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Theorem cdleme27cl 29314
Description: Part of proof of Lemma E in [Crawley] p. 113. Closure of  C. (Contributed by NM, 6-Feb-2013.)
Hypotheses
Ref Expression
cdleme26.b  |-  B  =  ( Base `  K
)
cdleme26.l  |-  .<_  =  ( le `  K )
cdleme26.j  |-  .\/  =  ( join `  K )
cdleme26.m  |-  ./\  =  ( meet `  K )
cdleme26.a  |-  A  =  ( Atoms `  K )
cdleme26.h  |-  H  =  ( LHyp `  K
)
cdleme27.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme27.f  |-  F  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
cdleme27.z  |-  Z  =  ( ( z  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W
) ) )
cdleme27.n  |-  N  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( s  .\/  z )  ./\  W
) ) )
cdleme27.d  |-  D  =  ( iota_ u  e.  B A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P  .\/  Q
) )  ->  u  =  N ) )
cdleme27.c  |-  C  =  if ( s  .<_  ( P  .\/  Q ) ,  D ,  F
)
Assertion
Ref Expression
cdleme27cl  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  P  =/=  Q
) )  ->  C  e.  B )
Distinct variable groups:    u, s,
z, A    B, s, u, z    u, F    H, s, z    .\/ , s, u, z    K, s, z    .<_ , s, u, z    ./\ , s, u, z    u, N    P, s, u, z    Q, s, u, z    U, s, u, z    W, s, u, z
Allowed substitution hints:    C( z, u, s)    D( z, u, s)    F( z, s)    H( u)    K( u)    N( z, s)    Z( z, u, s)

Proof of Theorem cdleme27cl
StepHypRef Expression
1 cdleme27.c . 2  |-  C  =  if ( s  .<_  ( P  .\/  Q ) ,  D ,  F
)
2 simpl1 963 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  P  =/=  Q ) )  /\  s  .<_  ( P  .\/  Q ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
3 simpl2l 1013 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  P  =/=  Q ) )  /\  s  .<_  ( P  .\/  Q ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
4 simpl2r 1014 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  P  =/=  Q ) )  /\  s  .<_  ( P  .\/  Q ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
5 simpl3l 1015 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  P  =/=  Q ) )  /\  s  .<_  ( P  .\/  Q ) )  ->  (
s  e.  A  /\  -.  s  .<_  W ) )
6 simpl3r 1016 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  P  =/=  Q ) )  /\  s  .<_  ( P  .\/  Q ) )  ->  P  =/=  Q )
7 simpr 449 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  P  =/=  Q ) )  /\  s  .<_  ( P  .\/  Q ) )  ->  s  .<_  ( P  .\/  Q
) )
8 cdleme26.b . . . . 5  |-  B  =  ( Base `  K
)
9 cdleme26.l . . . . 5  |-  .<_  =  ( le `  K )
10 cdleme26.j . . . . 5  |-  .\/  =  ( join `  K )
11 cdleme26.m . . . . 5  |-  ./\  =  ( meet `  K )
12 cdleme26.a . . . . 5  |-  A  =  ( Atoms `  K )
13 cdleme26.h . . . . 5  |-  H  =  ( LHyp `  K
)
14 cdleme27.u . . . . 5  |-  U  =  ( ( P  .\/  Q )  ./\  W )
15 cdleme27.z . . . . 5  |-  Z  =  ( ( z  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W
) ) )
16 cdleme27.n . . . . 5  |-  N  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( s  .\/  z )  ./\  W
) ) )
17 cdleme27.d . . . . 5  |-  D  =  ( iota_ u  e.  B A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P  .\/  Q
) )  ->  u  =  N ) )
188, 9, 10, 11, 12, 13, 14, 15, 16, 17cdleme25cl 29305 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( P  =/= 
Q  /\  s  .<_  ( P  .\/  Q ) ) )  ->  D  e.  B )
192, 3, 4, 5, 6, 7, 18syl312anc 1208 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  P  =/=  Q ) )  /\  s  .<_  ( P  .\/  Q ) )  ->  D  e.  B )
20 simp1l 984 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  P  =/=  Q
) )  ->  K  e.  HL )
21 simp1r 985 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  P  =/=  Q
) )  ->  W  e.  H )
22 simp2ll 1027 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  P  =/=  Q
) )  ->  P  e.  A )
23 simp2rl 1029 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  P  =/=  Q
) )  ->  Q  e.  A )
24 simp3ll 1031 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  P  =/=  Q
) )  ->  s  e.  A )
25 cdleme27.f . . . . . 6  |-  F  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
269, 10, 11, 12, 13, 14, 25, 8cdleme1b 29174 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  s  e.  A ) )  ->  F  e.  B )
2720, 21, 22, 23, 24, 26syl23anc 1194 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  P  =/=  Q
) )  ->  F  e.  B )
2827adantr 453 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  P  =/=  Q ) )  /\  -.  s  .<_  ( P 
.\/  Q ) )  ->  F  e.  B
)
2919, 28ifclda 3497 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  P  =/=  Q
) )  ->  if ( s  .<_  ( P 
.\/  Q ) ,  D ,  F )  e.  B )
301, 29syl5eqel 2337 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  P  =/=  Q
) )  ->  C  e.  B )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2412   A.wral 2509   ifcif 3470   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   iota_crio 6181   Basecbs 13022   lecple 13089   joincjn 13922   meetcmee 13923   Atomscatm 28212   HLchlt 28299   LHypclh 28932
This theorem is referenced by:  cdleme27N  29317  cdleme28a  29318  cdleme28b  29319  cdleme29ex  29322  cdleme32fva  29385  cdleme32c  29391  cdleme32e  29393
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-3or 940  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-iin 3806  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 28125  df-ol 28127  df-oml 28128  df-covers 28215  df-ats 28216  df-atl 28247  df-cvlat 28271  df-hlat 28300  df-llines 28446  df-lplanes 28447  df-lvols 28448  df-lines 28449  df-psubsp 28451  df-pmap 28452  df-padd 28744  df-lhyp 28936
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