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Theorem cdlemeg46fjgN 30769
Description: NOT NEEDED? TODO FIX COMMENT TODO eliminate eqcomd 2371? p. 116 2nd line. (Contributed by NM, 2-Apr-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlemef46g.b  |-  B  =  ( Base `  K
)
cdlemef46g.l  |-  .<_  =  ( le `  K )
cdlemef46g.j  |-  .\/  =  ( join `  K )
cdlemef46g.m  |-  ./\  =  ( meet `  K )
cdlemef46g.a  |-  A  =  ( Atoms `  K )
cdlemef46g.h  |-  H  =  ( LHyp `  K
)
cdlemef46g.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdlemef46g.d  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
cdlemefs46g.e  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
cdlemef46g.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 ) )
cdlemef46.v  |-  V  =  ( ( Q  .\/  P )  ./\  W )
cdlemef46.n  |-  N  =  ( ( v  .\/  V )  ./\  ( P  .\/  ( ( Q  .\/  v )  ./\  W
) ) )
cdlemefs46.o  |-  O  =  ( ( Q  .\/  P )  ./\  ( N  .\/  ( ( u  .\/  v )  ./\  W
) ) )
cdlemef46.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
cdlemeg46fjgN  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( ( F `  R )  .\/  ( F `  ( G `  S )
) )  =  ( F `  ( R 
.\/  ( G `  S ) ) ) )
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    R, s, t, x, y, z    U, s, t, x, y, z    W, s, t, x, y, z    S, s, t, x, y, z    a, b, c, u, v, A    B, a, b, c, u, v    v, D    G, s, t, x, y, z    H, a, b, c, u, v    .\/ , a, b, c, u, v    K, a, b, c, u, v    .<_ , a, b, c, u, v    ./\ , a, b, c, u, v    N, a, b, c    O, a, b, c    P, a, b, c, u, v    Q, a, b, c, u, v    R, a, b, c, u, v    S, a, b, c, u, v    V, a, b, c    W, a, b, c, u, v   
x, u, y, z, N    x, O, y, z    v, t    u, V    x, v, y, z, V    D, a, b, c    E, a, b, c    F, a, b, c, u, v   
t, N    U, a,
b, c, v    t, V    s, a, t, b, c
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 cdlemeg46fjgN
StepHypRef Expression
1 simp1 956 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
2 simp21 989 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  P  =/=  Q )
3 simp22 990 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( R  e.  A  /\  -.  R  .<_  W ) )
4 simp23 991 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( S  e.  A  /\  -.  S  .<_  W ) )
5 cdlemef46g.b . . . . 5  |-  B  =  ( Base `  K
)
6 cdlemef46g.l . . . . 5  |-  .<_  =  ( le `  K )
7 cdlemef46g.j . . . . 5  |-  .\/  =  ( join `  K )
8 cdlemef46g.m . . . . 5  |-  ./\  =  ( meet `  K )
9 cdlemef46g.a . . . . 5  |-  A  =  ( Atoms `  K )
10 cdlemef46g.h . . . . 5  |-  H  =  ( LHyp `  K
)
11 cdlemef46g.u . . . . 5  |-  U  =  ( ( P  .\/  Q )  ./\  W )
12 cdlemef46g.d . . . . 5  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
13 cdlemefs46g.e . . . . 5  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
14 cdlemef46g.f . . . . 5  |-  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 ) )
15 cdlemef46.v . . . . 5  |-  V  =  ( ( Q  .\/  P )  ./\  W )
16 cdlemef46.n . . . . 5  |-  N  =  ( ( v  .\/  V )  ./\  ( P  .\/  ( ( Q  .\/  v )  ./\  W
) ) )
17 cdlemefs46.o . . . . 5  |-  O  =  ( ( Q  .\/  P )  ./\  ( N  .\/  ( ( u  .\/  v )  ./\  W
) ) )
18 cdlemef46.g . . . . 5  |-  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 ) )
195, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18cdlemeg46fvaw 30764 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  P  =/=  Q
)  ->  ( ( G `  S )  e.  A  /\  -.  ( G `  S )  .<_  W ) )
201, 4, 2, 19syl3anc 1183 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( ( G `  S )  e.  A  /\  -.  ( G `  S )  .<_  W ) )
21 vex 2876 . . . . 5  |-  s  e. 
_V
22 eqid 2366 . . . . . 6  |-  ( ( s  .\/  U ) 
./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
2312, 22cdleme31sc 30632 . . . . 5  |-  ( s  e.  _V  ->  [_ s  /  t ]_ D  =  ( ( s 
.\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s ) 
./\  W ) ) ) )
2421, 23ax-mp 8 . . . 4  |-  [_ s  /  t ]_ D  =  ( ( s 
.\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s ) 
./\  W ) ) )
25 eqid 2366 . . . 4  |-  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P 
.\/  Q ) )  ->  y  =  E ) )  =  (
iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  E ) )
26 eqid 2366 . . . 4  |-  if ( s  .<_  ( P  .\/  Q ) ,  (
iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  E ) ) , 
[_ s  /  t ]_ D )  =  if ( s  .<_  ( P 
.\/  Q ) ,  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  E ) ) , 
[_ s  /  t ]_ D )
27 eqid 2366 . . . 4  |-  ( 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 ) ) ) )  =  ( 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 ) ) ) )
285, 6, 7, 8, 9, 10, 11, 24, 12, 13, 25, 26, 27, 14cdleme42mN 30735 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( ( G `
 S )  e.  A  /\  -.  ( G `  S )  .<_  W ) ) )  ->  ( F `  ( R  .\/  ( G `
 S ) ) )  =  ( ( F `  R ) 
.\/  ( F `  ( G `  S ) ) ) )
291, 2, 3, 20, 28syl13anc 1185 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( F `  ( R  .\/  ( G `  S )
) )  =  ( ( F `  R
)  .\/  ( F `  ( G `  S
) ) ) )
3029eqcomd 2371 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( ( F `  R )  .\/  ( F `  ( G `  S )
) )  =  ( F `  ( R 
.\/  ( G `  S ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715    =/= wne 2529   A.wral 2628   _Vcvv 2873   [_csb 3167   ifcif 3654   class class class wbr 4125    e. cmpt 4179   ` cfv 5358  (class class class)co 5981   iota_crio 6439   Basecbs 13356   lecple 13423   joincjn 14288   meetcmee 14289   Atomscatm 29512   HLchlt 29599   LHypclh 30232
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-iin 4010  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-undef 6440  df-riota 6446  df-poset 14290  df-plt 14302  df-lub 14318  df-glb 14319  df-join 14320  df-meet 14321  df-p0 14355  df-p1 14356  df-lat 14362  df-clat 14424  df-oposet 29425  df-ol 29427  df-oml 29428  df-covers 29515  df-ats 29516  df-atl 29547  df-cvlat 29571  df-hlat 29600  df-llines 29746  df-lplanes 29747  df-lvols 29748  df-lines 29749  df-psubsp 29751  df-pmap 29752  df-padd 30044  df-lhyp 30236
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