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Theorem cdlemg2ce 30757
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 958 . . 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 30756 . . . 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 978 . . 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 202 . 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 987 . . . . . 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 983 . . . . . . 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 993 . . . . . . 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 519 . . . . . 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 984 . . . . . . 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 994 . . . . . . 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 519 . . . . . 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 989 . . . . . 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 1187 . . . . 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 995 . . . . . 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 16 . . . . 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 224 . . . 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 1152 . . 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 2772 . 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 15 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2543   A.wral 2642   E.wrex 2643   [_csb 3187   ifcif 3675   class class class wbr 4146    e. cmpt 4200   ` cfv 5387  (class class class)co 6013   iota_crio 6471   Basecbs 13389   lecple 13456   joincjn 14321   meetcmee 14322   Atomscatm 29429   HLchlt 29516   LHypclh 30149   LTrncltrn 30266
This theorem is referenced by:  cdlemg2jlemOLDN  30758  cdlemg2fvlem  30759  cdlemg2klem  30760
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 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-iin 4031  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-undef 6472  df-riota 6478  df-map 6949  df-poset 14323  df-plt 14335  df-lub 14351  df-glb 14352  df-join 14353  df-meet 14354  df-p0 14388  df-p1 14389  df-lat 14395  df-clat 14457  df-oposet 29342  df-ol 29344  df-oml 29345  df-covers 29432  df-ats 29433  df-atl 29464  df-cvlat 29488  df-hlat 29517  df-llines 29663  df-lplanes 29664  df-lvols 29665  df-lines 29666  df-psubsp 29668  df-pmap 29669  df-padd 29961  df-lhyp 30153  df-laut 30154  df-ldil 30269  df-ltrn 30270  df-trl 30324
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