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Theorem lvolnlelln 34870
Description: A lattice line cannot majorize a lattice volume. (Contributed by NM, 14-Jul-2012.)
Hypotheses
Ref Expression
lvolnlelln.l |- .<_ = ( le ` K )
lvolnlelln.n |- N = ( LLines ` K )
lvolnlelln.v |- V = ( LVols ` K )
Assertion
Ref Expression
lvolnlelln |- ( ( K e. HL /\ X e. V /\ Y e. N ) -> -. X .<_ Y )
Proof of Theorem lvolnlelln
Dummy variables q p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp3 1063 . . 3 |- ( ( K e. HL /\ X e. V /\ Y e. N ) -> Y e. N )
2 eqid 2622 . . . . 5 |- ( Base ` K ) = ( Base ` K )
3 eqid 2622 . . . . 5 |- ( join ` K ) = ( join ` K )
4 eqid 2622 . . . . 5 |- ( Atoms ` K ) = ( Atoms ` K )
5 lvolnlelln.n . . . . 5 |- N = ( LLines ` K )
62, 3, 4, 5islln2 34797 . . . 4 |- ( K e. HL -> ( Y e. N <-> ( Y e. ( Base ` K ) /\ E. p e. ( Atoms ` K ) E. q e. ( Atoms ` K ) ( p =/= q /\ Y = ( p ( join ` K ) q ) ) ) ) )
763ad2ant1 1082 . . 3 |- ( ( K e. HL /\ X e. V /\ Y e. N ) -> ( Y e. N <-> ( Y e. ( Base ` K ) /\ E. p e. ( Atoms ` K ) E. q e. ( Atoms ` K ) ( p =/= q /\ Y = ( p ( join ` K ) q ) ) ) ) )
81, 7mpbid 222 . 2 |- ( ( K e. HL /\ X e. V /\ Y e. N ) -> ( Y e. ( Base ` K ) /\ E. p e. ( Atoms ` K ) E. q e. ( Atoms ` K ) ( p =/= q /\ Y = ( p ( join ` K ) q ) ) ) )
9 simp11 1091 . . . . . . 7 |- ( ( ( K e. HL /\ X e. V /\ Y e. N ) /\ ( p e. ( Atoms ` K ) /\ q e. ( Atoms ` K ) ) /\ ( p =/= q /\ Y = ( p ( join ` K ) q ) ) ) -> K e. HL )
10 simp12 1092 . . . . . . 7 |- ( ( ( K e. HL /\ X e. V /\ Y e. N ) /\ ( p e. ( Atoms ` K ) /\ q e. ( Atoms ` K ) ) /\ ( p =/= q /\ Y = ( p ( join ` K ) q ) ) ) -> X e. V )
11 simp2l 1087 . . . . . . 7 |- ( ( ( K e. HL /\ X e. V /\ Y e. N ) /\ ( p e. ( Atoms ` K ) /\ q e. ( Atoms ` K ) ) /\ ( p =/= q /\ Y = ( p ( join ` K ) q ) ) ) -> p e. ( Atoms ` K ) )
12 simp2r 1088 . . . . . . 7 |- ( ( ( K e. HL /\ X e. V /\ Y e. N ) /\ ( p e. ( Atoms ` K ) /\ q e. ( Atoms ` K ) ) /\ ( p =/= q /\ Y = ( p ( join ` K ) q ) ) ) -> q e. ( Atoms ` K ) )
13 lvolnlelln.l . . . . . . . 8 |- .<_ = ( le ` K )
14 lvolnlelln.v . . . . . . . 8 |- V = ( LVols ` K )
1513, 3, 4, 14lvolnle3at 34868 . . . . . . 7 |- ( ( ( K e. HL /\ X e. V ) /\ ( p e. ( Atoms ` K ) /\ p e. ( Atoms ` K ) /\ q e. ( Atoms ` K ) ) ) -> -. X .<_ ( ( p ( join ` K ) p ) ( join ` K ) q ) )
169, 10, 11, 11, 12, 15syl23anc 1333 . . . . . 6 |- ( ( ( K e. HL /\ X e. V /\ Y e. N ) /\ ( p e. ( Atoms ` K ) /\ q e. ( Atoms ` K ) ) /\ ( p =/= q /\ Y = ( p ( join ` K ) q ) ) ) -> -. X .<_ ( ( p ( join ` K ) p ) ( join ` K ) q ) )
17 simp3r 1090 . . . . . . . 8 |- ( ( ( K e. HL /\ X e. V /\ Y e. N ) /\ ( p e. ( Atoms ` K ) /\ q e. ( Atoms ` K ) ) /\ ( p =/= q /\ Y = ( p ( join ` K ) q ) ) ) -> Y = ( p ( join ` K ) q ) )
183, 4hlatjidm 34655 . . . . . . . . . 10 |- ( ( K e. HL /\ p e. ( Atoms ` K ) ) -> ( p ( join ` K ) p ) = p )
199, 11, 18syl2anc 693 . . . . . . . . 9 |- ( ( ( K e. HL /\ X e. V /\ Y e. N ) /\ ( p e. ( Atoms ` K ) /\ q e. ( Atoms ` K ) ) /\ ( p =/= q /\ Y = ( p ( join ` K ) q ) ) ) -> ( p ( join ` K ) p ) = p )
2019oveq1d 6665 . . . . . . . 8 |- ( ( ( K e. HL /\ X e. V /\ Y e. N ) /\ ( p e. ( Atoms ` K ) /\ q e. ( Atoms ` K ) ) /\ ( p =/= q /\ Y = ( p ( join ` K ) q ) ) ) -> ( ( p ( join ` K ) p ) ( join ` K ) q ) = ( p ( join ` K ) q ) )
2117, 20eqtr4d 2659 . . . . . . 7 |- ( ( ( K e. HL /\ X e. V /\ Y e. N ) /\ ( p e. ( Atoms ` K ) /\ q e. ( Atoms ` K ) ) /\ ( p =/= q /\ Y = ( p ( join ` K ) q ) ) ) -> Y = ( ( p ( join ` K ) p ) ( join ` K ) q ) )
2221breq2d 4665 . . . . . 6 |- ( ( ( K e. HL /\ X e. V /\ Y e. N ) /\ ( p e. ( Atoms ` K ) /\ q e. ( Atoms ` K ) ) /\ ( p =/= q /\ Y = ( p ( join ` K ) q ) ) ) -> ( X .<_ Y <-> X .<_ ( ( p ( join ` K ) p ) ( join ` K ) q ) ) )
2316, 22mtbird 315 . . . . 5 |- ( ( ( K e. HL /\ X e. V /\ Y e. N ) /\ ( p e. ( Atoms ` K ) /\ q e. ( Atoms ` K ) ) /\ ( p =/= q /\ Y = ( p ( join ` K ) q ) ) ) -> -. X .<_ Y )
24233exp 1264 . . . 4 |- ( ( K e. HL /\ X e. V /\ Y e. N ) -> ( ( p e. ( Atoms ` K ) /\ q e. ( Atoms ` K ) ) -> ( ( p =/= q /\ Y = ( p ( join ` K ) q ) ) -> -. X .<_ Y ) ) )
2524rexlimdvv 3037 . . 3 |- ( ( K e. HL /\ X e. V /\ Y e. N ) -> ( E. p e. ( Atoms ` K ) E. q e. ( Atoms ` K ) ( p =/= q /\ Y = ( p ( join ` K ) q ) ) -> -. X .<_ Y ) )
2625adantld 483 . 2 |- ( ( K e. HL /\ X e. V /\ Y e. N ) -> ( ( Y e. ( Base ` K ) /\ E. p e. ( Atoms ` K ) E. q e. ( Atoms ` K ) ( p =/= q /\ Y = ( p ( join ` K ) q ) ) ) -> -. X .<_ Y ) )
278, 26mpd 15 1 |- ( ( K e. HL /\ X e. V /\ Y e. N ) -> -. X .<_ Y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   Atomscatm 34550   HLchlt 34637   LLinesclln 34777   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-llines 34784  df-lplanes 34785  df-lvols 34786
This theorem is referenced by:  lvolnelln  34875
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