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Theorem cdleme50rnlem 31180
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
Dummy variables  e 
d are mutually distinct and distinct from all other variables.
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 31178 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  F : B
--> B )
12 frn 5588 . . 3  |-  ( F : B --> B  ->  ran  F  C_  B )
1311, 12syl 16 . 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 31142 . . . . . 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 31174 . . . . . 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 5719 . . . . . . . 8  |-  ( d  =  ( G `  e )  ->  ( F `  d )  =  ( F `  ( G `  e ) ) )
2120eqeq1d 2443 . . . . . . 7  |-  ( d  =  ( G `  e )  ->  (
( F `  d
)  =  e  <->  ( F `  ( G `  e
) )  =  e ) )
2221rspcev 3044 . . . . . 6  |-  ( ( ( G `  e
)  e.  B  /\  ( F `  ( G `
 e ) )  =  e )  ->  E. d  e.  B  ( F `  d )  =  e )
2318, 19, 22syl2anc 643 . . . . 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 452 . . . . . 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 5582 . . . . . 6  |-  ( F : B --> B  ->  F  Fn  B )
26 fvelrnb 5765 . . . . . 6  |-  ( F  Fn  B  ->  (
e  e.  ran  F  <->  E. d  e.  B  ( F `  d )  =  e ) )
2724, 25, 263syl 19 . . . . 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 224 . . . 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 424 . . 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 3346 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  B  C_  ran  F )
3113, 30eqssd 3357 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 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   [_csb 3243    C_ wss 3312   ifcif 3731   class class class wbr 4204    e. cmpt 4258   ran crn 4870    Fn wfn 5440   -->wf 5441   ` cfv 5445  (class class class)co 6072   iota_crio 6533   Basecbs 13457   lecple 13524   joincjn 14389   meetcmee 14390   Atomscatm 29900   HLchlt 29987   LHypclh 30620
This theorem is referenced by:  cdleme50rn  31181
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-undef 6534  df-riota 6540  df-poset 14391  df-plt 14403  df-lub 14419  df-glb 14420  df-join 14421  df-meet 14422  df-p0 14456  df-p1 14457  df-lat 14463  df-clat 14525  df-oposet 29813  df-ol 29815  df-oml 29816  df-covers 29903  df-ats 29904  df-atl 29935  df-cvlat 29959  df-hlat 29988  df-llines 30134  df-lplanes 30135  df-lvols 30136  df-lines 30137  df-psubsp 30139  df-pmap 30140  df-padd 30432  df-lhyp 30624
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