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Theorem cdlemg2fvlem 30708
Description: Lemma for cdlemg2fv 30713. (Contributed by NM, 23-Apr-2013.)
Hypotheses
Ref Expression
cdlemg2.b  |-  B  =  ( Base `  K
)
cdlemg2.l  |-  .<_  =  ( le `  K )
cdlemg2.j  |-  .\/  =  ( join `  K )
cdlemg2.m  |-  ./\  =  ( meet `  K )
cdlemg2.a  |-  A  =  ( Atoms `  K )
cdlemg2.h  |-  H  =  ( LHyp `  K
)
cdlemg2.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemg2ex.u  |-  U  =  ( ( p  .\/  q )  ./\  W
)
cdlemg2ex.d  |-  D  =  ( ( t  .\/  U )  ./\  ( q  .\/  ( ( p  .\/  t )  ./\  W
) ) )
cdlemg2ex.e  |-  E  =  ( ( p  .\/  q )  ./\  ( D  .\/  ( ( s 
.\/  t )  ./\  W ) ) )
cdlemg2ex.g  |-  G  =  ( x  e.  B  |->  if ( ( p  =/=  q  /\  -.  x  .<_  W ) ,  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( if ( s  .<_  ( p  .\/  q ) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( p 
.\/  q ) )  ->  y  =  E ) ) ,  [_ s  /  t ]_ D
)  .\/  ( x  ./\ 
W ) ) ) ) ,  x ) )
Assertion
Ref Expression
cdlemg2fvlem  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( F  e.  T  /\  ( P  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( F `  X )  =  ( ( F `
 P )  .\/  ( X  ./\  W ) ) )
Distinct variable groups:    t, s, x, y, z, A    B, s, t, x, y, z    D, s, x, y, z   
x, E, y, z    H, s, t, x, y, z    .\/ , s, t, x, y, z    K, s, t, x, y, z    .<_ , s, t, x, y, z    ./\ , s, t, x, y, z    P, s, t, x, y, z    U, s, t, x, y, z    W, s, t, x, y, z    X, s, t, x, y, z   
q, p, A    F, p, q    H, p, q    K, p, q    .<_ , p, q    T, p, q    W, p, q, s, t, x, y, z    .\/ , p, q    P, p, q    B, p, q    ./\ , p, q    X, p, q
Allowed substitution hints:    D( t, q, p)    T( x, y, z, t, s)    U( q, p)    E( t, s, q, p)    F( x, y, z, t, s)    G( x, y, z, t, s, q, p)

Proof of Theorem cdlemg2fvlem
StepHypRef Expression
1 simp1 957 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( F  e.  T  /\  ( P  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp3l 985 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( F  e.  T  /\  ( P  .\/  ( X  ./\  W ) )  =  X ) )  ->  F  e.  T )
3 simp2r 984 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( F  e.  T  /\  ( P  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( X  e.  B  /\  -.  X  .<_  W ) )
4 simp2l 983 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( F  e.  T  /\  ( P  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
5 simp3r 986 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( F  e.  T  /\  ( P  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( P  .\/  ( X  ./\  W ) )  =  X )
64, 5jca 519 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( F  e.  T  /\  ( P  .\/  ( X  ./\  W ) )  =  X ) )  ->  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( P  .\/  ( X  ./\  W ) )  =  X ) )
7 cdlemg2.b . . 3  |-  B  =  ( Base `  K
)
8 cdlemg2.l . . 3  |-  .<_  =  ( le `  K )
9 cdlemg2.j . . 3  |-  .\/  =  ( join `  K )
10 cdlemg2.m . . 3  |-  ./\  =  ( meet `  K )
11 cdlemg2.a . . 3  |-  A  =  ( Atoms `  K )
12 cdlemg2.h . . 3  |-  H  =  ( LHyp `  K
)
13 cdlemg2.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
14 cdlemg2ex.u . . 3  |-  U  =  ( ( p  .\/  q )  ./\  W
)
15 cdlemg2ex.d . . 3  |-  D  =  ( ( t  .\/  U )  ./\  ( q  .\/  ( ( p  .\/  t )  ./\  W
) ) )
16 cdlemg2ex.e . . 3  |-  E  =  ( ( p  .\/  q )  ./\  ( D  .\/  ( ( s 
.\/  t )  ./\  W ) ) )
17 cdlemg2ex.g . . 3  |-  G  =  ( x  e.  B  |->  if ( ( p  =/=  q  /\  -.  x  .<_  W ) ,  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( if ( s  .<_  ( p  .\/  q ) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( p 
.\/  q ) )  ->  y  =  E ) ) ,  [_ s  /  t ]_ D
)  .\/  ( x  ./\ 
W ) ) ) ) ,  x ) )
18 fveq1 5667 . . . 4  |-  ( F  =  G  ->  ( F `  X )  =  ( G `  X ) )
19 fveq1 5667 . . . . 5  |-  ( F  =  G  ->  ( F `  P )  =  ( G `  P ) )
2019oveq1d 6035 . . . 4  |-  ( F  =  G  ->  (
( F `  P
)  .\/  ( X  ./\ 
W ) )  =  ( ( G `  P )  .\/  ( X  ./\  W ) ) )
2118, 20eqeq12d 2401 . . 3  |-  ( F  =  G  ->  (
( F `  X
)  =  ( ( F `  P ) 
.\/  ( X  ./\  W ) )  <->  ( G `  X )  =  ( ( G `  P
)  .\/  ( X  ./\ 
W ) ) ) )
227, 8, 9, 10, 11, 12, 14, 15, 16, 17cdleme48fvg 30614 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  A  /\  -.  p  .<_  W )  /\  ( q  e.  A  /\  -.  q  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( P  .\/  ( X 
./\  W ) )  =  X ) )  ->  ( G `  X )  =  ( ( G `  P
)  .\/  ( X  ./\ 
W ) ) )
23223expb 1154 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  A  /\  -.  p  .<_  W )  /\  ( q  e.  A  /\  -.  q  .<_  W ) )  /\  ( ( X  e.  B  /\  -.  X  .<_  W )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( P  .\/  ( X  ./\  W ) )  =  X ) ) )  ->  ( G `  X )  =  ( ( G `
 P )  .\/  ( X  ./\  W ) ) )
247, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 21, 23cdlemg2ce 30706 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( X  e.  B  /\  -.  X  .<_  W )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( P  .\/  ( X  ./\  W ) )  =  X ) ) )  ->  ( F `  X )  =  ( ( F `
 P )  .\/  ( X  ./\  W ) ) )
251, 2, 3, 6, 24syl112anc 1188 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( F  e.  T  /\  ( P  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( F `  X )  =  ( ( F `
 P )  .\/  ( X  ./\  W ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   A.wral 2649   [_csb 3194   ifcif 3682   class class class wbr 4153    e. cmpt 4207   ` cfv 5394  (class class class)co 6020   iota_crio 6478   Basecbs 13396   lecple 13463   joincjn 14328   meetcmee 14329   Atomscatm 29378   HLchlt 29465   LHypclh 30098   LTrncltrn 30215
This theorem is referenced by:  cdlemg2fv  30713
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-undef 6479  df-riota 6485  df-map 6956  df-poset 14330  df-plt 14342  df-lub 14358  df-glb 14359  df-join 14360  df-meet 14361  df-p0 14395  df-p1 14396  df-lat 14402  df-clat 14464  df-oposet 29291  df-ol 29293  df-oml 29294  df-covers 29381  df-ats 29382  df-atl 29413  df-cvlat 29437  df-hlat 29466  df-llines 29612  df-lplanes 29613  df-lvols 29614  df-lines 29615  df-psubsp 29617  df-pmap 29618  df-padd 29910  df-lhyp 30102  df-laut 30103  df-ldil 30218  df-ltrn 30219  df-trl 30273
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