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Theorem cdleme17d2 29485
Description: Part of proof of Lemma E in [Crawley] p. 114, first part of 4th paragraph.  F,  G represent f(s), fs(p) respectively. We show, in their notation, fs(p)=q. TODO FIX COMMENT (Contributed by NM, 5-Apr-2013.)
Hypotheses
Ref Expression
cdlemef46.b  |-  B  =  ( Base `  K
)
cdlemef46.l  |-  .<_  =  ( le `  K )
cdlemef46.j  |-  .\/  =  ( join `  K )
cdlemef46.m  |-  ./\  =  ( meet `  K )
cdlemef46.a  |-  A  =  ( Atoms `  K )
cdlemef46.h  |-  H  =  ( LHyp `  K
)
cdlemef46.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdlemef46.d  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
cdlemefs46.e  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
cdlemef46.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 ) )
Assertion
Ref Expression
cdleme17d2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  ( F `  P )  =  Q )
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    Q, s, t, x, y, z    U, s, t, x, y, z    W, s, t, x, y, z    S, s, t, x, y, z
Allowed substitution hints:    D( t)    E( t, s)    F( x, y, z, t, s)

Proof of Theorem cdleme17d2
StepHypRef Expression
1 simp1 960 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  (
( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
2 simp2l 986 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  P  =/=  Q )
3 simp12 991 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
4 simp2r 987 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  ( S  e.  A  /\  -.  S  .<_  W ) )
5 simp11l 1071 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  K  e.  HL )
6 simp12l 1073 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  P  e.  A )
7 simp13l 1075 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  Q  e.  A )
8 cdlemef46.l . . . . 5  |-  .<_  =  ( le `  K )
9 cdlemef46.j . . . . 5  |-  .\/  =  ( join `  K )
10 cdlemef46.a . . . . 5  |-  A  =  ( Atoms `  K )
118, 9, 10hlatlej1 28365 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  P  .<_  ( P  .\/  Q ) )
125, 6, 7, 11syl3anc 1187 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  P  .<_  ( P  .\/  Q
) )
13 simp3 962 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  -.  S  .<_  ( P  .\/  Q ) )
14 cdlemef46.b . . . 4  |-  B  =  ( Base `  K
)
15 cdlemef46.m . . . 4  |-  ./\  =  ( meet `  K )
16 cdlemef46.h . . . 4  |-  H  =  ( LHyp `  K
)
17 cdlemef46.u . . . 4  |-  U  =  ( ( P  .\/  Q )  ./\  W )
18 cdlemef46.d . . . 4  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
19 cdlemef46.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 ) )
20 cdlemefs46.e . . . 4  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
2114, 8, 9, 15, 10, 16, 17, 18, 19, 20cdlemefs45 29419 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( F `  P )  =  [_ P  /  s ]_ [_ S  /  t ]_ E
)
221, 2, 3, 4, 12, 13, 21syl132anc 1205 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  ( F `  P )  =  [_ P  /  s ]_ [_ S  /  t ]_ E )
23 simp2rl 1029 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  S  e.  A )
24 eqid 2253 . . . 4  |-  ( ( S  .\/  U ) 
./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )  =  ( ( S  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
25 eqid 2253 . . . 4  |-  ( ( P  .\/  Q ) 
./\  ( ( ( S  .\/  U ) 
./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )  .\/  (
( P  .\/  S
)  ./\  W )
) )  =  ( ( P  .\/  Q
)  ./\  ( (
( S  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )  .\/  (
( P  .\/  S
)  ./\  W )
) )
2618, 20, 24, 25cdleme31sde 29375 . . 3  |-  ( ( P  e.  A  /\  S  e.  A )  ->  [_ P  /  s ]_ [_ S  /  t ]_ E  =  (
( P  .\/  Q
)  ./\  ( (
( S  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )  .\/  (
( P  .\/  S
)  ./\  W )
) ) )
276, 23, 26syl2anc 645 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  [_ P  /  s ]_ [_ S  /  t ]_ E  =  ( ( P 
.\/  Q )  ./\  ( ( ( S 
.\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S ) 
./\  W ) ) )  .\/  ( ( P  .\/  S ) 
./\  W ) ) ) )
28 simp11 990 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
298, 9, 15, 10, 16, 17, 24, 25cdleme17d1 29279 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  (
( P  .\/  Q
)  ./\  ( (
( S  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )  .\/  (
( P  .\/  S
)  ./\  W )
) )  =  Q )
3028, 3, 7, 4, 13, 29syl131anc 1200 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  (
( P  .\/  Q
)  ./\  ( (
( S  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )  .\/  (
( P  .\/  S
)  ./\  W )
) )  =  Q )
3122, 27, 303eqtrd 2289 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  ( F `  P )  =  Q )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2412   A.wral 2509   [_csb 3009   ifcif 3470   class class class wbr 3920    e. cmpt 3974   ` cfv 4592  (class class class)co 5710   iota_crio 6181   Basecbs 13022   lecple 13089   joincjn 13922   meetcmee 13923   Atomscatm 28254   HLchlt 28341   LHypclh 28974
This theorem is referenced by:  cdleme17d3  29486
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28167  df-ol 28169  df-oml 28170  df-covers 28257  df-ats 28258  df-atl 28289  df-cvlat 28313  df-hlat 28342  df-llines 28488  df-lplanes 28489  df-lvols 28490  df-lines 28491  df-psubsp 28493  df-pmap 28494  df-padd 28786  df-lhyp 28978
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