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Theorem 4atexlem7 30886
Description: Whenever there are at least 4 atoms under  P  .\/  Q (specifically,  P,  Q,  r, and  ( P  .\/  Q
)  ./\  W), there are also at least 4 atoms under  P  .\/  S. This proves the statement in Lemma E of [Crawley] p. 114, last line, "...p  \/ q/0 and hence p  \/ s/0 contains at least four atoms..." Note that by cvlsupr2 30155, our  ( P  .\/  r )  =  ( Q  .\/  r ) is a shorter way to express  r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ). With a longer proof, the condition  -.  S  .<_  ( P  .\/  Q ) could be eliminated (see 4atex 30887), although for some purposes this more restricted lemma may be adequate. (Contributed by NM, 25-Nov-2012.)
Hypotheses
Ref Expression
4that.l  |-  .<_  =  ( le `  K )
4that.j  |-  .\/  =  ( join `  K )
4that.a  |-  A  =  ( Atoms `  K )
4that.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
4atexlem7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )
Distinct variable groups:    z, r, A    H, r    .\/ , r,
z    K, r, z    .<_ , r, z    P, r, z    Q, r, z    S, r, z    W, r, z
Allowed substitution hint:    H( z)

Proof of Theorem 4atexlem7
StepHypRef Expression
1 simp11l 1066 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A ) )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) )  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp1r1 1051 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A ) )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
323ad2ant1 976 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A ) )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) )  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
4 simp1r2 1052 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A ) )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
543ad2ant1 976 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A ) )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) )  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
6 simp2 956 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A ) )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) )  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )  ->  r  e.  A
)
7 simp3l 983 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A ) )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) )  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )  ->  -.  r  .<_  W )
86, 7jca 518 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A ) )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) )  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )  ->  ( r  e.  A  /\  -.  r  .<_  W ) )
9 simp1r3 1053 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A ) )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) )  ->  S  e.  A )
1093ad2ant1 976 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A ) )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) )  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )  ->  S  e.  A
)
11 simp3r 984 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A ) )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) )  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )  ->  ( P  .\/  r )  =  ( Q  .\/  r ) )
12 simp12 986 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A ) )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) )  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )  ->  P  =/=  Q
)
13 simp13 987 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A ) )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) )  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )  ->  -.  S  .<_  ( P  .\/  Q ) )
14 4that.l . . . . . . 7  |-  .<_  =  ( le `  K )
15 4that.j . . . . . . 7  |-  .\/  =  ( join `  K )
16 eqid 2296 . . . . . . 7  |-  ( meet `  K )  =  (
meet `  K )
17 4that.a . . . . . . 7  |-  A  =  ( Atoms `  K )
18 4that.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
1914, 15, 16, 17, 184atexlemex6 30885 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( r  e.  A  /\  -.  r  .<_  W )  /\  S  e.  A )  /\  (
( P  .\/  r
)  =  ( Q 
.\/  r )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )
201, 3, 5, 8, 10, 11, 12, 13, 19syl323anc 1212 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A ) )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) )  /\  r  e.  A  /\  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )
2120rexlimdv3a 2682 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A ) )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) )  ->  ( E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z
) ) ) )
22213exp 1150 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A
) )  ->  ( P  =/=  Q  ->  ( -.  S  .<_  ( P 
.\/  Q )  -> 
( E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z
) ) ) ) ) )
23223impd 1165 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A
) )  ->  (
( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) ) )
24233impia 1148 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A
)  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ 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   lecple 13231   joincjn 14094   meetcmee 14095   Atomscatm 30075   HLchlt 30162   LHypclh 30795
This theorem is referenced by:  4atex  30887  cdleme21i  31146
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
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