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Theorem 4atexlemex4 34869
Description: Lemma for 4atexlem7 34871. Show that when  C  =  S,  D satisfies the existence condition of the consequent. (Contributed by NM, 26-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 ) )
4thatlem0.d  |-  D  =  ( ( R  .\/  T )  ./\  ( P  .\/  S ) )
Assertion
Ref Expression
4atexlemex4  |-  ( (
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    z, D
Allowed substitution hints:    ph( z)    Q( z)    R( z)    T( z)    U( z)    H( z)    K( z)   
./\ ( z)    V( z)

Proof of Theorem 4atexlemex4
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.a . . . 4  |-  A  =  ( Atoms `  K )
5 4thatlem0.u . . . 4  |-  U  =  ( ( P  .\/  Q )  ./\  W )
61, 2, 3, 4, 54atexlemswapqr 34859 . . 3  |-  ( ph  ->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( S  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W  /\  ( P  .\/  Q )  =  ( R  .\/  Q ) )  /\  ( T  e.  A  /\  ( ( ( P 
.\/  R )  ./\  W )  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  R  /\  -.  S  .<_  ( P 
.\/  R ) ) ) )
7 4thatlem0.m . . . . 5  |-  ./\  =  ( meet `  K )
8 4thatlem0.h . . . . 5  |-  H  =  ( LHyp `  K
)
9 4thatlem0.v . . . . 5  |-  V  =  ( ( P  .\/  S )  ./\  W )
10 4thatlem0.c . . . . 5  |-  C  =  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )
11 4thatlem0.d . . . . 5  |-  D  =  ( ( R  .\/  T )  ./\  ( P  .\/  S ) )
121, 2, 3, 7, 4, 8, 5, 9, 10, 114atexlemcnd 34868 . . . 4  |-  ( ph  ->  C  =/=  D )
13 pm13.18 2778 . . . . . 6  |-  ( ( C  =  S  /\  C  =/=  D )  ->  S  =/=  D )
1413necomd 2738 . . . . 5  |-  ( ( C  =  S  /\  C  =/=  D )  ->  D  =/=  S )
1514expcom 435 . . . 4  |-  ( C  =/=  D  ->  ( C  =  S  ->  D  =/=  S ) )
1612, 15syl 16 . . 3  |-  ( ph  ->  ( C  =  S  ->  D  =/=  S
) )
17 biid 236 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( S  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W  /\  ( P  .\/  Q )  =  ( R  .\/  Q ) )  /\  ( T  e.  A  /\  ( ( ( P 
.\/  R )  ./\  W )  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  R  /\  -.  S  .<_  ( P 
.\/  R ) ) )  <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( S  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W  /\  ( P  .\/  Q )  =  ( R  .\/  Q ) )  /\  ( T  e.  A  /\  ( ( ( P 
.\/  R )  ./\  W )  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  R  /\  -.  S  .<_  ( P 
.\/  R ) ) ) )
18 eqid 2467 . . . 4  |-  ( ( P  .\/  R ) 
./\  W )  =  ( ( P  .\/  R )  ./\  W )
1917, 2, 3, 7, 4, 8, 18, 9, 114atexlemex2 34867 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( S  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W  /\  ( P  .\/  Q )  =  ( R  .\/  Q ) )  /\  ( T  e.  A  /\  ( ( ( P 
.\/  R )  ./\  W )  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  R  /\  -.  S  .<_  ( P 
.\/  R ) ) )  /\  D  =/= 
S )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z
) ) )
206, 16, 19syl6an 545 . 2  |-  ( ph  ->  ( C  =  S  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) ) )
2120imp 429 1  |-  ( (
ph  /\  C  =  S )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z
) ) )
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   E.wrex 2815   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   lecple 14558   joincjn 15427   meetcmee 15428   Atomscatm 34060   HLchlt 34147   LHypclh 34780
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 6574
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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-poset 15429  df-plt 15441  df-lub 15457  df-glb 15458  df-join 15459  df-meet 15460  df-p0 15522  df-p1 15523  df-lat 15529  df-clat 15591  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-llines 34294  df-lplanes 34295  df-lhyp 34784
This theorem is referenced by:  4atexlemex6  34870
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