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Theorem cdleme0ex1N 33701
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 1005 . . 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 1031 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
3 simp2r 1032 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  Q  e.  A )
4 simp3 1007 . . 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 33522 . . 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 1268 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  U  e.  A )
13 simp2ll 1072 . . 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 33698 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A ) )  ->  U  .<_  ( P  .\/  Q ) )
151, 13, 3, 14syl12anc 1262 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  U  .<_  ( P  .\/  Q ) )
16 simp1l 1029 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  K  e.  HL )
17 hllat 32841 . . . . 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 2423 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
2019, 6, 8hlatjcl 32844 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
2116, 13, 3, 20syl3anc 1264 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  Q )  e.  (
Base `  K )
)
22 simp1r 1030 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  W  e.  H )
2319, 9lhpbase 33475 . . . . 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 16261 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
2618, 21, 24, 25syl3anc 1264 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
2710, 26syl5eqbr 4395 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  U  .<_  W )
28 breq1 4364 . . . 4  |-  ( u  =  U  ->  (
u  .<_  ( P  .\/  Q )  <->  U  .<_  ( P 
.\/  Q ) ) )
29 breq1 4364 . . . 4  |-  ( u  =  U  ->  (
u  .<_  W  <->  U  .<_  W ) )
3028, 29anbi12d 715 . . 3  |-  ( u  =  U  ->  (
( u  .<_  ( P 
.\/  Q )  /\  u  .<_  W )  <->  ( U  .<_  ( P  .\/  Q
)  /\  U  .<_  W ) ) )
3130rspcev 3120 . 2  |-  ( ( U  e.  A  /\  ( U  .<_  ( P 
.\/  Q )  /\  U  .<_  W ) )  ->  E. u  e.  A  ( u  .<_  ( P 
.\/  Q )  /\  u  .<_  W ) )
3212, 15, 27, 31syl12anc 1262 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 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2594   E.wrex 2710   class class class wbr 4361   ` cfv 5539  (class class class)co 6244   Basecbs 15059   lecple 15135   joincjn 16127   meetcmee 16128   Latclat 16229   Atomscatm 32741   HLchlt 32828   LHypclh 33461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-rep 4474  ax-sep 4484  ax-nul 4493  ax-pow 4540  ax-pr 4598  ax-un 6536
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-ral 2714  df-rex 2715  df-reu 2716  df-rab 2718  df-v 3019  df-sbc 3238  df-csb 3334  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-op 3943  df-uni 4158  df-iun 4239  df-br 4362  df-opab 4421  df-mpt 4422  df-id 4706  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-rn 4802  df-res 4803  df-ima 4804  df-iota 5503  df-fun 5541  df-fn 5542  df-f 5543  df-f1 5544  df-fo 5545  df-f1o 5546  df-fv 5547  df-riota 6206  df-ov 6247  df-oprab 6248  df-preset 16111  df-poset 16129  df-plt 16142  df-lub 16158  df-glb 16159  df-join 16160  df-meet 16161  df-p0 16223  df-p1 16224  df-lat 16230  df-clat 16292  df-oposet 32654  df-ol 32656  df-oml 32657  df-covers 32744  df-ats 32745  df-atl 32776  df-cvlat 32800  df-hlat 32829  df-lhyp 33465
This theorem is referenced by:  cdleme0ex2N  33702
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