Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme27b Structured version   Unicode version

Theorem cdleme27b 33387
Description: Lemma for cdleme27N 33388. (Contributed by NM, 3-Feb-2013.)
Hypotheses
Ref Expression
cdleme26.b  |-  B  =  ( Base `  K
)
cdleme26.l  |-  .<_  =  ( le `  K )
cdleme26.j  |-  .\/  =  ( join `  K )
cdleme26.m  |-  ./\  =  ( meet `  K )
cdleme26.a  |-  A  =  ( Atoms `  K )
cdleme26.h  |-  H  =  ( LHyp `  K
)
cdleme27.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme27.f  |-  F  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
cdleme27.z  |-  Z  =  ( ( z  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W
) ) )
cdleme27.n  |-  N  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( s  .\/  z )  ./\  W
) ) )
cdleme27.d  |-  D  =  ( iota_ u  e.  B  A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P  .\/  Q
) )  ->  u  =  N ) )
cdleme27.c  |-  C  =  if ( s  .<_  ( P  .\/  Q ) ,  D ,  F
)
cdleme27.g  |-  G  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
cdleme27.o  |-  O  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( t  .\/  z )  ./\  W
) ) )
cdleme27.e  |-  E  =  ( iota_ u  e.  B  A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P  .\/  Q
) )  ->  u  =  O ) )
cdleme27.y  |-  Y  =  if ( t  .<_  ( P  .\/  Q ) ,  E ,  G
)
Assertion
Ref Expression
cdleme27b  |-  ( s  =  t  ->  C  =  Y )
Distinct variable groups:    t, s, u, z, A    B, s,
t, u, z    u, F    u, G    H, s,
t, z    .\/ , s, t, u, z    K, s, t, z    .<_ , s, t, u, z    ./\ , s,
t, u, z    t, N, u    O, s, u    P, s, t, u, z    Q, s, t, u, z    U, s, t, u, z    W, s, t, u, z
Allowed substitution hints:    C( z, u, t, s)    D( z, u, t, s)    E( z, u, t, s)    F( z, t, s)    G( z, t, s)    H( u)    K( u)    N( z, s)    O( z, t)    Y( z, u, t, s)    Z( z, u, t, s)

Proof of Theorem cdleme27b
StepHypRef Expression
1 breq1 4398 . . 3  |-  ( s  =  t  ->  (
s  .<_  ( P  .\/  Q )  <->  t  .<_  ( P 
.\/  Q ) ) )
2 oveq1 6285 . . . . . . . . . . . 12  |-  ( s  =  t  ->  (
s  .\/  z )  =  ( t  .\/  z ) )
32oveq1d 6293 . . . . . . . . . . 11  |-  ( s  =  t  ->  (
( s  .\/  z
)  ./\  W )  =  ( ( t 
.\/  z )  ./\  W ) )
43oveq2d 6294 . . . . . . . . . 10  |-  ( s  =  t  ->  ( Z  .\/  ( ( s 
.\/  z )  ./\  W ) )  =  ( Z  .\/  ( ( t  .\/  z ) 
./\  W ) ) )
54oveq2d 6294 . . . . . . . . 9  |-  ( s  =  t  ->  (
( P  .\/  Q
)  ./\  ( Z  .\/  ( ( s  .\/  z )  ./\  W
) ) )  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( t  .\/  z )  ./\  W
) ) ) )
6 cdleme27.n . . . . . . . . 9  |-  N  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( s  .\/  z )  ./\  W
) ) )
7 cdleme27.o . . . . . . . . 9  |-  O  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( t  .\/  z )  ./\  W
) ) )
85, 6, 73eqtr4g 2468 . . . . . . . 8  |-  ( s  =  t  ->  N  =  O )
98eqeq2d 2416 . . . . . . 7  |-  ( s  =  t  ->  (
u  =  N  <->  u  =  O ) )
109imbi2d 314 . . . . . 6  |-  ( s  =  t  ->  (
( ( -.  z  .<_  W  /\  -.  z  .<_  ( P  .\/  Q
) )  ->  u  =  N )  <->  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P 
.\/  Q ) )  ->  u  =  O ) ) )
1110ralbidv 2843 . . . . 5  |-  ( s  =  t  ->  ( A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P  .\/  Q
) )  ->  u  =  N )  <->  A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P 
.\/  Q ) )  ->  u  =  O ) ) )
1211riotabidv 6242 . . . 4  |-  ( s  =  t  ->  ( iota_ u  e.  B  A. z  e.  A  (
( -.  z  .<_  W  /\  -.  z  .<_  ( P  .\/  Q ) )  ->  u  =  N ) )  =  ( iota_ u  e.  B  A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P  .\/  Q
) )  ->  u  =  O ) ) )
13 cdleme27.d . . . 4  |-  D  =  ( iota_ u  e.  B  A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P  .\/  Q
) )  ->  u  =  N ) )
14 cdleme27.e . . . 4  |-  E  =  ( iota_ u  e.  B  A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P  .\/  Q
) )  ->  u  =  O ) )
1512, 13, 143eqtr4g 2468 . . 3  |-  ( s  =  t  ->  D  =  E )
16 oveq1 6285 . . . . 5  |-  ( s  =  t  ->  (
s  .\/  U )  =  ( t  .\/  U ) )
17 oveq2 6286 . . . . . . 7  |-  ( s  =  t  ->  ( P  .\/  s )  =  ( P  .\/  t
) )
1817oveq1d 6293 . . . . . 6  |-  ( s  =  t  ->  (
( P  .\/  s
)  ./\  W )  =  ( ( P 
.\/  t )  ./\  W ) )
1918oveq2d 6294 . . . . 5  |-  ( s  =  t  ->  ( Q  .\/  ( ( P 
.\/  s )  ./\  W ) )  =  ( Q  .\/  ( ( P  .\/  t ) 
./\  W ) ) )
2016, 19oveq12d 6296 . . . 4  |-  ( s  =  t  ->  (
( s  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) ) )
21 cdleme27.f . . . 4  |-  F  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
22 cdleme27.g . . . 4  |-  G  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
2320, 21, 223eqtr4g 2468 . . 3  |-  ( s  =  t  ->  F  =  G )
241, 15, 23ifbieq12d 3912 . 2  |-  ( s  =  t  ->  if ( s  .<_  ( P 
.\/  Q ) ,  D ,  F )  =  if ( t 
.<_  ( P  .\/  Q
) ,  E ,  G ) )
25 cdleme27.c . 2  |-  C  =  if ( s  .<_  ( P  .\/  Q ) ,  D ,  F
)
26 cdleme27.y . 2  |-  Y  =  if ( t  .<_  ( P  .\/  Q ) ,  E ,  G
)
2724, 25, 263eqtr4g 2468 1  |-  ( s  =  t  ->  C  =  Y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1405   A.wral 2754   ifcif 3885   class class class wbr 4395   ` cfv 5569   iota_crio 6239  (class class class)co 6278   Basecbs 14841   lecple 14916   joincjn 15897   meetcmee 15898   Atomscatm 32281   LHypclh 33001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-iota 5533  df-fv 5577  df-riota 6240  df-ov 6281
This theorem is referenced by:  cdleme27N  33388  cdleme28c  33391
  Copyright terms: Public domain W3C validator