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Theorem 4atexlemunv 30863
Description: Lemma for 4atexlem7 30872. (Contributed by NM, 21-Nov-2012.)
Hypotheses
Ref Expression
4thatlem.ph  |-  ( ph  <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) ) )
4thatlem0.l  |-  .<_  =  ( le `  K )
4thatlem0.j  |-  .\/  =  ( join `  K )
4thatlem0.m  |-  ./\  =  ( meet `  K )
4thatlem0.a  |-  A  =  ( Atoms `  K )
4thatlem0.h  |-  H  =  ( LHyp `  K
)
4thatlem0.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
4thatlem0.v  |-  V  =  ( ( P  .\/  S )  ./\  W )
Assertion
Ref Expression
4atexlemunv  |-  ( ph  ->  U  =/=  V )

Proof of Theorem 4atexlemunv
StepHypRef Expression
1 4thatlem.ph . . 3  |-  ( ph  <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) ) )
214atexlemnslpq 30853 . 2  |-  ( ph  ->  -.  S  .<_  ( P 
.\/  Q ) )
314atexlemk 30844 . . . . . . 7  |-  ( ph  ->  K  e.  HL )
414atexlemp 30847 . . . . . . 7  |-  ( ph  ->  P  e.  A )
514atexlems 30849 . . . . . . 7  |-  ( ph  ->  S  e.  A )
6 4thatlem0.l . . . . . . . 8  |-  .<_  =  ( le `  K )
7 4thatlem0.j . . . . . . . 8  |-  .\/  =  ( join `  K )
8 4thatlem0.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
96, 7, 8hlatlej2 30173 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  S  .<_  ( P  .\/  S ) )
103, 4, 5, 9syl3anc 1184 . . . . . 6  |-  ( ph  ->  S  .<_  ( P  .\/  S ) )
1110adantr 452 . . . . 5  |-  ( (
ph  /\  U  =  V )  ->  S  .<_  ( P  .\/  S
) )
12 4thatlem0.v . . . . . . . . 9  |-  V  =  ( ( P  .\/  S )  ./\  W )
1314atexlemkl 30854 . . . . . . . . . 10  |-  ( ph  ->  K  e.  Lat )
141, 7, 84atexlempsb 30857 . . . . . . . . . 10  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
15 4thatlem0.h . . . . . . . . . . 11  |-  H  =  ( LHyp `  K
)
161, 154atexlemwb 30856 . . . . . . . . . 10  |-  ( ph  ->  W  e.  ( Base `  K ) )
17 eqid 2436 . . . . . . . . . . 11  |-  ( Base `  K )  =  (
Base `  K )
18 4thatlem0.m . . . . . . . . . . 11  |-  ./\  =  ( meet `  K )
1917, 6, 18latmle1 14505 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  ./\  W )  .<_  ( P  .\/  S ) )
2013, 14, 16, 19syl3anc 1184 . . . . . . . . 9  |-  ( ph  ->  ( ( P  .\/  S )  ./\  W )  .<_  ( P  .\/  S
) )
2112, 20syl5eqbr 4245 . . . . . . . 8  |-  ( ph  ->  V  .<_  ( P  .\/  S ) )
2214atexlemkc 30855 . . . . . . . . 9  |-  ( ph  ->  K  e.  CvLat )
23 4thatlem0.u . . . . . . . . . 10  |-  U  =  ( ( P  .\/  Q )  ./\  W )
241, 6, 7, 18, 8, 15, 23, 124atexlemv 30862 . . . . . . . . 9  |-  ( ph  ->  V  e.  A )
2517, 6, 18latmle2 14506 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  ./\  W )  .<_  W )
2613, 14, 16, 25syl3anc 1184 . . . . . . . . . . 11  |-  ( ph  ->  ( ( P  .\/  S )  ./\  W )  .<_  W )
2712, 26syl5eqbr 4245 . . . . . . . . . 10  |-  ( ph  ->  V  .<_  W )
2814atexlempw 30846 . . . . . . . . . . 11  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
2928simprd 450 . . . . . . . . . 10  |-  ( ph  ->  -.  P  .<_  W )
30 nbrne2 4230 . . . . . . . . . 10  |-  ( ( V  .<_  W  /\  -.  P  .<_  W )  ->  V  =/=  P
)
3127, 29, 30syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  V  =/=  P )
326, 7, 8cvlatexchb1 30132 . . . . . . . . 9  |-  ( ( K  e.  CvLat  /\  ( V  e.  A  /\  S  e.  A  /\  P  e.  A )  /\  V  =/=  P
)  ->  ( V  .<_  ( P  .\/  S
)  <->  ( P  .\/  V )  =  ( P 
.\/  S ) ) )
3322, 24, 5, 4, 31, 32syl131anc 1197 . . . . . . . 8  |-  ( ph  ->  ( V  .<_  ( P 
.\/  S )  <->  ( P  .\/  V )  =  ( P  .\/  S ) ) )
3421, 33mpbid 202 . . . . . . 7  |-  ( ph  ->  ( P  .\/  V
)  =  ( P 
.\/  S ) )
3534adantr 452 . . . . . 6  |-  ( (
ph  /\  U  =  V )  ->  ( P  .\/  V )  =  ( P  .\/  S
) )
36 oveq2 6089 . . . . . . . 8  |-  ( U  =  V  ->  ( P  .\/  U )  =  ( P  .\/  V
) )
3736eqcomd 2441 . . . . . . 7  |-  ( U  =  V  ->  ( P  .\/  V )  =  ( P  .\/  U
) )
3814atexlemq 30848 . . . . . . . . . . 11  |-  ( ph  ->  Q  e.  A )
3917, 7, 8hlatjcl 30164 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
403, 4, 38, 39syl3anc 1184 . . . . . . . . . 10  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
4117, 6, 18latmle1 14505 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  ( P  .\/  Q ) )
4213, 40, 16, 41syl3anc 1184 . . . . . . . . 9  |-  ( ph  ->  ( ( P  .\/  Q )  ./\  W )  .<_  ( P  .\/  Q
) )
4323, 42syl5eqbr 4245 . . . . . . . 8  |-  ( ph  ->  U  .<_  ( P  .\/  Q ) )
441, 6, 7, 18, 8, 15, 234atexlemu 30861 . . . . . . . . 9  |-  ( ph  ->  U  e.  A )
4517, 6, 18latmle2 14506 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
4613, 40, 16, 45syl3anc 1184 . . . . . . . . . . 11  |-  ( ph  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
4723, 46syl5eqbr 4245 . . . . . . . . . 10  |-  ( ph  ->  U  .<_  W )
48 nbrne2 4230 . . . . . . . . . 10  |-  ( ( U  .<_  W  /\  -.  P  .<_  W )  ->  U  =/=  P
)
4947, 29, 48syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  U  =/=  P )
506, 7, 8cvlatexchb1 30132 . . . . . . . . 9  |-  ( ( K  e.  CvLat  /\  ( U  e.  A  /\  Q  e.  A  /\  P  e.  A )  /\  U  =/=  P
)  ->  ( U  .<_  ( P  .\/  Q
)  <->  ( P  .\/  U )  =  ( P 
.\/  Q ) ) )
5122, 44, 38, 4, 49, 50syl131anc 1197 . . . . . . . 8  |-  ( ph  ->  ( U  .<_  ( P 
.\/  Q )  <->  ( P  .\/  U )  =  ( P  .\/  Q ) ) )
5243, 51mpbid 202 . . . . . . 7  |-  ( ph  ->  ( P  .\/  U
)  =  ( P 
.\/  Q ) )
5337, 52sylan9eqr 2490 . . . . . 6  |-  ( (
ph  /\  U  =  V )  ->  ( P  .\/  V )  =  ( P  .\/  Q
) )
5435, 53eqtr3d 2470 . . . . 5  |-  ( (
ph  /\  U  =  V )  ->  ( P  .\/  S )  =  ( P  .\/  Q
) )
5511, 54breqtrd 4236 . . . 4  |-  ( (
ph  /\  U  =  V )  ->  S  .<_  ( P  .\/  Q
) )
5655ex 424 . . 3  |-  ( ph  ->  ( U  =  V  ->  S  .<_  ( P 
.\/  Q ) ) )
5756necon3bd 2638 . 2  |-  ( ph  ->  ( -.  S  .<_  ( P  .\/  Q )  ->  U  =/=  V
) )
582, 57mpd 15 1  |-  ( ph  ->  U  =/=  V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   lecple 13536   joincjn 14401   meetcmee 14402   Latclat 14474   Atomscatm 30061   CvLatclc 30063   HLchlt 30148   LHypclh 30781
This theorem is referenced by:  4atexlemtlw  30864  4atexlemntlpq  30865  4atexlemc  30866  4atexlemnclw  30867
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-p1 14469  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-lhyp 30785
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