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Theorem 4atexlem7 29168
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 28437, 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 29169), 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 1071 . . . . . 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 1056 . . . . . . 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 981 . . . . . 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 1057 . . . . . . 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 981 . . . . . 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 961 . . . . . . 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 988 . . . . . . 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 520 . . . . . 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 1058 . . . . . . 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 981 . . . . . 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 989 . . . . . 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 991 . . . . . 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 992 . . . . . 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 2253 . . . . . . 7  |-  ( meet `  K )  =  (
meet `  K )
17 4that.a . . . . . . 7  |-  A  =  ( Atoms `  K )
18 4that.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
1914, 15, 16, 17, 184atexlemex6 29167 . . . . . 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 1217 . . . . 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 2631 . . . 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 1155 . . 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 1170 . 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 1153 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 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2412   E.wrex 2510   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   lecple 13089   joincjn 13922   meetcmee 13923   Atomscatm 28357   HLchlt 28444   LHypclh 29077
This theorem is referenced by:  4atex  29169  cdleme21i  29428
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28270  df-ol 28272  df-oml 28273  df-covers 28360  df-ats 28361  df-atl 28392  df-cvlat 28416  df-hlat 28445  df-llines 28591  df-lplanes 28592  df-lhyp 29081
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