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Theorem cdlemefs29cpre1N 30659
Description: TODO FIX COMMENT (Contributed by NM, 26-Mar-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlemefs32.b  |-  B  =  ( Base `  K
)
cdlemefs32.l  |-  .<_  =  ( le `  K )
cdlemefs32.j  |-  .\/  =  ( join `  K )
cdlemefs32.m  |-  ./\  =  ( meet `  K )
cdlemefs32.a  |-  A  =  ( Atoms `  K )
cdlemefs32.h  |-  H  =  ( LHyp `  K
)
cdlemefs32.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdlemefs32.d  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
cdlemefs32.e  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
cdlemefs32.i  |-  I  =  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  E ) )
cdlemefs32.n  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  C
)
Assertion
Ref Expression
cdlemefs29cpre1N  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  R  .<_  ( P  .\/  Q ) )  ->  E! z  e.  B  A. s  e.  A  ( (
( -.  s  .<_  W  /\  s  .<_  ( P 
.\/  Q ) )  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) ) )
Distinct variable groups:    t, s,
y, z, A    B, s, t, y, z    y, D    y, E    H, s,
t, y    .\/ , s, t, y, z    K, s, t, y    .<_ , s, t, y, z    ./\ , s,
t, y, z    z, N    P, s, t, y, z    Q, s, t, y, z    R, s, t, y   
t, U, y    W, s, t, y, z    D, s    z, H    z, K    z, R
Allowed substitution hints:    C( y, z, t, s)    D( z, t)    U( z, s)    E( z, t, s)    I( y, z, t, s)    N( y, t, s)

Proof of Theorem cdlemefs29cpre1N
StepHypRef Expression
1 cdlemefs32.b . 2  |-  B  =  ( Base `  K
)
2 cdlemefs32.l . 2  |-  .<_  =  ( le `  K )
3 cdlemefs32.j . 2  |-  .\/  =  ( join `  K )
4 cdlemefs32.m . 2  |-  ./\  =  ( meet `  K )
5 cdlemefs32.a . 2  |-  A  =  ( Atoms `  K )
6 cdlemefs32.h . 2  |-  H  =  ( LHyp `  K
)
7 breq1 4105 . 2  |-  ( s  =  R  ->  (
s  .<_  ( P  .\/  Q )  <->  R  .<_  ( P 
.\/  Q ) ) )
8 simp1 955 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  (
s  e.  A  /\  ( -.  s  .<_  W  /\  s  .<_  ( P 
.\/  Q ) ) ) )  ->  (
( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
9 simp3l 983 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  (
s  e.  A  /\  ( -.  s  .<_  W  /\  s  .<_  ( P 
.\/  Q ) ) ) )  ->  s  e.  A )
10 simp3rl 1028 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  (
s  e.  A  /\  ( -.  s  .<_  W  /\  s  .<_  ( P 
.\/  Q ) ) ) )  ->  -.  s  .<_  W )
119, 10jca 518 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  (
s  e.  A  /\  ( -.  s  .<_  W  /\  s  .<_  ( P 
.\/  Q ) ) ) )  ->  (
s  e.  A  /\  -.  s  .<_  W ) )
12 simp3rr 1029 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  (
s  e.  A  /\  ( -.  s  .<_  W  /\  s  .<_  ( P 
.\/  Q ) ) ) )  ->  s  .<_  ( P  .\/  Q
) )
13 simp2 956 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  (
s  e.  A  /\  ( -.  s  .<_  W  /\  s  .<_  ( P 
.\/  Q ) ) ) )  ->  P  =/=  Q )
14 cdlemefs32.u . . . 4  |-  U  =  ( ( P  .\/  Q )  ./\  W )
15 cdlemefs32.d . . . 4  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
16 cdlemefs32.e . . . 4  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
17 cdlemefs32.i . . . 4  |-  I  =  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  E ) )
18 cdlemefs32.n . . . 4  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  C
)
191, 2, 3, 4, 5, 6, 14, 15, 16, 17, 18cdlemefs27cl 30654 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  s  .<_  ( P  .\/  Q
)  /\  P  =/=  Q ) )  ->  N  e.  B )
208, 11, 12, 13, 19syl13anc 1184 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  (
s  e.  A  /\  ( -.  s  .<_  W  /\  s  .<_  ( P 
.\/  Q ) ) ) )  ->  N  e.  B )
211, 2, 3, 4, 5, 6, 14, 15, 16, 17, 18cdlemefs32snb 30656 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  R  .<_  ( P  .\/  Q ) )  ->  [_ R  / 
s ]_ N  e.  B
)
221, 2, 3, 4, 5, 6, 7, 20, 21cdlemefrs29cpre1 30639 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  R  .<_  ( P  .\/  Q ) )  ->  E! z  e.  B  A. s  e.  A  ( (
( -.  s  .<_  W  /\  s  .<_  ( P 
.\/  Q ) )  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R 
./\  W ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710    =/= wne 2521   A.wral 2619   E!wreu 2621   ifcif 3641   class class class wbr 4102   ` cfv 5334  (class class class)co 5942   iota_crio 6381   Basecbs 13239   lecple 13306   joincjn 14171   meetcmee 14172   Atomscatm 29505   HLchlt 29592   LHypclh 30225
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-iin 3987  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-undef 6382  df-riota 6388  df-poset 14173  df-plt 14185  df-lub 14201  df-glb 14202  df-join 14203  df-meet 14204  df-p0 14238  df-p1 14239  df-lat 14245  df-clat 14307  df-oposet 29418  df-ol 29420  df-oml 29421  df-covers 29508  df-ats 29509  df-atl 29540  df-cvlat 29564  df-hlat 29593  df-llines 29739  df-lplanes 29740  df-lvols 29741  df-lines 29742  df-psubsp 29744  df-pmap 29745  df-padd 30037  df-lhyp 30229
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