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Theorem cdleme0ex1N 33241
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 997 . . 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 1023 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
3 simp2r 1024 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  Q  e.  A )
4 simp3 999 . . 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 33062 . . 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 1234 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  U  e.  A )
13 simp2ll 1064 . . 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 33238 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A ) )  ->  U  .<_  ( P  .\/  Q ) )
151, 13, 3, 14syl12anc 1228 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  U  .<_  ( P  .\/  Q ) )
16 simp1l 1021 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  K  e.  HL )
17 hllat 32381 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
1816, 17syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  K  e.  Lat )
19 eqid 2402 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
2019, 6, 8hlatjcl 32384 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
2116, 13, 3, 20syl3anc 1230 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  Q )  e.  (
Base `  K )
)
22 simp1r 1022 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  W  e.  H )
2319, 9lhpbase 33015 . . . . 5  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
2422, 23syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  W  e.  ( Base `  K )
)
2519, 5, 7latmle2 16031 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
2618, 21, 24, 25syl3anc 1230 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
2710, 26syl5eqbr 4428 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  U  .<_  W )
28 breq1 4398 . . . 4  |-  ( u  =  U  ->  (
u  .<_  ( P  .\/  Q )  <->  U  .<_  ( P 
.\/  Q ) ) )
29 breq1 4398 . . . 4  |-  ( u  =  U  ->  (
u  .<_  W  <->  U  .<_  W ) )
3028, 29anbi12d 709 . . 3  |-  ( u  =  U  ->  (
( u  .<_  ( P 
.\/  Q )  /\  u  .<_  W )  <->  ( U  .<_  ( P  .\/  Q
)  /\  U  .<_  W ) ) )
3130rspcev 3160 . 2  |-  ( ( U  e.  A  /\  ( U  .<_  ( P 
.\/  Q )  /\  U  .<_  W ) )  ->  E. u  e.  A  ( u  .<_  ( P 
.\/  Q )  /\  u  .<_  W ) )
3212, 15, 27, 31syl12anc 1228 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   E.wrex 2755   class class class wbr 4395   ` cfv 5569  (class class class)co 6278   Basecbs 14841   lecple 14916   joincjn 15897   meetcmee 15898   Latclat 15999   Atomscatm 32281   HLchlt 32368   LHypclh 33001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-preset 15881  df-poset 15899  df-plt 15912  df-lub 15928  df-glb 15929  df-join 15930  df-meet 15931  df-p0 15993  df-p1 15994  df-lat 16000  df-clat 16062  df-oposet 32194  df-ol 32196  df-oml 32197  df-covers 32284  df-ats 32285  df-atl 32316  df-cvlat 32340  df-hlat 32369  df-lhyp 33005
This theorem is referenced by:  cdleme0ex2N  33242
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