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Theorem cdleme11dN 30748
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 30756. (Contributed by NM, 13-Jun-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdleme11.l  |-  .<_  =  ( le `  K )
cdleme11.j  |-  .\/  =  ( join `  K )
cdleme11.m  |-  ./\  =  ( meet `  K )
cdleme11.a  |-  A  =  ( Atoms `  K )
cdleme11.h  |-  H  =  ( LHyp `  K
)
cdleme11.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
Assertion
Ref Expression
cdleme11dN  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  ( P  .\/  S )  =/=  ( P  .\/  T
) )

Proof of Theorem cdleme11dN
StepHypRef Expression
1 simp1 957 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  (
( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A ) )
2 simp2 958 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  (
( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
) )
3 simp32 994 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  -.  S  .<_  ( P  .\/  Q ) )
4 simp33 995 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  U  .<_  ( S  .\/  T
) )
5 cdleme11.l . . . 4  |-  .<_  =  ( le `  K )
6 cdleme11.j . . . 4  |-  .\/  =  ( join `  K )
7 cdleme11.m . . . 4  |-  ./\  =  ( meet `  K )
8 cdleme11.a . . . 4  |-  A  =  ( Atoms `  K )
9 cdleme11.h . . . 4  |-  H  =  ( LHyp `  K
)
10 cdleme11.u . . . 4  |-  U  =  ( ( P  .\/  Q )  ./\  W )
115, 6, 7, 8, 9, 10cdleme11c 30747 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  U  .<_  ( S  .\/  T ) ) )  ->  -.  P  .<_  ( S  .\/  T ) )
121, 2, 3, 4, 11syl112anc 1188 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  -.  P  .<_  ( S  .\/  T ) )
13 simp11l 1068 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  K  e.  HL )
14 simp12l 1070 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  P  e.  A )
15 simp21l 1074 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  S  e.  A )
165, 6, 8hlatlej2 29862 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  S  .<_  ( P  .\/  S ) )
1713, 14, 15, 16syl3anc 1184 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  S  .<_  ( P  .\/  S
) )
18 breq2 4180 . . . . 5  |-  ( ( P  .\/  S )  =  ( P  .\/  T )  ->  ( S  .<_  ( P  .\/  S
)  <->  S  .<_  ( P 
.\/  T ) ) )
1917, 18syl5ibcom 212 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  (
( P  .\/  S
)  =  ( P 
.\/  T )  ->  S  .<_  ( P  .\/  T ) ) )
20 simp22 991 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  T  e.  A )
21 simp31 993 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  S  =/=  T )
225, 6, 8hlatexch2 29882 . . . . 5  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  P  e.  A  /\  T  e.  A
)  /\  S  =/=  T )  ->  ( S  .<_  ( P  .\/  T
)  ->  P  .<_  ( S  .\/  T ) ) )
2313, 15, 14, 20, 21, 22syl131anc 1197 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  ( S  .<_  ( P  .\/  T )  ->  P  .<_  ( S  .\/  T ) ) )
2419, 23syld 42 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  (
( P  .\/  S
)  =  ( P 
.\/  T )  ->  P  .<_  ( S  .\/  T ) ) )
2524necon3bd 2608 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  ( -.  P  .<_  ( S 
.\/  T )  -> 
( P  .\/  S
)  =/=  ( P 
.\/  T ) ) )
2612, 25mpd 15 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q
)  /\  ( S  =/=  T  /\  -.  S  .<_  ( P  .\/  Q
)  /\  U  .<_  ( S  .\/  T ) ) )  ->  ( P  .\/  S )  =/=  ( P  .\/  T
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2571   class class class wbr 4176   ` cfv 5417  (class class class)co 6044   lecple 13495   joincjn 14360   meetcmee 14361   Atomscatm 29750   HLchlt 29837   LHypclh 30470
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 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-iin 4060  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-undef 6506  df-riota 6512  df-poset 14362  df-plt 14374  df-lub 14390  df-glb 14391  df-join 14392  df-meet 14393  df-p0 14427  df-p1 14428  df-lat 14434  df-clat 14496  df-oposet 29663  df-ol 29665  df-oml 29666  df-covers 29753  df-ats 29754  df-atl 29785  df-cvlat 29809  df-hlat 29838  df-psubsp 29989  df-pmap 29990  df-padd 30282  df-lhyp 30474
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