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Theorem cdlemg2fvlem 31329
Description: Lemma for cdlemg2fv 31334. (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 5720 . . . 4  |-  ( F  =  G  ->  ( F `  X )  =  ( G `  X ) )
19 fveq1 5720 . . . . 5  |-  ( F  =  G  ->  ( F `  P )  =  ( G `  P ) )
2019oveq1d 6089 . . . 4  |-  ( F  =  G  ->  (
( F `  P
)  .\/  ( X  ./\ 
W ) )  =  ( ( G `  P )  .\/  ( X  ./\  W ) ) )
2118, 20eqeq12d 2450 . . 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 31235 . . . 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 31327 . 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 1652    e. wcel 1725    =/= wne 2599   A.wral 2698   [_csb 3244   ifcif 3732   class class class wbr 4205    e. cmpt 4259   ` cfv 5447  (class class class)co 6074   iota_crio 6535   Basecbs 13462   lecple 13529   joincjn 14394   meetcmee 14395   Atomscatm 29999   HLchlt 30086   LHypclh 30719   LTrncltrn 30836
This theorem is referenced by:  cdlemg2fv  31334
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-iun 4088  df-iin 4089  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-undef 6536  df-riota 6542  df-map 7013  df-poset 14396  df-plt 14408  df-lub 14424  df-glb 14425  df-join 14426  df-meet 14427  df-p0 14461  df-p1 14462  df-lat 14468  df-clat 14530  df-oposet 29912  df-ol 29914  df-oml 29915  df-covers 30002  df-ats 30003  df-atl 30034  df-cvlat 30058  df-hlat 30087  df-llines 30233  df-lplanes 30234  df-lvols 30235  df-lines 30236  df-psubsp 30238  df-pmap 30239  df-padd 30531  df-lhyp 30723  df-laut 30724  df-ldil 30839  df-ltrn 30840  df-trl 30894
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