Theorem dalem15 33630

Description: Lemma for dath 33688. 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 2451	. . . 4 |- ( LVols ` K ) = ( LVols ` K )
8		dalem15.y	. . . 4 |- Y = ( ( P .\/ Q ) .\/ R )
9		dalem15.z	. . . 4 |- Z = ( ( S .\/ T ) .\/ U )
10		eqid 2451	. . . 4 |- ( Y .\/ C ) = ( Y .\/ C )
11	2, 3, 4, 5, 6, 7, 8, 9, 10	dalem14 33629	. . 3 |- ( ( ph /\ Y =/= Z ) -> ( Y .\/ Z ) e. ( LVols ` K ) )
12	2	dalemkehl 33575	. . . . 5 |- ( ph -> K e. HL )
13	2	dalemyeo 33584	. . . . 5 |- ( ph -> Y e. O )
14	2	dalemzeo 33585	. . . . 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 33574	. . . . 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 1219	. . . 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 2543	1 |- ( ( ph /\ Y =/= Z ) -> X e. N )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   =/= wne 2644   class class class wbr 4392   ` cfv 5518  (class class class)co 6192   Basecbs 14278   lecple 14349   joincjn 15218   meetcmee 15219   Atomscatm 33216   HLchlt 33303   LLinesclln 33443   LPlanesclpl 33444   LVolsclvol 33445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-poset 15220  df-plt 15232  df-lub 15248  df-glb 15249  df-join 15250  df-meet 15251  df-p0 15313  df-lat 15320  df-clat 15382  df-oposet 33129  df-ol 33131  df-oml 33132  df-covers 33219  df-ats 33220  df-atl 33251  df-cvlat 33275  df-hlat 33304  df-llines 33450  df-lplanes 33451  df-lvols 33452

This theorem is referenced by:  dalem16  33631  dalem53  33677