Theorem 2lplnm2N 34907

Description: The meet of two different lattice planes in a lattice volume is a lattice line. (Contributed by NM, 12-Jul-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
2lplnm2.l	|- .<_ = ( le ` K )
2lplnm2.m	|- ./\ = ( meet ` K )
2lplnm2.a	|- N = ( LLines ` K )
2lplnm2.p	|- P = ( LPlanes ` K )
2lplnm2.v	|- V = ( LVols ` K )

Assertion

Ref	Expression
2lplnm2N	|- ( ( K e. HL /\ ( X e. P /\ Y e. P /\ W e. V ) /\ ( X .<_ W /\ Y .<_ W /\ X =/= Y ) ) -> ( X ./\ Y ) e. N )

Proof of Theorem 2lplnm2N

Step	Hyp	Ref	Expression
1		simp22 1095	. 2 |- ( ( K e. HL /\ ( X e. P /\ Y e. P /\ W e. V ) /\ ( X .<_ W /\ Y .<_ W /\ X =/= Y ) ) -> Y e. P )
2		simp1 1061	. . 3 |- ( ( K e. HL /\ ( X e. P /\ Y e. P /\ W e. V ) /\ ( X .<_ W /\ Y .<_ W /\ X =/= Y ) ) -> K e. HL )
3		hllat 34650	. . . . 5 |- ( K e. HL -> K e. Lat )
4	3	3ad2ant1 1082	. . . 4 |- ( ( K e. HL /\ ( X e. P /\ Y e. P /\ W e. V ) /\ ( X .<_ W /\ Y .<_ W /\ X =/= Y ) ) -> K e. Lat )
5		simp21 1094	. . . . 5 |- ( ( K e. HL /\ ( X e. P /\ Y e. P /\ W e. V ) /\ ( X .<_ W /\ Y .<_ W /\ X =/= Y ) ) -> X e. P )
6		eqid 2622	. . . . . 6 |- ( Base ` K ) = ( Base ` K )
7		2lplnm2.p	. . . . . 6 |- P = ( LPlanes ` K )
8	6, 7	lplnbase 34820	. . . . 5 |- ( X e. P -> X e. ( Base ` K ) )
9	5, 8	syl 17	. . . 4 |- ( ( K e. HL /\ ( X e. P /\ Y e. P /\ W e. V ) /\ ( X .<_ W /\ Y .<_ W /\ X =/= Y ) ) -> X e. ( Base ` K ) )
10	6, 7	lplnbase 34820	. . . . 5 |- ( Y e. P -> Y e. ( Base ` K ) )
11	1, 10	syl 17	. . . 4 |- ( ( K e. HL /\ ( X e. P /\ Y e. P /\ W e. V ) /\ ( X .<_ W /\ Y .<_ W /\ X =/= Y ) ) -> Y e. ( Base ` K ) )
12		2lplnm2.m	. . . . 5 |- ./\ = ( meet ` K )
13	6, 12	latmcl 17052	. . . 4 |- ( ( K e. Lat /\ X e. ( Base ` K ) /\ Y e. ( Base ` K ) ) -> ( X ./\ Y ) e. ( Base ` K ) )
14	4, 9, 11, 13	syl3anc 1326	. . 3 |- ( ( K e. HL /\ ( X e. P /\ Y e. P /\ W e. V ) /\ ( X .<_ W /\ Y .<_ W /\ X =/= Y ) ) -> ( X ./\ Y ) e. ( Base ` K ) )
15		2lplnm2.l	. . . . . . 7 |- .<_ = ( le ` K )
16		eqid 2622	. . . . . . 7 |- ( join ` K ) = ( join ` K )
17		2lplnm2.v	. . . . . . 7 |- V = ( LVols ` K )
18	15, 16, 7, 17	2lplnj 34906	. . . . . 6 |- ( ( K e. HL /\ ( X e. P /\ Y e. P /\ W e. V ) /\ ( X .<_ W /\ Y .<_ W /\ X =/= Y ) ) -> ( X ( join ` K ) Y ) = W )
19		simp23 1096	. . . . . 6 |- ( ( K e. HL /\ ( X e. P /\ Y e. P /\ W e. V ) /\ ( X .<_ W /\ Y .<_ W /\ X =/= Y ) ) -> W e. V )
20	18, 19	eqeltrd 2701	. . . . 5 |- ( ( K e. HL /\ ( X e. P /\ Y e. P /\ W e. V ) /\ ( X .<_ W /\ Y .<_ W /\ X =/= Y ) ) -> ( X ( join ` K ) Y ) e. V )
21	6, 15, 16	latlej1 17060	. . . . . 6 |- ( ( K e. Lat /\ X e. ( Base ` K ) /\ Y e. ( Base ` K ) ) -> X .<_ ( X ( join ` K ) Y ) )
22	4, 9, 11, 21	syl3anc 1326	. . . . 5 |- ( ( K e. HL /\ ( X e. P /\ Y e. P /\ W e. V ) /\ ( X .<_ W /\ Y .<_ W /\ X =/= Y ) ) -> X .<_ ( X ( join ` K ) Y ) )
23		eqid 2622	. . . . . 6 |- ( <o ` K ) = ( <o ` K )
24	15, 23, 7, 17	lplncvrlvol2 34901	. . . . 5 |- ( ( ( K e. HL /\ X e. P /\ ( X ( join ` K ) Y ) e. V ) /\ X .<_ ( X ( join ` K ) Y ) ) -> X ( <o ` K ) ( X ( join ` K ) Y ) )
25	2, 5, 20, 22, 24	syl31anc 1329	. . . 4 |- ( ( K e. HL /\ ( X e. P /\ Y e. P /\ W e. V ) /\ ( X .<_ W /\ Y .<_ W /\ X =/= Y ) ) -> X ( <o ` K ) ( X ( join ` K ) Y ) )
26	6, 16, 12, 23	cvrexch 34706	. . . . 5 |- ( ( K e. HL /\ X e. ( Base ` K ) /\ Y e. ( Base ` K ) ) -> ( ( X ./\ Y ) ( <o ` K ) Y <-> X ( <o ` K ) ( X ( join ` K ) Y ) ) )
27	2, 9, 11, 26	syl3anc 1326	. . . 4 |- ( ( K e. HL /\ ( X e. P /\ Y e. P /\ W e. V ) /\ ( X .<_ W /\ Y .<_ W /\ X =/= Y ) ) -> ( ( X ./\ Y ) ( <o ` K ) Y <-> X ( <o ` K ) ( X ( join ` K ) Y ) ) )
28	25, 27	mpbird 247	. . 3 |- ( ( K e. HL /\ ( X e. P /\ Y e. P /\ W e. V ) /\ ( X .<_ W /\ Y .<_ W /\ X =/= Y ) ) -> ( X ./\ Y ) ( <o ` K ) Y )
29		2lplnm2.a	. . . 4 |- N = ( LLines ` K )
30	6, 23, 29, 7	llncvrlpln 34844	. . 3 |- ( ( ( K e. HL /\ ( X ./\ Y ) e. ( Base ` K ) /\ Y e. ( Base ` K ) ) /\ ( X ./\ Y ) ( <o ` K ) Y ) -> ( ( X ./\ Y ) e. N <-> Y e. P ) )
31	2, 14, 11, 28, 30	syl31anc 1329	. 2 |- ( ( K e. HL /\ ( X e. P /\ Y e. P /\ W e. V ) /\ ( X .<_ W /\ Y .<_ W /\ X =/= Y ) ) -> ( ( X ./\ Y ) e. N <-> Y e. P ) )
32	1, 31	mpbird 247	1 |- ( ( K e. HL /\ ( X e. P /\ Y e. P /\ W e. V ) /\ ( X .<_ W /\ Y .<_ W /\ X =/= Y ) ) -> ( X ./\ Y ) e. N )

Syntax hints:   -> wi 4   <-> wb 196   /\ 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   meetcmee 16945   Latclat 17045   <o ccvr 34549   HLchlt 34637   LLinesclln 34777   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-3or 1038  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: (None)