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Theorem cdleme0ex1N 30705
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Nov-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdleme0.l  |-  .<_  =  ( le `  K )
cdleme0.j  |-  .\/  =  ( join `  K )
cdleme0.m  |-  ./\  =  ( meet `  K )
cdleme0.a  |-  A  =  ( Atoms `  K )
cdleme0.h  |-  H  =  ( LHyp `  K
)
cdleme0.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
Assertion
Ref Expression
cdleme0ex1N  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  E. u  e.  A  ( u  .<_  ( P  .\/  Q
)  /\  u  .<_  W ) )
Distinct variable groups:    u, A    u, 
.\/    u,  .<_    u, P    u, Q    u, U    u, W
Allowed substitution hints:    H( u)    K( u)   
./\ ( u)

Proof of Theorem cdleme0ex1N
StepHypRef Expression
1 simp1 957 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp2l 983 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
3 simp2r 984 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  Q  e.  A )
4 simp3 959 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  P  =/=  Q )
5 cdleme0.l . . . 4  |-  .<_  =  ( le `  K )
6 cdleme0.j . . . 4  |-  .\/  =  ( join `  K )
7 cdleme0.m . . . 4  |-  ./\  =  ( meet `  K )
8 cdleme0.a . . . 4  |-  A  =  ( Atoms `  K )
9 cdleme0.h . . . 4  |-  H  =  ( LHyp `  K
)
10 cdleme0.u . . . 4  |-  U  =  ( ( P  .\/  Q )  ./\  W )
115, 6, 7, 8, 9, 10lhpat2 30527 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  U  e.  A
)
121, 2, 3, 4, 11syl112anc 1188 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  U  e.  A )
13 simp2ll 1024 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  P  e.  A )
145, 6, 7, 8, 9, 10cdlemeulpq 30702 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A ) )  ->  U  .<_  ( P  .\/  Q ) )
151, 13, 3, 14syl12anc 1182 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  U  .<_  ( P  .\/  Q ) )
16 simp1l 981 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  K  e.  HL )
17 hllat 29846 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
1816, 17syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  K  e.  Lat )
19 eqid 2404 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
2019, 6, 8hlatjcl 29849 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
2116, 13, 3, 20syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  Q )  e.  (
Base `  K )
)
22 simp1r 982 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  W  e.  H )
2319, 9lhpbase 30480 . . . . 5  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
2422, 23syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  W  e.  ( Base `  K )
)
2519, 5, 7latmle2 14461 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
2618, 21, 24, 25syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
2710, 26syl5eqbr 4205 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  U  .<_  W )
28 breq1 4175 . . . 4  |-  ( u  =  U  ->  (
u  .<_  ( P  .\/  Q )  <->  U  .<_  ( P 
.\/  Q ) ) )
29 breq1 4175 . . . 4  |-  ( u  =  U  ->  (
u  .<_  W  <->  U  .<_  W ) )
3028, 29anbi12d 692 . . 3  |-  ( u  =  U  ->  (
( u  .<_  ( P 
.\/  Q )  /\  u  .<_  W )  <->  ( U  .<_  ( P  .\/  Q
)  /\  U  .<_  W ) ) )
3130rspcev 3012 . 2  |-  ( ( U  e.  A  /\  ( U  .<_  ( P 
.\/  Q )  /\  U  .<_  W ) )  ->  E. u  e.  A  ( u  .<_  ( P 
.\/  Q )  /\  u  .<_  W ) )
3212, 15, 27, 31syl12anc 1182 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  E. u  e.  A  ( u  .<_  ( P  .\/  Q
)  /\  u  .<_  W ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   Latclat 14429   Atomscatm 29746   HLchlt 29833   LHypclh 30466
This theorem is referenced by:  cdleme0ex2N  30706
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-lhyp 30470
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