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Theorem 4atexlemunv 30255
Description: Lemma for 4atexlem7 30264. (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 30245 . 2  |-  ( ph  ->  -.  S  .<_  ( P 
.\/  Q ) )
314atexlemk 30236 . . . . . . 7  |-  ( ph  ->  K  e.  HL )
414atexlemp 30239 . . . . . . 7  |-  ( ph  ->  P  e.  A )
514atexlems 30241 . . . . . . 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 29565 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  S  .<_  ( P  .\/  S ) )
103, 4, 5, 9syl3anc 1182 . . . . . 6  |-  ( ph  ->  S  .<_  ( P  .\/  S ) )
1110adantr 451 . . . . 5  |-  ( (
ph  /\  U  =  V )  ->  S  .<_  ( P  .\/  S
) )
12 4thatlem0.v . . . . . . . . 9  |-  V  =  ( ( P  .\/  S )  ./\  W )
1314atexlemkl 30246 . . . . . . . . . 10  |-  ( ph  ->  K  e.  Lat )
141, 7, 84atexlempsb 30249 . . . . . . . . . 10  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
15 4thatlem0.h . . . . . . . . . . 11  |-  H  =  ( LHyp `  K
)
161, 154atexlemwb 30248 . . . . . . . . . 10  |-  ( ph  ->  W  e.  ( Base `  K ) )
17 eqid 2283 . . . . . . . . . . 11  |-  ( Base `  K )  =  (
Base `  K )
18 4thatlem0.m . . . . . . . . . . 11  |-  ./\  =  ( meet `  K )
1917, 6, 18latmle1 14182 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  ./\  W )  .<_  ( P  .\/  S ) )
2013, 14, 16, 19syl3anc 1182 . . . . . . . . 9  |-  ( ph  ->  ( ( P  .\/  S )  ./\  W )  .<_  ( P  .\/  S
) )
2112, 20syl5eqbr 4056 . . . . . . . 8  |-  ( ph  ->  V  .<_  ( P  .\/  S ) )
2214atexlemkc 30247 . . . . . . . . 9  |-  ( ph  ->  K  e.  CvLat )
23 4thatlem0.u . . . . . . . . . 10  |-  U  =  ( ( P  .\/  Q )  ./\  W )
241, 6, 7, 18, 8, 15, 23, 124atexlemv 30254 . . . . . . . . 9  |-  ( ph  ->  V  e.  A )
2517, 6, 18latmle2 14183 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  ./\  W )  .<_  W )
2613, 14, 16, 25syl3anc 1182 . . . . . . . . . . 11  |-  ( ph  ->  ( ( P  .\/  S )  ./\  W )  .<_  W )
2712, 26syl5eqbr 4056 . . . . . . . . . 10  |-  ( ph  ->  V  .<_  W )
2814atexlempw 30238 . . . . . . . . . . 11  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
2928simprd 449 . . . . . . . . . 10  |-  ( ph  ->  -.  P  .<_  W )
30 nbrne2 4041 . . . . . . . . . 10  |-  ( ( V  .<_  W  /\  -.  P  .<_  W )  ->  V  =/=  P
)
3127, 29, 30syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  V  =/=  P )
326, 7, 8cvlatexchb1 29524 . . . . . . . . 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 1195 . . . . . . . 8  |-  ( ph  ->  ( V  .<_  ( P 
.\/  S )  <->  ( P  .\/  V )  =  ( P  .\/  S ) ) )
3421, 33mpbid 201 . . . . . . 7  |-  ( ph  ->  ( P  .\/  V
)  =  ( P 
.\/  S ) )
3534adantr 451 . . . . . 6  |-  ( (
ph  /\  U  =  V )  ->  ( P  .\/  V )  =  ( P  .\/  S
) )
36 oveq2 5866 . . . . . . . 8  |-  ( U  =  V  ->  ( P  .\/  U )  =  ( P  .\/  V
) )
3736eqcomd 2288 . . . . . . 7  |-  ( U  =  V  ->  ( P  .\/  V )  =  ( P  .\/  U
) )
3814atexlemq 30240 . . . . . . . . . . 11  |-  ( ph  ->  Q  e.  A )
3917, 7, 8hlatjcl 29556 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
403, 4, 38, 39syl3anc 1182 . . . . . . . . . 10  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
4117, 6, 18latmle1 14182 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  ( P  .\/  Q ) )
4213, 40, 16, 41syl3anc 1182 . . . . . . . . 9  |-  ( ph  ->  ( ( P  .\/  Q )  ./\  W )  .<_  ( P  .\/  Q
) )
4323, 42syl5eqbr 4056 . . . . . . . 8  |-  ( ph  ->  U  .<_  ( P  .\/  Q ) )
441, 6, 7, 18, 8, 15, 234atexlemu 30253 . . . . . . . . 9  |-  ( ph  ->  U  e.  A )
4517, 6, 18latmle2 14183 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
4613, 40, 16, 45syl3anc 1182 . . . . . . . . . . 11  |-  ( ph  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
4723, 46syl5eqbr 4056 . . . . . . . . . 10  |-  ( ph  ->  U  .<_  W )
48 nbrne2 4041 . . . . . . . . . 10  |-  ( ( U  .<_  W  /\  -.  P  .<_  W )  ->  U  =/=  P
)
4947, 29, 48syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  U  =/=  P )
506, 7, 8cvlatexchb1 29524 . . . . . . . . 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 1195 . . . . . . . 8  |-  ( ph  ->  ( U  .<_  ( P 
.\/  Q )  <->  ( P  .\/  U )  =  ( P  .\/  Q ) ) )
5243, 51mpbid 201 . . . . . . 7  |-  ( ph  ->  ( P  .\/  U
)  =  ( P 
.\/  Q ) )
5337, 52sylan9eqr 2337 . . . . . 6  |-  ( (
ph  /\  U  =  V )  ->  ( P  .\/  V )  =  ( P  .\/  Q
) )
5435, 53eqtr3d 2317 . . . . 5  |-  ( (
ph  /\  U  =  V )  ->  ( P  .\/  S )  =  ( P  .\/  Q
) )
5511, 54breqtrd 4047 . . . 4  |-  ( (
ph  /\  U  =  V )  ->  S  .<_  ( P  .\/  Q
) )
5655ex 423 . . 3  |-  ( ph  ->  ( U  =  V  ->  S  .<_  ( P 
.\/  Q ) ) )
5756necon3bd 2483 . 2  |-  ( ph  ->  ( -.  S  .<_  ( P  .\/  Q )  ->  U  =/=  V
) )
582, 57mpd 14 1  |-  ( ph  ->  U  =/=  V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Latclat 14151   Atomscatm 29453   CvLatclc 29455   HLchlt 29540   LHypclh 30173
This theorem is referenced by:  4atexlemtlw  30256  4atexlemntlpq  30257  4atexlemc  30258  4atexlemnclw  30259
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-lhyp 30177
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