Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemefrs32fva1 Structured version   Visualization version   Unicode version

Theorem cdlemefrs32fva1 34039
Description: Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT. (Contributed by NM, 29-Mar-2013.)
Hypotheses
Ref Expression
cdlemefrs27.b  |-  B  =  ( Base `  K
)
cdlemefrs27.l  |-  .<_  =  ( le `  K )
cdlemefrs27.j  |-  .\/  =  ( join `  K )
cdlemefrs27.m  |-  ./\  =  ( meet `  K )
cdlemefrs27.a  |-  A  =  ( Atoms `  K )
cdlemefrs27.h  |-  H  =  ( LHyp `  K
)
cdlemefrs27.eq  |-  ( s  =  R  ->  ( ph 
<->  ps ) )
cdlemefrs27.nb  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  (
s  e.  A  /\  ( -.  s  .<_  W  /\  ph ) ) )  ->  N  e.  B )
cdlemefrs27.rnb  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  [_ R  /  s ]_ N  e.  B
)
cdleme29frs.o  |-  O  =  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) ) )
cdleme29frs.f  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) )
Assertion
Ref Expression
cdlemefrs32fva1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  ( F `  R
)  =  [_ R  /  s ]_ N
)
Distinct variable groups:    z, s, A    H, s    .\/ , s    K, s    .<_ , s    P, s    Q, s    R, s    W, s    ps, s    z, A    z, B    z, H    z, K    z, 
.<_    z, N    z, P    z, Q    z, R    z, W    ps, z    B, s   
z,  .\/    ./\ , s, z    ph, z    x, z, A   
x, B    x,  .\/    x, 
.<_    x,  ./\    x, N    x, s, R    x, W    x, P    x, Q
Allowed substitution hints:    ph( x, s)    ps( x)    F( x, z, s)    H( x)    K( x)    N( s)    O( x, z, s)

Proof of Theorem cdlemefrs32fva1
StepHypRef Expression
1 simp2rl 1099 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  R  e.  A )
2 cdlemefrs27.b . . . . 5  |-  B  =  ( Base `  K
)
3 cdlemefrs27.a . . . . 5  |-  A  =  ( Atoms `  K )
42, 3atbase 32926 . . . 4  |-  ( R  e.  A  ->  R  e.  B )
51, 4syl 17 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  R  e.  B )
6 simp2l 1056 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  P  =/=  Q )
7 simp2rr 1100 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  -.  R  .<_  W )
8 cdleme29frs.o . . . 4  |-  O  =  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) ) )
9 cdleme29frs.f . . . 4  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) )
108, 9cdleme31fv1s 34030 . . 3  |-  ( ( R  e.  B  /\  ( P  =/=  Q  /\  -.  R  .<_  W ) )  ->  ( F `  R )  =  [_ R  /  x ]_ O
)
115, 6, 7, 10syl12anc 1290 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  ( F `  R
)  =  [_ R  /  x ]_ O )
12 cdlemefrs27.l . . 3  |-  .<_  =  ( le `  K )
13 cdlemefrs27.j . . 3  |-  .\/  =  ( join `  K )
14 cdlemefrs27.m . . 3  |-  ./\  =  ( meet `  K )
15 cdlemefrs27.h . . 3  |-  H  =  ( LHyp `  K
)
16 cdlemefrs27.eq . . 3  |-  ( s  =  R  ->  ( ph 
<->  ps ) )
17 cdlemefrs27.nb . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  (
s  e.  A  /\  ( -.  s  .<_  W  /\  ph ) ) )  ->  N  e.  B )
18 cdlemefrs27.rnb . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  [_ R  /  s ]_ N  e.  B
)
192, 12, 13, 14, 3, 15, 16, 17, 18, 8cdlemefrs32fva 34038 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  [_ R  /  x ]_ O  =  [_ R  /  s ]_ N
)
2011, 19eqtrd 2505 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  ( F `  R
)  =  [_ R  /  s ]_ N
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   [_csb 3349   ifcif 3872   class class class wbr 4395    |-> cmpt 4454   ` cfv 5589   iota_crio 6269  (class class class)co 6308   Basecbs 15199   lecple 15275   joincjn 16267   meetcmee 16268   Atomscatm 32900   HLchlt 32987   LHypclh 33620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-preset 16251  df-poset 16269  df-plt 16282  df-lub 16298  df-glb 16299  df-join 16300  df-meet 16301  df-p0 16363  df-p1 16364  df-lat 16370  df-clat 16432  df-oposet 32813  df-ol 32815  df-oml 32816  df-covers 32903  df-ats 32904  df-atl 32935  df-cvlat 32959  df-hlat 32988  df-lhyp 33624
This theorem is referenced by:  cdlemefr32fva1  34048  cdlemefs32fva1  34061
  Copyright terms: Public domain W3C validator