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Theorem cdlemg2ce 31403
Description: Utility theorem to eliminate p,q when converting theorems with explicit f. TODO: fix comment. (Contributed by NM, 22-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 ) )
cdlemg2ce.p  |-  ( F  =  G  ->  ( ps 
<->  ch ) )
cdlemg2ce.c  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  A  /\  -.  p  .<_  W )  /\  ( q  e.  A  /\  -.  q  .<_  W ) )  /\  ph )  ->  ch )
Assertion
Ref Expression
cdlemg2ce  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ph )  ->  ps )
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    U, s, t, x, y, z    W, 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    ph, p, q    ps, p, q
Allowed substitution hints:    ph( x, y, z, t, s)    ps( x, y, z, t, s)    ch( x, y, z, t, s, q, p)    B( q, p)    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)    .\/ ( q, p)    ./\ ( q, p)

Proof of Theorem cdlemg2ce
StepHypRef Expression
1 simp2 956 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ph )  ->  F  e.  T )
2 cdlemg2.b . . . . 5  |-  B  =  ( Base `  K
)
3 cdlemg2.l . . . . 5  |-  .<_  =  ( le `  K )
4 cdlemg2.j . . . . 5  |-  .\/  =  ( join `  K )
5 cdlemg2.m . . . . 5  |-  ./\  =  ( meet `  K )
6 cdlemg2.a . . . . 5  |-  A  =  ( Atoms `  K )
7 cdlemg2.h . . . . 5  |-  H  =  ( LHyp `  K
)
8 cdlemg2.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
9 cdlemg2ex.u . . . . 5  |-  U  =  ( ( p  .\/  q )  ./\  W
)
10 cdlemg2ex.d . . . . 5  |-  D  =  ( ( t  .\/  U )  ./\  ( q  .\/  ( ( p  .\/  t )  ./\  W
) ) )
11 cdlemg2ex.e . . . . 5  |-  E  =  ( ( p  .\/  q )  ./\  ( D  .\/  ( ( s 
.\/  t )  ./\  W ) ) )
12 cdlemg2ex.g . . . . 5  |-  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 ) )
132, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12cdlemg2cex 31402 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( F  e.  T  <->  E. p  e.  A  E. q  e.  A  ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  F  =  G )
) )
14133ad2ant1 976 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ph )  ->  ( F  e.  T  <->  E. p  e.  A  E. q  e.  A  ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  F  =  G ) ) )
151, 14mpbid 201 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ph )  ->  E. p  e.  A  E. q  e.  A  ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  F  =  G ) )
16 simp11 985 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ph )  /\  ( p  e.  A  /\  q  e.  A
)  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  F  =  G ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
17 simp2l 981 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ph )  /\  ( p  e.  A  /\  q  e.  A
)  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  F  =  G ) )  ->  p  e.  A )
18 simp31 991 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ph )  /\  ( p  e.  A  /\  q  e.  A
)  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  F  =  G ) )  ->  -.  p  .<_  W )
1917, 18jca 518 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ph )  /\  ( p  e.  A  /\  q  e.  A
)  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  F  =  G ) )  -> 
( p  e.  A  /\  -.  p  .<_  W ) )
20 simp2r 982 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ph )  /\  ( p  e.  A  /\  q  e.  A
)  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  F  =  G ) )  -> 
q  e.  A )
21 simp32 992 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ph )  /\  ( p  e.  A  /\  q  e.  A
)  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  F  =  G ) )  ->  -.  q  .<_  W )
2220, 21jca 518 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ph )  /\  ( p  e.  A  /\  q  e.  A
)  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  F  =  G ) )  -> 
( q  e.  A  /\  -.  q  .<_  W ) )
23 simp13 987 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ph )  /\  ( p  e.  A  /\  q  e.  A
)  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  F  =  G ) )  ->  ph )
24 cdlemg2ce.c . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
p  e.  A  /\  -.  p  .<_  W )  /\  ( q  e.  A  /\  -.  q  .<_  W ) )  /\  ph )  ->  ch )
2516, 19, 22, 23, 24syl31anc 1185 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ph )  /\  ( p  e.  A  /\  q  e.  A
)  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  F  =  G ) )  ->  ch )
26 simp33 993 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ph )  /\  ( p  e.  A  /\  q  e.  A
)  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  F  =  G ) )  ->  F  =  G )
27 cdlemg2ce.p . . . . . 6  |-  ( F  =  G  ->  ( ps 
<->  ch ) )
2826, 27syl 15 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ph )  /\  ( p  e.  A  /\  q  e.  A
)  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  F  =  G ) )  -> 
( ps  <->  ch )
)
2925, 28mpbird 223 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ph )  /\  ( p  e.  A  /\  q  e.  A
)  /\  ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  F  =  G ) )  ->  ps )
30293exp 1150 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ph )  ->  (
( p  e.  A  /\  q  e.  A
)  ->  ( ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  F  =  G )  ->  ps ) ) )
3130rexlimdvv 2686 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ph )  ->  ( E. p  e.  A  E. q  e.  A  ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  F  =  G )  ->  ps )
)
3215, 31mpd 14 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ph )  ->  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   [_csb 3094   ifcif 3578   class class class wbr 4039    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   iota_crio 6313   Basecbs 13164   lecple 13231   joincjn 14094   meetcmee 14095   Atomscatm 30075   HLchlt 30162   LHypclh 30795   LTrncltrn 30912
This theorem is referenced by:  cdlemg2jlemOLDN  31404  cdlemg2fvlem  31405  cdlemg2klem  31406
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-3or 935  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-rmo 2564  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-iin 3924  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-map 6790  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-lvols 30311  df-lines 30312  df-psubsp 30314  df-pmap 30315  df-padd 30607  df-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916  df-trl 30970
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