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Theorem lhpexle2lem 35295
Description: Lemma for lhpexle2 35296. (Contributed by NM, 19-Jun-2013.)
Hypotheses
Ref Expression
lhpex1.l |- .<_ = ( le ` K )
lhpex1.a |- A = ( Atoms ` K )
lhpex1.h |- H = ( LHyp ` K )
Assertion
Ref Expression
lhpexle2lem |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) -> E. p e. A ( p .<_ W /\ p =/= X /\ p =/= Y ) )
Distinct variable groups:   .<_ , p   A, p   H, p   K, p   W, p   X, p   Y, p
Proof of Theorem lhpexle2lem
StepHypRef Expression
1 simpl1 1064 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ X = Y ) -> ( K e. HL /\ W e. H ) )
2 lhpex1.l . . . . 5 |- .<_ = ( le ` K )
3 lhpex1.a . . . . 5 |- A = ( Atoms ` K )
4 lhpex1.h . . . . 5 |- H = ( LHyp ` K )
52, 3, 4lhpexle1 35294 . . . 4 |- ( ( K e. HL /\ W e. H ) -> E. p e. A ( p .<_ W /\ p =/= X ) )
61, 5syl 17 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ X = Y ) -> E. p e. A ( p .<_ W /\ p =/= X ) )
7 simp3l 1089 . . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ X = Y /\ ( p .<_ W /\ p =/= X ) ) -> p .<_ W )
8 simp3r 1090 . . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ X = Y /\ ( p .<_ W /\ p =/= X ) ) -> p =/= X )
9 simp2 1062 . . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ X = Y /\ ( p .<_ W /\ p =/= X ) ) -> X = Y )
108, 9neeqtrd 2863 . . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ X = Y /\ ( p .<_ W /\ p =/= X ) ) -> p =/= Y )
117, 8, 103jca 1242 . . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ X = Y /\ ( p .<_ W /\ p =/= X ) ) -> ( p .<_ W /\ p =/= X /\ p =/= Y ) )
12113expia 1267 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ X = Y ) -> ( ( p .<_ W /\ p =/= X ) -> ( p .<_ W /\ p =/= X /\ p =/= Y ) ) )
1312reximdv 3016 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ X = Y ) -> ( E. p e. A ( p .<_ W /\ p =/= X ) -> E. p e. A ( p .<_ W /\ p =/= X /\ p =/= Y ) ) )
146, 13mpd 15 . 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ X = Y ) -> E. p e. A ( p .<_ W /\ p =/= X /\ p =/= Y ) )
15 simpl1l 1112 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ X =/= Y ) -> K e. HL )
16 simpl2l 1114 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ X =/= Y ) -> X e. A )
17 simpl3l 1116 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ X =/= Y ) -> Y e. A )
18 simpr 477 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ X =/= Y ) -> X =/= Y )
19 eqid 2622 . . . . 5 |- ( join ` K ) = ( join ` K )
202, 19, 3hlsupr 34672 . . . 4 |- ( ( ( K e. HL /\ X e. A /\ Y e. A ) /\ X =/= Y ) -> E. p e. A ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) )
2115, 16, 17, 18, 20syl31anc 1329 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ X =/= Y ) -> E. p e. A ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) )
22 eqid 2622 . . . . . . . 8 |- ( Base ` K ) = ( Base ` K )
23 simpl1l 1112 . . . . . . . . 9 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ ( ( X =/= Y /\ p e. A ) /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> K e. HL )
24 hllat 34650 . . . . . . . . 9 |- ( K e. HL -> K e. Lat )
2523, 24syl 17 . . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ ( ( X =/= Y /\ p e. A ) /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> K e. Lat )
26 simprlr 803 . . . . . . . . 9 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ ( ( X =/= Y /\ p e. A ) /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> p e. A )
2722, 3atbase 34576 . . . . . . . . 9 |- ( p e. A -> p e. ( Base ` K ) )
2826, 27syl 17 . . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ ( ( X =/= Y /\ p e. A ) /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> p e. ( Base ` K ) )
29 simpl2l 1114 . . . . . . . . 9 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ ( ( X =/= Y /\ p e. A ) /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> X e. A )
30 simpl3l 1116 . . . . . . . . 9 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ ( ( X =/= Y /\ p e. A ) /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> Y e. A )
3122, 19, 3hlatjcl 34653 . . . . . . . . 9 |- ( ( K e. HL /\ X e. A /\ Y e. A ) -> ( X ( join ` K ) Y ) e. ( Base ` K ) )
3223, 29, 30, 31syl3anc 1326 . . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ ( ( X =/= Y /\ p e. A ) /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> ( X ( join ` K ) Y ) e. ( Base ` K ) )
33 simpl1r 1113 . . . . . . . . 9 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ ( ( X =/= Y /\ p e. A ) /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> W e. H )
3422, 4lhpbase 35284 . . . . . . . . 9 |- ( W e. H -> W e. ( Base ` K ) )
3533, 34syl 17 . . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ ( ( X =/= Y /\ p e. A ) /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> W e. ( Base ` K ) )
36 simprr3 1111 . . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ ( ( X =/= Y /\ p e. A ) /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> p .<_ ( X ( join ` K ) Y ) )
37 simpl2r 1115 . . . . . . . . 9 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ ( ( X =/= Y /\ p e. A ) /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> X .<_ W )
38 simpl3r 1117 . . . . . . . . 9 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ ( ( X =/= Y /\ p e. A ) /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> Y .<_ W )
3922, 3atbase 34576 . . . . . . . . . . 11 |- ( X e. A -> X e. ( Base ` K ) )
4029, 39syl 17 . . . . . . . . . 10 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ ( ( X =/= Y /\ p e. A ) /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> X e. ( Base ` K ) )
4122, 3atbase 34576 . . . . . . . . . . 11 |- ( Y e. A -> Y e. ( Base ` K ) )
4230, 41syl 17 . . . . . . . . . 10 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ ( ( X =/= Y /\ p e. A ) /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> Y e. ( Base ` K ) )
4322, 2, 19latjle12 17062 . . . . . . . . . 10 |- ( ( K e. Lat /\ ( X e. ( Base ` K ) /\ Y e. ( Base ` K ) /\ W e. ( Base ` K ) ) ) -> ( ( X .<_ W /\ Y .<_ W ) <-> ( X ( join ` K ) Y ) .<_ W ) )
4425, 40, 42, 35, 43syl13anc 1328 . . . . . . . . 9 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ ( ( X =/= Y /\ p e. A ) /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> ( ( X .<_ W /\ Y .<_ W ) <-> ( X ( join ` K ) Y ) .<_ W ) )
4537, 38, 44mpbi2and 956 . . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ ( ( X =/= Y /\ p e. A ) /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> ( X ( join ` K ) Y ) .<_ W )
4622, 2, 25, 28, 32, 35, 36, 45lattrd 17058 . . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ ( ( X =/= Y /\ p e. A ) /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> p .<_ W )
47 simprr1 1109 . . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ ( ( X =/= Y /\ p e. A ) /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> p =/= X )
48 simprr2 1110 . . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ ( ( X =/= Y /\ p e. A ) /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> p =/= Y )
4946, 47, 483jca 1242 . . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ ( ( X =/= Y /\ p e. A ) /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> ( p .<_ W /\ p =/= X /\ p =/= Y ) )
5049exp44 641 . . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) -> ( X =/= Y -> ( p e. A -> ( ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) -> ( p .<_ W /\ p =/= X /\ p =/= Y ) ) ) ) )
5150imp31 448 . . . 4 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ X =/= Y ) /\ p e. A ) -> ( ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) -> ( p .<_ W /\ p =/= X /\ p =/= Y ) ) )
5251reximdva 3017 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ X =/= Y ) -> ( E. p e. A ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) -> E. p e. A ( p .<_ W /\ p =/= X /\ p =/= Y ) ) )
5321, 52mpd 15 . 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) /\ X =/= Y ) -> E. p e. A ( p .<_ W /\ p =/= X /\ p =/= Y ) )
5414, 53pm2.61dane 2881 1 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) -> E. p e. A ( p .<_ W /\ p =/= X /\ p =/= Y ) )
Colors of variables: wff setvar class
Syntax hints:   -> 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   Latclat 17045   Atomscatm 34550   HLchlt 34637   LHypclh 35270
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-p1 17040  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-lhyp 35274
This theorem is referenced by:  lhpexle2  35296
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