Theorem lvoli3 34863

Description: Condition implying a 3-dim lattice volume. (Contributed by NM, 2-Aug-2012.)

Hypotheses

Ref	Expression
lvoli3.l	|- .<_ = ( le ` K )
lvoli3.j	|- .\/ = ( join ` K )
lvoli3.a	|- A = ( Atoms ` K )
lvoli3.p	|- P = ( LPlanes ` K )
lvoli3.v	|- V = ( LVols ` K )

Assertion

Ref	Expression
lvoli3	|- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ -. Q .<_ X ) -> ( X .\/ Q ) e. V )

Proof of Theorem lvoli3

Dummy variables y r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl2 1065	. . 3 |- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ -. Q .<_ X ) -> X e. P )
2		simpl3 1066	. . 3 |- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ -. Q .<_ X ) -> Q e. A )
3		simpr 477	. . 3 |- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ -. Q .<_ X ) -> -. Q .<_ X )
4		eqidd 2623	. . 3 |- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ -. Q .<_ X ) -> ( X .\/ Q ) = ( X .\/ Q ) )
5		breq2 4657	. . . . . 6 |- ( y = X -> ( r .<_ y <-> r .<_ X ) )
6	5	notbid 308	. . . . 5 |- ( y = X -> ( -. r .<_ y <-> -. r .<_ X ) )
7		oveq1 6657	. . . . . 6 |- ( y = X -> ( y .\/ r ) = ( X .\/ r ) )
8	7	eqeq2d 2632	. . . . 5 |- ( y = X -> ( ( X .\/ Q ) = ( y .\/ r ) <-> ( X .\/ Q ) = ( X .\/ r ) ) )
9	6, 8	anbi12d 747	. . . 4 |- ( y = X -> ( ( -. r .<_ y /\ ( X .\/ Q ) = ( y .\/ r ) ) <-> ( -. r .<_ X /\ ( X .\/ Q ) = ( X .\/ r ) ) ) )
10		breq1 4656	. . . . . 6 |- ( r = Q -> ( r .<_ X <-> Q .<_ X ) )
11	10	notbid 308	. . . . 5 |- ( r = Q -> ( -. r .<_ X <-> -. Q .<_ X ) )
12		oveq2 6658	. . . . . 6 |- ( r = Q -> ( X .\/ r ) = ( X .\/ Q ) )
13	12	eqeq2d 2632	. . . . 5 |- ( r = Q -> ( ( X .\/ Q ) = ( X .\/ r ) <-> ( X .\/ Q ) = ( X .\/ Q ) ) )
14	11, 13	anbi12d 747	. . . 4 |- ( r = Q -> ( ( -. r .<_ X /\ ( X .\/ Q ) = ( X .\/ r ) ) <-> ( -. Q .<_ X /\ ( X .\/ Q ) = ( X .\/ Q ) ) ) )
15	9, 14	rspc2ev 3324	. . 3 |- ( ( X e. P /\ Q e. A /\ ( -. Q .<_ X /\ ( X .\/ Q ) = ( X .\/ Q ) ) ) -> E. y e. P E. r e. A ( -. r .<_ y /\ ( X .\/ Q ) = ( y .\/ r ) ) )
16	1, 2, 3, 4, 15	syl112anc 1330	. 2 |- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ -. Q .<_ X ) -> E. y e. P E. r e. A ( -. r .<_ y /\ ( X .\/ Q ) = ( y .\/ r ) ) )
17		simpl1 1064	. . 3 |- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ -. Q .<_ X ) -> K e. HL )
18		hllat 34650	. . . . 5 |- ( K e. HL -> K e. Lat )
19	17, 18	syl 17	. . . 4 |- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ -. Q .<_ X ) -> K e. Lat )
20		eqid 2622	. . . . . 6 |- ( Base ` K ) = ( Base ` K )
21		lvoli3.p	. . . . . 6 |- P = ( LPlanes ` K )
22	20, 21	lplnbase 34820	. . . . 5 |- ( X e. P -> X e. ( Base ` K ) )
23	1, 22	syl 17	. . . 4 |- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ -. Q .<_ X ) -> X e. ( Base ` K ) )
24		lvoli3.a	. . . . . 6 |- A = ( Atoms ` K )
25	20, 24	atbase 34576	. . . . 5 |- ( Q e. A -> Q e. ( Base ` K ) )
26	2, 25	syl 17	. . . 4 |- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ -. Q .<_ X ) -> Q e. ( Base ` K ) )
27		lvoli3.j	. . . . 5 |- .\/ = ( join ` K )
28	20, 27	latjcl 17051	. . . 4 |- ( ( K e. Lat /\ X e. ( Base ` K ) /\ Q e. ( Base ` K ) ) -> ( X .\/ Q ) e. ( Base ` K ) )
29	19, 23, 26, 28	syl3anc 1326	. . 3 |- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ -. Q .<_ X ) -> ( X .\/ Q ) e. ( Base ` K ) )
30		lvoli3.l	. . . 4 |- .<_ = ( le ` K )
31		lvoli3.v	. . . 4 |- V = ( LVols ` K )
32	20, 30, 27, 24, 21, 31	islvol3 34862	. . 3 |- ( ( K e. HL /\ ( X .\/ Q ) e. ( Base ` K ) ) -> ( ( X .\/ Q ) e. V <-> E. y e. P E. r e. A ( -. r .<_ y /\ ( X .\/ Q ) = ( y .\/ r ) ) ) )
33	17, 29, 32	syl2anc 693	. 2 |- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ -. Q .<_ X ) -> ( ( X .\/ Q ) e. V <-> E. y e. P E. r e. A ( -. r .<_ y /\ ( X .\/ Q ) = ( y .\/ r ) ) ) )
34	16, 33	mpbird 247	1 |- ( ( ( K e. HL /\ X e. P /\ Q e. A ) /\ -. Q .<_ X ) -> ( X .\/ Q ) e. V )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   Latclat 17045   Atomscatm 34550   HLchlt 34637   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-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-lplanes 34785  df-lvols 34786

This theorem is referenced by:  dalem9  34958  dalem39  34997