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Theorem cdleme27b 30557
Description: Lemma for cdleme27N 30558. (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 4026 . . 3  |-  ( s  =  t  ->  (
s  .<_  ( P  .\/  Q )  <->  t  .<_  ( P 
.\/  Q ) ) )
2 oveq1 5865 . . . . . . . . . . . 12  |-  ( s  =  t  ->  (
s  .\/  z )  =  ( t  .\/  z ) )
32oveq1d 5873 . . . . . . . . . . 11  |-  ( s  =  t  ->  (
( s  .\/  z
)  ./\  W )  =  ( ( t 
.\/  z )  ./\  W ) )
43oveq2d 5874 . . . . . . . . . 10  |-  ( s  =  t  ->  ( Z  .\/  ( ( s 
.\/  z )  ./\  W ) )  =  ( Z  .\/  ( ( t  .\/  z ) 
./\  W ) ) )
54oveq2d 5874 . . . . . . . . 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 2340 . . . . . . . 8  |-  ( s  =  t  ->  N  =  O )
98eqeq2d 2294 . . . . . . 7  |-  ( s  =  t  ->  (
u  =  N  <->  u  =  O ) )
109imbi2d 307 . . . . . 6  |-  ( s  =  t  ->  (
( ( -.  z  .<_  W  /\  -.  z  .<_  ( P  .\/  Q
) )  ->  u  =  N )  <->  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P 
.\/  Q ) )  ->  u  =  O ) ) )
1110ralbidv 2563 . . . . 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 6306 . . . 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 2340 . . 3  |-  ( s  =  t  ->  D  =  E )
16 oveq1 5865 . . . . 5  |-  ( s  =  t  ->  (
s  .\/  U )  =  ( t  .\/  U ) )
17 oveq2 5866 . . . . . . 7  |-  ( s  =  t  ->  ( P  .\/  s )  =  ( P  .\/  t
) )
1817oveq1d 5873 . . . . . 6  |-  ( s  =  t  ->  (
( P  .\/  s
)  ./\  W )  =  ( ( P 
.\/  t )  ./\  W ) )
1918oveq2d 5874 . . . . 5  |-  ( s  =  t  ->  ( Q  .\/  ( ( P 
.\/  s )  ./\  W ) )  =  ( Q  .\/  ( ( P  .\/  t ) 
./\  W ) ) )
2016, 19oveq12d 5876 . . . 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 2340 . . 3  |-  ( s  =  t  ->  F  =  G )
241, 15, 23ifbieq12d 3587 . 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 2340 1  |-  ( s  =  t  ->  C  =  Y )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623   A.wral 2543   ifcif 3565   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   iota_crio 6297   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Atomscatm 29453   LHypclh 30173
This theorem is referenced by:  cdleme27N  30558  cdleme28c  30561
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-riota 6304
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