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Theorem 4atexlemunv 33343
Description: Lemma for 4atexlem7 33352. (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 33333 . 2  |-  ( ph  ->  -.  S  .<_  ( P 
.\/  Q ) )
314atexlemk 33324 . . . . . . 7  |-  ( ph  ->  K  e.  HL )
414atexlemp 33327 . . . . . . 7  |-  ( ph  ->  P  e.  A )
514atexlems 33329 . . . . . . 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 32653 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  S  .<_  ( P  .\/  S ) )
103, 4, 5, 9syl3anc 1264 . . . . . 6  |-  ( ph  ->  S  .<_  ( P  .\/  S ) )
1110adantr 466 . . . . 5  |-  ( (
ph  /\  U  =  V )  ->  S  .<_  ( P  .\/  S
) )
12 4thatlem0.v . . . . . . . . 9  |-  V  =  ( ( P  .\/  S )  ./\  W )
1314atexlemkl 33334 . . . . . . . . . 10  |-  ( ph  ->  K  e.  Lat )
141, 7, 84atexlempsb 33337 . . . . . . . . . 10  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
15 4thatlem0.h . . . . . . . . . . 11  |-  H  =  ( LHyp `  K
)
161, 154atexlemwb 33336 . . . . . . . . . 10  |-  ( ph  ->  W  e.  ( Base `  K ) )
17 eqid 2429 . . . . . . . . . . 11  |-  ( Base `  K )  =  (
Base `  K )
18 4thatlem0.m . . . . . . . . . . 11  |-  ./\  =  ( meet `  K )
1917, 6, 18latmle1 16273 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  ./\  W )  .<_  ( P  .\/  S ) )
2013, 14, 16, 19syl3anc 1264 . . . . . . . . 9  |-  ( ph  ->  ( ( P  .\/  S )  ./\  W )  .<_  ( P  .\/  S
) )
2112, 20syl5eqbr 4459 . . . . . . . 8  |-  ( ph  ->  V  .<_  ( P  .\/  S ) )
2214atexlemkc 33335 . . . . . . . . 9  |-  ( ph  ->  K  e.  CvLat )
23 4thatlem0.u . . . . . . . . . 10  |-  U  =  ( ( P  .\/  Q )  ./\  W )
241, 6, 7, 18, 8, 15, 23, 124atexlemv 33342 . . . . . . . . 9  |-  ( ph  ->  V  e.  A )
2517, 6, 18latmle2 16274 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  ./\  W )  .<_  W )
2613, 14, 16, 25syl3anc 1264 . . . . . . . . . . 11  |-  ( ph  ->  ( ( P  .\/  S )  ./\  W )  .<_  W )
2712, 26syl5eqbr 4459 . . . . . . . . . 10  |-  ( ph  ->  V  .<_  W )
2814atexlempw 33326 . . . . . . . . . . 11  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
2928simprd 464 . . . . . . . . . 10  |-  ( ph  ->  -.  P  .<_  W )
30 nbrne2 4444 . . . . . . . . . 10  |-  ( ( V  .<_  W  /\  -.  P  .<_  W )  ->  V  =/=  P
)
3127, 29, 30syl2anc 665 . . . . . . . . 9  |-  ( ph  ->  V  =/=  P )
326, 7, 8cvlatexchb1 32612 . . . . . . . . 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 1277 . . . . . . . 8  |-  ( ph  ->  ( V  .<_  ( P 
.\/  S )  <->  ( P  .\/  V )  =  ( P  .\/  S ) ) )
3421, 33mpbid 213 . . . . . . 7  |-  ( ph  ->  ( P  .\/  V
)  =  ( P 
.\/  S ) )
3534adantr 466 . . . . . 6  |-  ( (
ph  /\  U  =  V )  ->  ( P  .\/  V )  =  ( P  .\/  S
) )
36 oveq2 6313 . . . . . . . 8  |-  ( U  =  V  ->  ( P  .\/  U )  =  ( P  .\/  V
) )
3736eqcomd 2437 . . . . . . 7  |-  ( U  =  V  ->  ( P  .\/  V )  =  ( P  .\/  U
) )
3814atexlemq 33328 . . . . . . . . . . 11  |-  ( ph  ->  Q  e.  A )
3917, 7, 8hlatjcl 32644 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
403, 4, 38, 39syl3anc 1264 . . . . . . . . . 10  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
4117, 6, 18latmle1 16273 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  ( P  .\/  Q ) )
4213, 40, 16, 41syl3anc 1264 . . . . . . . . 9  |-  ( ph  ->  ( ( P  .\/  Q )  ./\  W )  .<_  ( P  .\/  Q
) )
4323, 42syl5eqbr 4459 . . . . . . . 8  |-  ( ph  ->  U  .<_  ( P  .\/  Q ) )
441, 6, 7, 18, 8, 15, 234atexlemu 33341 . . . . . . . . 9  |-  ( ph  ->  U  e.  A )
4517, 6, 18latmle2 16274 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
4613, 40, 16, 45syl3anc 1264 . . . . . . . . . . 11  |-  ( ph  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
4723, 46syl5eqbr 4459 . . . . . . . . . 10  |-  ( ph  ->  U  .<_  W )
48 nbrne2 4444 . . . . . . . . . 10  |-  ( ( U  .<_  W  /\  -.  P  .<_  W )  ->  U  =/=  P
)
4947, 29, 48syl2anc 665 . . . . . . . . 9  |-  ( ph  ->  U  =/=  P )
506, 7, 8cvlatexchb1 32612 . . . . . . . . 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 1277 . . . . . . . 8  |-  ( ph  ->  ( U  .<_  ( P 
.\/  Q )  <->  ( P  .\/  U )  =  ( P  .\/  Q ) ) )
5243, 51mpbid 213 . . . . . . 7  |-  ( ph  ->  ( P  .\/  U
)  =  ( P 
.\/  Q ) )
5337, 52sylan9eqr 2492 . . . . . 6  |-  ( (
ph  /\  U  =  V )  ->  ( P  .\/  V )  =  ( P  .\/  Q
) )
5435, 53eqtr3d 2472 . . . . 5  |-  ( (
ph  /\  U  =  V )  ->  ( P  .\/  S )  =  ( P  .\/  Q
) )
5511, 54breqtrd 4450 . . . 4  |-  ( (
ph  /\  U  =  V )  ->  S  .<_  ( P  .\/  Q
) )
5655ex 435 . . 3  |-  ( ph  ->  ( U  =  V  ->  S  .<_  ( P 
.\/  Q ) ) )
5756necon3bd 2643 . 2  |-  ( ph  ->  ( -.  S  .<_  ( P  .\/  Q )  ->  U  =/=  V
) )
582, 57mpd 15 1  |-  ( ph  ->  U  =/=  V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Basecbs 15084   lecple 15159   joincjn 16140   meetcmee 16141   Latclat 16242   Atomscatm 32541   CvLatclc 32543   HLchlt 32628   LHypclh 33261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-preset 16124  df-poset 16142  df-plt 16155  df-lub 16171  df-glb 16172  df-join 16173  df-meet 16174  df-p0 16236  df-p1 16237  df-lat 16243  df-clat 16305  df-oposet 32454  df-ol 32456  df-oml 32457  df-covers 32544  df-ats 32545  df-atl 32576  df-cvlat 32600  df-hlat 32629  df-lhyp 33265
This theorem is referenced by:  4atexlemtlw  33344  4atexlemntlpq  33345  4atexlemc  33346  4atexlemnclw  33347
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