Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  4atexlemex2 Unicode version

Theorem 4atexlemex2 30882
Description: Lemma for 4atexlem7 30886. Show that when  C  =/=  S,  C satisfies the existence condition of the consequent. (Contributed by NM, 25-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 )
4thatlem0.c  |-  C  =  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )
Assertion
Ref Expression
4atexlemex2  |-  ( (
ph  /\  C  =/=  S )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z
) ) )
Distinct variable groups:    z, A    z, C    z,  .\/    z,  .<_    z, P    z, S    z, W
Allowed substitution hints:    ph( z)    Q( z)    R( z)    T( z)    U( z)    H( z)    K( z)   
./\ ( z)    V( z)

Proof of Theorem 4atexlemex2
StepHypRef Expression
1 4thatlem.ph . . . 4  |-  ( 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 ) ) ) )
2 4thatlem0.l . . . 4  |-  .<_  =  ( le `  K )
3 4thatlem0.j . . . 4  |-  .\/  =  ( join `  K )
4 4thatlem0.m . . . 4  |-  ./\  =  ( meet `  K )
5 4thatlem0.a . . . 4  |-  A  =  ( Atoms `  K )
6 4thatlem0.h . . . 4  |-  H  =  ( LHyp `  K
)
7 4thatlem0.u . . . 4  |-  U  =  ( ( P  .\/  Q )  ./\  W )
8 4thatlem0.v . . . 4  |-  V  =  ( ( P  .\/  S )  ./\  W )
9 4thatlem0.c . . . 4  |-  C  =  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )
101, 2, 3, 4, 5, 6, 7, 8, 94atexlemc 30880 . . 3  |-  ( ph  ->  C  e.  A )
1110adantr 451 . 2  |-  ( (
ph  /\  C  =/=  S )  ->  C  e.  A )
121, 2, 3, 4, 5, 6, 7, 8, 94atexlemnclw 30881 . . 3  |-  ( ph  ->  -.  C  .<_  W )
1312adantr 451 . 2  |-  ( (
ph  /\  C  =/=  S )  ->  -.  C  .<_  W )
141, 2, 3, 4, 5, 6, 7, 84atexlemntlpq 30879 . . . . 5  |-  ( ph  ->  -.  T  .<_  ( P 
.\/  Q ) )
15 id 19 . . . . . . . . . . 11  |-  ( C  =  P  ->  C  =  P )
169, 15syl5eqr 2342 . . . . . . . . . 10  |-  ( C  =  P  ->  (
( Q  .\/  T
)  ./\  ( P  .\/  S ) )  =  P )
1716adantl 452 . . . . . . . . 9  |-  ( (
ph  /\  C  =  P )  ->  (
( Q  .\/  T
)  ./\  ( P  .\/  S ) )  =  P )
1814atexlemkl 30868 . . . . . . . . . . . 12  |-  ( ph  ->  K  e.  Lat )
191, 3, 54atexlemqtb 30872 . . . . . . . . . . . 12  |-  ( ph  ->  ( Q  .\/  T
)  e.  ( Base `  K ) )
201, 3, 54atexlempsb 30871 . . . . . . . . . . . 12  |-  ( ph  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
21 eqid 2296 . . . . . . . . . . . . 13  |-  ( Base `  K )  =  (
Base `  K )
2221, 2, 4latmle1 14198 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( Q  .\/  T ) )
2318, 19, 20, 22syl3anc 1182 . . . . . . . . . . 11  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( Q  .\/  T ) )
2414atexlemk 30858 . . . . . . . . . . . 12  |-  ( ph  ->  K  e.  HL )
2514atexlemq 30862 . . . . . . . . . . . 12  |-  ( ph  ->  Q  e.  A )
2614atexlemt 30864 . . . . . . . . . . . 12  |-  ( ph  ->  T  e.  A )
273, 5hlatjcom 30179 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  ->  ( Q  .\/  T
)  =  ( T 
.\/  Q ) )
2824, 25, 26, 27syl3anc 1182 . . . . . . . . . . 11  |-  ( ph  ->  ( Q  .\/  T
)  =  ( T 
.\/  Q ) )
2923, 28breqtrd 4063 . . . . . . . . . 10  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( T  .\/  Q ) )
3029adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  C  =  P )  ->  (
( Q  .\/  T
)  ./\  ( P  .\/  S ) )  .<_  ( T  .\/  Q ) )
3117, 30eqbrtrrd 4061 . . . . . . . 8  |-  ( (
ph  /\  C  =  P )  ->  P  .<_  ( T  .\/  Q
) )
3214atexlemkc 30869 . . . . . . . . . 10  |-  ( ph  ->  K  e.  CvLat )
3314atexlemp 30861 . . . . . . . . . 10  |-  ( ph  ->  P  e.  A )
3414atexlempnq 30866 . . . . . . . . . 10  |-  ( ph  ->  P  =/=  Q )
352, 3, 5cvlatexch2 30149 . . . . . . . . . 10  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  T  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .<_  ( T  .\/  Q
)  ->  T  .<_  ( P  .\/  Q ) ) )
3632, 33, 26, 25, 34, 35syl131anc 1195 . . . . . . . . 9  |-  ( ph  ->  ( P  .<_  ( T 
.\/  Q )  ->  T  .<_  ( P  .\/  Q ) ) )
3736adantr 451 . . . . . . . 8  |-  ( (
ph  /\  C  =  P )  ->  ( P  .<_  ( T  .\/  Q )  ->  T  .<_  ( P  .\/  Q ) ) )
3831, 37mpd 14 . . . . . . 7  |-  ( (
ph  /\  C  =  P )  ->  T  .<_  ( P  .\/  Q
) )
3938ex 423 . . . . . 6  |-  ( ph  ->  ( C  =  P  ->  T  .<_  ( P 
.\/  Q ) ) )
4039necon3bd 2496 . . . . 5  |-  ( ph  ->  ( -.  T  .<_  ( P  .\/  Q )  ->  C  =/=  P
) )
4114, 40mpd 14 . . . 4  |-  ( ph  ->  C  =/=  P )
4241adantr 451 . . 3  |-  ( (
ph  /\  C  =/=  S )  ->  C  =/=  P )
43 simpr 447 . . 3  |-  ( (
ph  /\  C  =/=  S )  ->  C  =/=  S )
4421, 2, 4latmle2 14199 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( P  .\/  S ) )
4518, 19, 20, 44syl3anc 1182 . . . . 5  |-  ( ph  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( P  .\/  S ) )
469, 45syl5eqbr 4072 . . . 4  |-  ( ph  ->  C  .<_  ( P  .\/  S ) )
4746adantr 451 . . 3  |-  ( (
ph  /\  C  =/=  S )  ->  C  .<_  ( P  .\/  S ) )
4814atexlems 30863 . . . . 5  |-  ( ph  ->  S  e.  A )
491, 2, 3, 54atexlempns 30873 . . . . 5  |-  ( ph  ->  P  =/=  S )
505, 2, 3cvlsupr2 30155 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  S  e.  A  /\  C  e.  A )  /\  P  =/=  S
)  ->  ( ( P  .\/  C )  =  ( S  .\/  C
)  <->  ( C  =/= 
P  /\  C  =/=  S  /\  C  .<_  ( P 
.\/  S ) ) ) )
5132, 33, 48, 10, 49, 50syl131anc 1195 . . . 4  |-  ( ph  ->  ( ( P  .\/  C )  =  ( S 
.\/  C )  <->  ( C  =/=  P  /\  C  =/= 
S  /\  C  .<_  ( P  .\/  S ) ) ) )
5251adantr 451 . . 3  |-  ( (
ph  /\  C  =/=  S )  ->  ( ( P  .\/  C )  =  ( S  .\/  C
)  <->  ( C  =/= 
P  /\  C  =/=  S  /\  C  .<_  ( P 
.\/  S ) ) ) )
5342, 43, 47, 52mpbir3and 1135 . 2  |-  ( (
ph  /\  C  =/=  S )  ->  ( P  .\/  C )  =  ( S  .\/  C ) )
54 breq1 4042 . . . . 5  |-  ( z  =  C  ->  (
z  .<_  W  <->  C  .<_  W ) )
5554notbid 285 . . . 4  |-  ( z  =  C  ->  ( -.  z  .<_  W  <->  -.  C  .<_  W ) )
56 oveq2 5882 . . . . 5  |-  ( z  =  C  ->  ( P  .\/  z )  =  ( P  .\/  C
) )
57 oveq2 5882 . . . . 5  |-  ( z  =  C  ->  ( S  .\/  z )  =  ( S  .\/  C
) )
5856, 57eqeq12d 2310 . . . 4  |-  ( z  =  C  ->  (
( P  .\/  z
)  =  ( S 
.\/  z )  <->  ( P  .\/  C )  =  ( S  .\/  C ) ) )
5955, 58anbi12d 691 . . 3  |-  ( z  =  C  ->  (
( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) )  <->  ( -.  C  .<_  W  /\  ( P 
.\/  C )  =  ( S  .\/  C
) ) ) )
6059rspcev 2897 . 2  |-  ( ( C  e.  A  /\  ( -.  C  .<_  W  /\  ( P  .\/  C )  =  ( S 
.\/  C ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z
) ) )
6111, 13, 53, 60syl12anc 1180 1  |-  ( (
ph  /\  C  =/=  S )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   meetcmee 14095   Latclat 14167   Atomscatm 30075   CvLatclc 30077   HLchlt 30162   LHypclh 30795
This theorem is referenced by:  4atexlemex4  30884  4atexlemex6  30885
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310  df-lhyp 30799
  Copyright terms: Public domain W3C validator