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Theorem cdleme50rnlem 29884
Description: Part of proof of Lemma D in [Crawley] p. 113. TODO: fix comment TODO: can we get rid of  G stuff if we show  G  =  `' F earlier? (Contributed by NM, 9-Apr-2013.)
Hypotheses
Ref Expression
cdlemef50.b  |-  B  =  ( Base `  K
)
cdlemef50.l  |-  .<_  =  ( le `  K )
cdlemef50.j  |-  .\/  =  ( join `  K )
cdlemef50.m  |-  ./\  =  ( meet `  K )
cdlemef50.a  |-  A  =  ( Atoms `  K )
cdlemef50.h  |-  H  =  ( LHyp `  K
)
cdlemef50.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdlemef50.d  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
cdlemefs50.e  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
cdlemef50.f  |-  F  =  ( 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 ) )
cdlemef50.v  |-  V  =  ( ( Q  .\/  P )  ./\  W )
cdlemef50.n  |-  N  =  ( ( v  .\/  V )  ./\  ( P  .\/  ( ( Q  .\/  v )  ./\  W
) ) )
cdlemefs50.o  |-  O  =  ( ( Q  .\/  P )  ./\  ( N  .\/  ( ( u  .\/  v )  ./\  W
) ) )
cdlemef50.g  |-  G  =  ( a  e.  B  |->  if ( ( Q  =/=  P  /\  -.  a  .<_  W ) ,  ( iota_ c  e.  B A. u  e.  A  ( ( -.  u  .<_  W  /\  ( u 
.\/  ( a  ./\  W ) )  =  a )  ->  c  =  ( if ( u  .<_  ( Q  .\/  P ) ,  ( iota_ b  e.  B A. v  e.  A  ( ( -.  v  .<_  W  /\  -.  v  .<_  ( Q 
.\/  P ) )  ->  b  =  O ) ) ,  [_ u  /  v ]_ N
)  .\/  ( a  ./\  W ) ) ) ) ,  a ) )
Assertion
Ref Expression
cdleme50rnlem  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ran  F  =  B )
Distinct variable groups:    a, b,
c, s, t, u, v, x, y, z, 
./\    .\/ , a, b, c, s, t, u, v, x, y, z    .<_ , a, b, c, s, t, u, v, x, y, z    A, a, b, c, s, t, u, v, x, y, z    B, a, b, c, s, t, u, v, x, y, z    D, a, b, c, s, v, x, y, z    E, a, b, c, x, y, z    F, a, b, c, u, v    H, a, b, c, s, t, u, v, x, y, z    K, a, b, c, s, t, u, v, x, y, z    P, a, b, c, s, t, u, v, x, y, z    Q, a, b, c, s, t, u, v, x, y, z    U, a, b, c, s, t, v, x, y, z    W, a, b, c, s, t, u, v, x, y, z    G, s, t, x, y, z    N, a, b, c, t, u, x, y, z    O, a, b, c, x, y, z    V, a, b, c, t, u, v, x, y, z
Allowed substitution hints:    D( u, t)    U( u)    E( v, u, t, s)    F( x, y, z, t, s)    G( v, u, a, b, c)    N( v, s)    O( v, u, t, s)    V( s)

Proof of Theorem cdleme50rnlem
StepHypRef Expression
1 cdlemef50.b . . . 4  |-  B  =  ( Base `  K
)
2 cdlemef50.l . . . 4  |-  .<_  =  ( le `  K )
3 cdlemef50.j . . . 4  |-  .\/  =  ( join `  K )
4 cdlemef50.m . . . 4  |-  ./\  =  ( meet `  K )
5 cdlemef50.a . . . 4  |-  A  =  ( Atoms `  K )
6 cdlemef50.h . . . 4  |-  H  =  ( LHyp `  K
)
7 cdlemef50.u . . . 4  |-  U  =  ( ( P  .\/  Q )  ./\  W )
8 cdlemef50.d . . . 4  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
9 cdlemefs50.e . . . 4  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
10 cdlemef50.f . . . 4  |-  F  =  ( 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 ) )
111, 2, 3, 4, 5, 6, 7, 8, 9, 10cdleme50f 29882 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  F : B
--> B )
12 frn 5319 . . 3  |-  ( F : B --> B  ->  ran  F  C_  B )
1311, 12syl 17 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ran  F  C_  B )
14 cdlemef50.v . . . . . . 7  |-  V  =  ( ( Q  .\/  P )  ./\  W )
15 cdlemef50.n . . . . . . 7  |-  N  =  ( ( v  .\/  V )  ./\  ( P  .\/  ( ( Q  .\/  v )  ./\  W
) ) )
16 cdlemefs50.o . . . . . . 7  |-  O  =  ( ( Q  .\/  P )  ./\  ( N  .\/  ( ( u  .\/  v )  ./\  W
) ) )
17 cdlemef50.g . . . . . . 7  |-  G  =  ( a  e.  B  |->  if ( ( Q  =/=  P  /\  -.  a  .<_  W ) ,  ( iota_ c  e.  B A. u  e.  A  ( ( -.  u  .<_  W  /\  ( u 
.\/  ( a  ./\  W ) )  =  a )  ->  c  =  ( if ( u  .<_  ( Q  .\/  P ) ,  ( iota_ b  e.  B A. v  e.  A  ( ( -.  v  .<_  W  /\  -.  v  .<_  ( Q 
.\/  P ) )  ->  b  =  O ) ) ,  [_ u  /  v ]_ N
)  .\/  ( a  ./\  W ) ) ) ) ,  a ) )
181, 2, 3, 4, 5, 6, 14, 15, 16, 17cdlemeg46fvcl 29846 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  e  e.  B )  ->  ( G `  e
)  e.  B )
191, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15, 16, 17cdleme48fgv 29878 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  e  e.  B )  ->  ( F `  ( G `  e )
)  =  e )
20 fveq2 5444 . . . . . . . 8  |-  ( d  =  ( G `  e )  ->  ( F `  d )  =  ( F `  ( G `  e ) ) )
2120eqeq1d 2264 . . . . . . 7  |-  ( d  =  ( G `  e )  ->  (
( F `  d
)  =  e  <->  ( F `  ( G `  e
) )  =  e ) )
2221rcla4ev 2852 . . . . . 6  |-  ( ( ( G `  e
)  e.  B  /\  ( F `  ( G `
 e ) )  =  e )  ->  E. d  e.  B  ( F `  d )  =  e )
2318, 19, 22syl2anc 645 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  e  e.  B )  ->  E. d  e.  B  ( F `  d )  =  e )
2411adantr 453 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  e  e.  B )  ->  F : B --> B )
25 ffn 5313 . . . . . 6  |-  ( F : B --> B  ->  F  Fn  B )
26 fvelrnb 5490 . . . . . 6  |-  ( F  Fn  B  ->  (
e  e.  ran  F  <->  E. d  e.  B  ( F `  d )  =  e ) )
2724, 25, 263syl 20 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  e  e.  B )  ->  ( e  e.  ran  F  <->  E. d  e.  B  ( F `  d )  =  e ) )
2823, 27mpbird 225 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  e  e.  B )  ->  e  e.  ran  F
)
2928ex 425 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( e  e.  B  ->  e  e. 
ran  F ) )
3029ssrdv 3146 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  B  C_  ran  F )
3113, 30eqssd 3157 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ran  F  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2419   A.wral 2516   E.wrex 2517   [_csb 3042    C_ wss 3113   ifcif 3525   class class class wbr 3983    e. cmpt 4037   ran crn 4648    Fn wfn 4654   -->wf 4655   ` cfv 4659  (class class class)co 5778   iota_crio 6249   Basecbs 13096   lecple 13163   joincjn 14026   meetcmee 14027   Atomscatm 28604   HLchlt 28691   LHypclh 29324
This theorem is referenced by:  cdleme50rn  29885
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-iin 3868  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-undef 6250  df-riota 6258  df-poset 14028  df-plt 14040  df-lub 14056  df-glb 14057  df-join 14058  df-meet 14059  df-p0 14093  df-p1 14094  df-lat 14100  df-clat 14162  df-oposet 28517  df-ol 28519  df-oml 28520  df-covers 28607  df-ats 28608  df-atl 28639  df-cvlat 28663  df-hlat 28692  df-llines 28838  df-lplanes 28839  df-lvols 28840  df-lines 28841  df-psubsp 28843  df-pmap 28844  df-padd 29136  df-lhyp 29328
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