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Theorem 4atexlemunv 35079
Description: Lemma for 4atexlem7 35088. (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 35069 . 2  |-  ( ph  ->  -.  S  .<_  ( P 
.\/  Q ) )
314atexlemk 35060 . . . . . . 7  |-  ( ph  ->  K  e.  HL )
414atexlemp 35063 . . . . . . 7  |-  ( ph  ->  P  e.  A )
514atexlems 35065 . . . . . . 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 34389 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  S  .<_  ( P  .\/  S ) )
103, 4, 5, 9syl3anc 1228 . . . . . 6  |-  ( ph  ->  S  .<_  ( P  .\/  S ) )
1110adantr 465 . . . . 5  |-  ( (
ph  /\  U  =  V )  ->  S  .<_  ( P  .\/  S
) )
12 4thatlem0.v . . . . . . . . 9  |-  V  =  ( ( P  .\/  S )  ./\  W )
1314atexlemkl 35070 . . . . . . . . . 10  |-  ( ph  ->  K  e.  Lat )
141, 7, 84atexlempsb 35073 . . . . . . . . . 10  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
15 4thatlem0.h . . . . . . . . . . 11  |-  H  =  ( LHyp `  K
)
161, 154atexlemwb 35072 . . . . . . . . . 10  |-  ( ph  ->  W  e.  ( Base `  K ) )
17 eqid 2467 . . . . . . . . . . 11  |-  ( Base `  K )  =  (
Base `  K )
18 4thatlem0.m . . . . . . . . . . 11  |-  ./\  =  ( meet `  K )
1917, 6, 18latmle1 15566 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  ./\  W )  .<_  ( P  .\/  S ) )
2013, 14, 16, 19syl3anc 1228 . . . . . . . . 9  |-  ( ph  ->  ( ( P  .\/  S )  ./\  W )  .<_  ( P  .\/  S
) )
2112, 20syl5eqbr 4480 . . . . . . . 8  |-  ( ph  ->  V  .<_  ( P  .\/  S ) )
2214atexlemkc 35071 . . . . . . . . 9  |-  ( ph  ->  K  e.  CvLat )
23 4thatlem0.u . . . . . . . . . 10  |-  U  =  ( ( P  .\/  Q )  ./\  W )
241, 6, 7, 18, 8, 15, 23, 124atexlemv 35078 . . . . . . . . 9  |-  ( ph  ->  V  e.  A )
2517, 6, 18latmle2 15567 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  ./\  W )  .<_  W )
2613, 14, 16, 25syl3anc 1228 . . . . . . . . . . 11  |-  ( ph  ->  ( ( P  .\/  S )  ./\  W )  .<_  W )
2712, 26syl5eqbr 4480 . . . . . . . . . 10  |-  ( ph  ->  V  .<_  W )
2814atexlempw 35062 . . . . . . . . . . 11  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
2928simprd 463 . . . . . . . . . 10  |-  ( ph  ->  -.  P  .<_  W )
30 nbrne2 4465 . . . . . . . . . 10  |-  ( ( V  .<_  W  /\  -.  P  .<_  W )  ->  V  =/=  P
)
3127, 29, 30syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  V  =/=  P )
326, 7, 8cvlatexchb1 34348 . . . . . . . . 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 1241 . . . . . . . 8  |-  ( ph  ->  ( V  .<_  ( P 
.\/  S )  <->  ( P  .\/  V )  =  ( P  .\/  S ) ) )
3421, 33mpbid 210 . . . . . . 7  |-  ( ph  ->  ( P  .\/  V
)  =  ( P 
.\/  S ) )
3534adantr 465 . . . . . 6  |-  ( (
ph  /\  U  =  V )  ->  ( P  .\/  V )  =  ( P  .\/  S
) )
36 oveq2 6293 . . . . . . . 8  |-  ( U  =  V  ->  ( P  .\/  U )  =  ( P  .\/  V
) )
3736eqcomd 2475 . . . . . . 7  |-  ( U  =  V  ->  ( P  .\/  V )  =  ( P  .\/  U
) )
3814atexlemq 35064 . . . . . . . . . . 11  |-  ( ph  ->  Q  e.  A )
3917, 7, 8hlatjcl 34380 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
403, 4, 38, 39syl3anc 1228 . . . . . . . . . 10  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
4117, 6, 18latmle1 15566 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  ( P  .\/  Q ) )
4213, 40, 16, 41syl3anc 1228 . . . . . . . . 9  |-  ( ph  ->  ( ( P  .\/  Q )  ./\  W )  .<_  ( P  .\/  Q
) )
4323, 42syl5eqbr 4480 . . . . . . . 8  |-  ( ph  ->  U  .<_  ( P  .\/  Q ) )
441, 6, 7, 18, 8, 15, 234atexlemu 35077 . . . . . . . . 9  |-  ( ph  ->  U  e.  A )
4517, 6, 18latmle2 15567 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
4613, 40, 16, 45syl3anc 1228 . . . . . . . . . . 11  |-  ( ph  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
4723, 46syl5eqbr 4480 . . . . . . . . . 10  |-  ( ph  ->  U  .<_  W )
48 nbrne2 4465 . . . . . . . . . 10  |-  ( ( U  .<_  W  /\  -.  P  .<_  W )  ->  U  =/=  P
)
4947, 29, 48syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  U  =/=  P )
506, 7, 8cvlatexchb1 34348 . . . . . . . . 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 1241 . . . . . . . 8  |-  ( ph  ->  ( U  .<_  ( P 
.\/  Q )  <->  ( P  .\/  U )  =  ( P  .\/  Q ) ) )
5243, 51mpbid 210 . . . . . . 7  |-  ( ph  ->  ( P  .\/  U
)  =  ( P 
.\/  Q ) )
5337, 52sylan9eqr 2530 . . . . . 6  |-  ( (
ph  /\  U  =  V )  ->  ( P  .\/  V )  =  ( P  .\/  Q
) )
5435, 53eqtr3d 2510 . . . . 5  |-  ( (
ph  /\  U  =  V )  ->  ( P  .\/  S )  =  ( P  .\/  Q
) )
5511, 54breqtrd 4471 . . . 4  |-  ( (
ph  /\  U  =  V )  ->  S  .<_  ( P  .\/  Q
) )
5655ex 434 . . 3  |-  ( ph  ->  ( U  =  V  ->  S  .<_  ( P 
.\/  Q ) ) )
5756necon3bd 2679 . 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 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447   ` cfv 5588  (class class class)co 6285   Basecbs 14493   lecple 14565   joincjn 15434   meetcmee 15435   Latclat 15535   Atomscatm 34277   CvLatclc 34279   HLchlt 34364   LHypclh 34997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-poset 15436  df-plt 15448  df-lub 15464  df-glb 15465  df-join 15466  df-meet 15467  df-p0 15529  df-p1 15530  df-lat 15536  df-clat 15598  df-oposet 34190  df-ol 34192  df-oml 34193  df-covers 34280  df-ats 34281  df-atl 34312  df-cvlat 34336  df-hlat 34365  df-lhyp 35001
This theorem is referenced by:  4atexlemtlw  35080  4atexlemntlpq  35081  4atexlemc  35082  4atexlemnclw  35083
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