ` K ) )
12	2	dalemkehl 35087	. . . . 5 |- ( ph -> K e. HL )
13	2	dalemyeo 35096	. . . . 5 |- ( ph -> Y e. O )
14	2	dalemzeo 35097	. . . . 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 35086	. . . . 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 1229	. . . 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 2535	1 |- ( ( ph /\ Y =/= Z ) -> X e. N )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 974   = wceq 1383   e. wcel 1804   =/= wne 2638   class class class wbr 4437   ` cfv 5578  (class class class)co 6281   Basecbs 14509   lecple 14581   joincjn 15447   meetcmee 15448   Atomscatm 34728   HLchlt 34815   LLinesclln 34955   LPlanesclpl 34956   LVolsclvol 34957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-preset 15431  df-poset 15449  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-lat 15550  df-clat 15612  df-oposet 34641  df-ol 34643  df-oml 34644  df-covers 34731  df-ats 34732  df-atl 34763  df-cvlat 34787  df-hlat 34816  df-llines 34962  df-lplanes 34963  df-lvols 34964

This theorem is referenced by:  dalem16  35143  dalem53  35189