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Theorem cdleme40v 29347
Description: Part of proof of Lemma E in [Crawley] p. 113. Change bound variables in  [_ S  /  u ]_ V (but we use  [_ R  /  u ]_ V for convenience since we have its hypotheses available) . (Contributed by NM, 18-Mar-2013.)
Hypotheses
Ref Expression
cdleme40.b  |-  B  =  ( Base `  K
)
cdleme40.l  |-  .<_  =  ( le `  K )
cdleme40.j  |-  .\/  =  ( join `  K )
cdleme40.m  |-  ./\  =  ( meet `  K )
cdleme40.a  |-  A  =  ( Atoms `  K )
cdleme40.h  |-  H  =  ( LHyp `  K
)
cdleme40.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme40.e  |-  E  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
cdleme40.g  |-  G  =  ( ( P  .\/  Q )  ./\  ( E  .\/  ( ( s  .\/  t )  ./\  W
) ) )
cdleme40.i  |-  I  =  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  G ) )
cdleme40.n  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  D
)
cdleme40.d  |-  D  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
cdleme40r.y  |-  Y  =  ( ( u  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  u )  ./\  W
) ) )
cdleme40r.t  |-  T  =  ( ( v  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  v )  ./\  W
) ) )
cdleme40r.x  |-  X  =  ( ( P  .\/  Q )  ./\  ( T  .\/  ( ( u  .\/  v )  ./\  W
) ) )
cdleme40r.o  |-  O  =  ( iota_ z  e.  B A. v  e.  A  ( ( -.  v  .<_  W  /\  -.  v  .<_  ( P  .\/  Q
) )  ->  z  =  X ) )
cdleme40r.v  |-  V  =  if ( u  .<_  ( P  .\/  Q ) ,  O ,  Y
)
Assertion
Ref Expression
cdleme40v  |-  ( R  e.  A  ->  [_ R  /  s ]_ N  =  [_ R  /  u ]_ V )
Distinct variable groups:    v, u, z, A    u, B, v, z    v, H, z   
u,  .\/ , v, z    v, K, z    u,  .<_ , v, z    u,  ./\ , v, z   
u, P, v, z   
u, Q, v, z   
v, R, z    u, T    v, U, z    u, W, v, z, s, t, y    A, s    y, t, A    B, s, t, y    E, s    t, H, y    .\/ , s, t, y    t, K, y    .<_ , s, t,
y    ./\ , s, t, y    P, s, t, y    Q, s, t, y    R, s, t, y    t, U, y    W, s, t, y   
y, Y    v, t,
y    T, s, t, y   
v, E, z    u, N, v    u, R    V, s    t, X, y    u, s, z, t, y
Allowed substitution hints:    D( y, z, v, u, t, s)    T( z, v)    U( u, s)    E( y, u, t)    G( y, z, v, u, t, s)    H( u, s)    I( y, z, v, u, t, s)    K( u, s)    N( y, z, t, s)    O( y, z, v, u, t, s)    V( y, z, v, u, t)    X( z, v, u, s)    Y( z, v, u, t, s)

Proof of Theorem cdleme40v
StepHypRef Expression
1 breq1 3923 . . . . 5  |-  ( s  =  u  ->  (
s  .<_  ( P  .\/  Q )  <->  u  .<_  ( P 
.\/  Q ) ) )
2 cdleme40.g . . . . . . . . . . . 12  |-  G  =  ( ( P  .\/  Q )  ./\  ( E  .\/  ( ( s  .\/  t )  ./\  W
) ) )
3 oveq1 5717 . . . . . . . . . . . . . . 15  |-  ( s  =  u  ->  (
s  .\/  t )  =  ( u  .\/  t ) )
43oveq1d 5725 . . . . . . . . . . . . . 14  |-  ( s  =  u  ->  (
( s  .\/  t
)  ./\  W )  =  ( ( u 
.\/  t )  ./\  W ) )
54oveq2d 5726 . . . . . . . . . . . . 13  |-  ( s  =  u  ->  ( E  .\/  ( ( s 
.\/  t )  ./\  W ) )  =  ( E  .\/  ( ( u  .\/  t ) 
./\  W ) ) )
65oveq2d 5726 . . . . . . . . . . . 12  |-  ( s  =  u  ->  (
( P  .\/  Q
)  ./\  ( E  .\/  ( ( s  .\/  t )  ./\  W
) ) )  =  ( ( P  .\/  Q )  ./\  ( E  .\/  ( ( u  .\/  t )  ./\  W
) ) ) )
72, 6syl5eq 2297 . . . . . . . . . . 11  |-  ( s  =  u  ->  G  =  ( ( P 
.\/  Q )  ./\  ( E  .\/  ( ( u  .\/  t ) 
./\  W ) ) ) )
87eqeq2d 2264 . . . . . . . . . 10  |-  ( s  =  u  ->  (
y  =  G  <->  y  =  ( ( P  .\/  Q )  ./\  ( E  .\/  ( ( u  .\/  t )  ./\  W
) ) ) ) )
98imbi2d 309 . . . . . . . . 9  |-  ( s  =  u  ->  (
( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  G )  <->  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P 
.\/  Q ) )  ->  y  =  ( ( P  .\/  Q
)  ./\  ( E  .\/  ( ( u  .\/  t )  ./\  W
) ) ) ) ) )
109ralbidv 2527 . . . . . . . 8  |-  ( s  =  u  ->  ( A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  G )  <->  A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P 
.\/  Q ) )  ->  y  =  ( ( P  .\/  Q
)  ./\  ( E  .\/  ( ( u  .\/  t )  ./\  W
) ) ) ) ) )
1110riotabidv 6192 . . . . . . 7  |-  ( s  =  u  ->  ( iota_ y  e.  B A. t  e.  A  (
( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q ) )  ->  y  =  G ) )  =  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  ( ( P 
.\/  Q )  ./\  ( E  .\/  ( ( u  .\/  t ) 
./\  W ) ) ) ) ) )
12 eqeq1 2259 . . . . . . . . . . 11  |-  ( y  =  z  ->  (
y  =  ( ( P  .\/  Q ) 
./\  ( E  .\/  ( ( u  .\/  t )  ./\  W
) ) )  <->  z  =  ( ( P  .\/  Q )  ./\  ( E  .\/  ( ( u  .\/  t )  ./\  W
) ) ) ) )
1312imbi2d 309 . . . . . . . . . 10  |-  ( y  =  z  ->  (
( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  ( ( P 
.\/  Q )  ./\  ( E  .\/  ( ( u  .\/  t ) 
./\  W ) ) ) )  <->  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P 
.\/  Q ) )  ->  z  =  ( ( P  .\/  Q
)  ./\  ( E  .\/  ( ( u  .\/  t )  ./\  W
) ) ) ) ) )
1413ralbidv 2527 . . . . . . . . 9  |-  ( y  =  z  ->  ( A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  ( ( P 
.\/  Q )  ./\  ( E  .\/  ( ( u  .\/  t ) 
./\  W ) ) ) )  <->  A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P 
.\/  Q ) )  ->  z  =  ( ( P  .\/  Q
)  ./\  ( E  .\/  ( ( u  .\/  t )  ./\  W
) ) ) ) ) )
15 breq1 3923 . . . . . . . . . . . . 13  |-  ( t  =  v  ->  (
t  .<_  W  <->  v  .<_  W ) )
1615notbid 287 . . . . . . . . . . . 12  |-  ( t  =  v  ->  ( -.  t  .<_  W  <->  -.  v  .<_  W ) )
17 breq1 3923 . . . . . . . . . . . . 13  |-  ( t  =  v  ->  (
t  .<_  ( P  .\/  Q )  <->  v  .<_  ( P 
.\/  Q ) ) )
1817notbid 287 . . . . . . . . . . . 12  |-  ( t  =  v  ->  ( -.  t  .<_  ( P 
.\/  Q )  <->  -.  v  .<_  ( P  .\/  Q
) ) )
1916, 18anbi12d 694 . . . . . . . . . . 11  |-  ( t  =  v  ->  (
( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q ) )  <->  ( -.  v  .<_  W  /\  -.  v  .<_  ( P  .\/  Q
) ) ) )
20 oveq1 5717 . . . . . . . . . . . . . . . . 17  |-  ( t  =  v  ->  (
t  .\/  U )  =  ( v  .\/  U ) )
21 oveq2 5718 . . . . . . . . . . . . . . . . . . 19  |-  ( t  =  v  ->  ( P  .\/  t )  =  ( P  .\/  v
) )
2221oveq1d 5725 . . . . . . . . . . . . . . . . . 18  |-  ( t  =  v  ->  (
( P  .\/  t
)  ./\  W )  =  ( ( P 
.\/  v )  ./\  W ) )
2322oveq2d 5726 . . . . . . . . . . . . . . . . 17  |-  ( t  =  v  ->  ( Q  .\/  ( ( P 
.\/  t )  ./\  W ) )  =  ( Q  .\/  ( ( P  .\/  v ) 
./\  W ) ) )
2420, 23oveq12d 5728 . . . . . . . . . . . . . . . 16  |-  ( t  =  v  ->  (
( t  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )  =  ( ( v  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  v )  ./\  W
) ) ) )
25 cdleme40.e . . . . . . . . . . . . . . . 16  |-  E  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
26 cdleme40r.t . . . . . . . . . . . . . . . 16  |-  T  =  ( ( v  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  v )  ./\  W
) ) )
2724, 25, 263eqtr4g 2310 . . . . . . . . . . . . . . 15  |-  ( t  =  v  ->  E  =  T )
28 oveq2 5718 . . . . . . . . . . . . . . . 16  |-  ( t  =  v  ->  (
u  .\/  t )  =  ( u  .\/  v ) )
2928oveq1d 5725 . . . . . . . . . . . . . . 15  |-  ( t  =  v  ->  (
( u  .\/  t
)  ./\  W )  =  ( ( u 
.\/  v )  ./\  W ) )
3027, 29oveq12d 5728 . . . . . . . . . . . . . 14  |-  ( t  =  v  ->  ( E  .\/  ( ( u 
.\/  t )  ./\  W ) )  =  ( T  .\/  ( ( u  .\/  v ) 
./\  W ) ) )
3130oveq2d 5726 . . . . . . . . . . . . 13  |-  ( t  =  v  ->  (
( P  .\/  Q
)  ./\  ( E  .\/  ( ( u  .\/  t )  ./\  W
) ) )  =  ( ( P  .\/  Q )  ./\  ( T  .\/  ( ( u  .\/  v )  ./\  W
) ) ) )
32 cdleme40r.x . . . . . . . . . . . . 13  |-  X  =  ( ( P  .\/  Q )  ./\  ( T  .\/  ( ( u  .\/  v )  ./\  W
) ) )
3331, 32syl6eqr 2303 . . . . . . . . . . . 12  |-  ( t  =  v  ->  (
( P  .\/  Q
)  ./\  ( E  .\/  ( ( u  .\/  t )  ./\  W
) ) )  =  X )
3433eqeq2d 2264 . . . . . . . . . . 11  |-  ( t  =  v  ->  (
z  =  ( ( P  .\/  Q ) 
./\  ( E  .\/  ( ( u  .\/  t )  ./\  W
) ) )  <->  z  =  X ) )
3519, 34imbi12d 313 . . . . . . . . . 10  |-  ( t  =  v  ->  (
( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  z  =  ( ( P 
.\/  Q )  ./\  ( E  .\/  ( ( u  .\/  t ) 
./\  W ) ) ) )  <->  ( ( -.  v  .<_  W  /\  -.  v  .<_  ( P 
.\/  Q ) )  ->  z  =  X ) ) )
3635cbvralv 2708 . . . . . . . . 9  |-  ( A. t  e.  A  (
( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q ) )  ->  z  =  ( ( P  .\/  Q )  ./\  ( E  .\/  ( ( u  .\/  t )  ./\  W
) ) ) )  <->  A. v  e.  A  ( ( -.  v  .<_  W  /\  -.  v  .<_  ( P  .\/  Q
) )  ->  z  =  X ) )
3714, 36syl6bb 254 . . . . . . . 8  |-  ( y  =  z  ->  ( A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  ( ( P 
.\/  Q )  ./\  ( E  .\/  ( ( u  .\/  t ) 
./\  W ) ) ) )  <->  A. v  e.  A  ( ( -.  v  .<_  W  /\  -.  v  .<_  ( P 
.\/  Q ) )  ->  z  =  X ) ) )
3837cbvriotav 6202 . . . . . . 7  |-  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P 
.\/  Q ) )  ->  y  =  ( ( P  .\/  Q
)  ./\  ( E  .\/  ( ( u  .\/  t )  ./\  W
) ) ) ) )  =  ( iota_ z  e.  B A. v  e.  A  ( ( -.  v  .<_  W  /\  -.  v  .<_  ( P 
.\/  Q ) )  ->  z  =  X ) )
3911, 38syl6eq 2301 . . . . . 6  |-  ( s  =  u  ->  ( iota_ y  e.  B A. t  e.  A  (
( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q ) )  ->  y  =  G ) )  =  ( iota_ z  e.  B A. v  e.  A  ( ( -.  v  .<_  W  /\  -.  v  .<_  ( P  .\/  Q
) )  ->  z  =  X ) ) )
40 cdleme40.i . . . . . 6  |-  I  =  ( iota_ y  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  G ) )
41 cdleme40r.o . . . . . 6  |-  O  =  ( iota_ z  e.  B A. v  e.  A  ( ( -.  v  .<_  W  /\  -.  v  .<_  ( P  .\/  Q
) )  ->  z  =  X ) )
4239, 40, 413eqtr4g 2310 . . . . 5  |-  ( s  =  u  ->  I  =  O )
43 oveq1 5717 . . . . . . 7  |-  ( s  =  u  ->  (
s  .\/  U )  =  ( u  .\/  U ) )
44 oveq2 5718 . . . . . . . . 9  |-  ( s  =  u  ->  ( P  .\/  s )  =  ( P  .\/  u
) )
4544oveq1d 5725 . . . . . . . 8  |-  ( s  =  u  ->  (
( P  .\/  s
)  ./\  W )  =  ( ( P 
.\/  u )  ./\  W ) )
4645oveq2d 5726 . . . . . . 7  |-  ( s  =  u  ->  ( Q  .\/  ( ( P 
.\/  s )  ./\  W ) )  =  ( Q  .\/  ( ( P  .\/  u ) 
./\  W ) ) )
4743, 46oveq12d 5728 . . . . . 6  |-  ( s  =  u  ->  (
( s  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  =  ( ( u  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  u )  ./\  W
) ) ) )
48 cdleme40.d . . . . . 6  |-  D  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
49 cdleme40r.y . . . . . 6  |-  Y  =  ( ( u  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  u )  ./\  W
) ) )
5047, 48, 493eqtr4g 2310 . . . . 5  |-  ( s  =  u  ->  D  =  Y )
511, 42, 50ifbieq12d 3492 . . . 4  |-  ( s  =  u  ->  if ( s  .<_  ( P 
.\/  Q ) ,  I ,  D )  =  if ( u 
.<_  ( P  .\/  Q
) ,  O ,  Y ) )
52 cdleme40.n . . . 4  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  D
)
53 cdleme40r.v . . . 4  |-  V  =  if ( u  .<_  ( P  .\/  Q ) ,  O ,  Y
)
5451, 52, 533eqtr4g 2310 . . 3  |-  ( s  =  u  ->  N  =  V )
5554cbvcsbv 3014 . 2  |-  [_ R  /  s ]_ N  =  [_ R  /  u ]_ V
5655a1i 12 1  |-  ( R  e.  A  ->  [_ R  /  s ]_ N  =  [_ R  /  u ]_ V )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2509   [_csb 3009   ifcif 3470   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   iota_crio 6181   Basecbs 13022   lecple 13089   joincjn 13922   meetcmee 13923   Atomscatm 28142   LHypclh 28862
This theorem is referenced by:  cdleme40w  29348
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-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  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-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-xp 4594  df-cnv 4596  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fv 4608  df-ov 5713  df-iota 6143  df-riota 6190
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