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Theorem dalem54 35593
Description: Lemma for dath 35603. Line  G H intersects the auxiliary axis of perspectivity  B. (Contributed by NM, 8-Aug-2012.)
Hypotheses
Ref Expression
dalem.ph  |-  ( ph  <->  ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  /\  ( Y  e.  O  /\  Z  e.  O )  /\  ( ( -.  C  .<_  ( P  .\/  Q
)  /\  -.  C  .<_  ( Q  .\/  R
)  /\  -.  C  .<_  ( R  .\/  P
) )  /\  ( -.  C  .<_  ( S 
.\/  T )  /\  -.  C  .<_  ( T 
.\/  U )  /\  -.  C  .<_  ( U 
.\/  S ) )  /\  ( C  .<_  ( P  .\/  S )  /\  C  .<_  ( Q 
.\/  T )  /\  C  .<_  ( R  .\/  U ) ) ) ) )
dalem.l  |-  .<_  =  ( le `  K )
dalem.j  |-  .\/  =  ( join `  K )
dalem.a  |-  A  =  ( Atoms `  K )
dalem.ps  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
dalem54.m  |-  ./\  =  ( meet `  K )
dalem54.o  |-  O  =  ( LPlanes `  K )
dalem54.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem54.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
dalem54.g  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
dalem54.h  |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )
dalem54.i  |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )
dalem54.b1  |-  B  =  ( ( ( G 
.\/  H )  .\/  I )  ./\  Y
)
Assertion
Ref Expression
dalem54  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  e.  A )

Proof of Theorem dalem54
StepHypRef Expression
1 dalem.ph . . . 4  |-  ( ph  <->  ( ( ( K  e.  HL  /\  C  e.  ( Base `  K
) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  /\  ( Y  e.  O  /\  Z  e.  O )  /\  ( ( -.  C  .<_  ( P  .\/  Q
)  /\  -.  C  .<_  ( Q  .\/  R
)  /\  -.  C  .<_  ( R  .\/  P
) )  /\  ( -.  C  .<_  ( S 
.\/  T )  /\  -.  C  .<_  ( T 
.\/  U )  /\  -.  C  .<_  ( U 
.\/  S ) )  /\  ( C  .<_  ( P  .\/  S )  /\  C  .<_  ( Q 
.\/  T )  /\  C  .<_  ( R  .\/  U ) ) ) ) )
21dalemkehl 35490 . . 3  |-  ( ph  ->  K  e.  HL )
323ad2ant1 1017 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
4 dalem.l . . . 4  |-  .<_  =  ( le `  K )
5 dalem.j . . . 4  |-  .\/  =  ( join `  K )
6 dalem.a . . . 4  |-  A  =  ( Atoms `  K )
7 dalem.ps . . . 4  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
8 dalem54.m . . . 4  |-  ./\  =  ( meet `  K )
9 dalem54.o . . . 4  |-  O  =  ( LPlanes `  K )
10 dalem54.y . . . 4  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
11 dalem54.z . . . 4  |-  Z  =  ( ( S  .\/  T )  .\/  U )
12 dalem54.g . . . 4  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
131, 4, 5, 6, 7, 8, 9, 10, 11, 12dalem23 35563 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  A )
14 dalem54.h . . . 4  |-  H  =  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) )
151, 4, 5, 6, 7, 8, 9, 10, 11, 14dalem29 35568 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  H  e.  A )
16 dalem54.i . . . 4  |-  I  =  ( ( c  .\/  R )  ./\  ( d  .\/  U ) )
171, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16dalem41 35580 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  =/=  H )
18 eqid 2457 . . . 4  |-  ( LLines `  K )  =  (
LLines `  K )
195, 6, 18llni2 35379 . . 3  |-  ( ( ( K  e.  HL  /\  G  e.  A  /\  H  e.  A )  /\  G  =/=  H
)  ->  ( G  .\/  H )  e.  (
LLines `  K ) )
203, 13, 15, 17, 19syl31anc 1231 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  .\/  H
)  e.  ( LLines `  K ) )
21 dalem54.b1 . . 3  |-  B  =  ( ( ( G 
.\/  H )  .\/  I )  ./\  Y
)
221, 4, 5, 6, 7, 8, 18, 9, 10, 11, 12, 14, 16, 21dalem53 35592 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  e.  ( LLines `  K ) )
231dalemkelat 35491 . . . . . . 7  |-  ( ph  ->  K  e.  Lat )
24233ad2ant1 1017 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  Lat )
25 eqid 2457 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
2625, 18llnbase 35376 . . . . . . . 8  |-  ( ( G  .\/  H )  e.  ( LLines `  K
)  ->  ( G  .\/  H )  e.  (
Base `  K )
)
2720, 26syl 16 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  .\/  H
)  e.  ( Base `  K ) )
281, 4, 5, 6, 7, 8, 9, 10, 11, 16dalem34 35573 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  e.  A )
2925, 6atbase 35157 . . . . . . . 8  |-  ( I  e.  A  ->  I  e.  ( Base `  K
) )
3028, 29syl 16 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  e.  ( Base `  K ) )
3125, 5latjcl 15808 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  I  e.  ( Base `  K )
)  ->  ( ( G  .\/  H )  .\/  I )  e.  (
Base `  K )
)
3224, 27, 30, 31syl3anc 1228 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  I )  e.  ( Base `  K
) )
331, 9dalemyeb 35516 . . . . . . 7  |-  ( ph  ->  Y  e.  ( Base `  K ) )
34333ad2ant1 1017 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  e.  ( Base `  K ) )
3525, 4, 8latmle2 15834 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( ( G  .\/  H )  .\/  I )  e.  ( Base `  K
)  /\  Y  e.  ( Base `  K )
)  ->  ( (
( G  .\/  H
)  .\/  I )  ./\  Y )  .<_  Y )
3624, 32, 34, 35syl3anc 1228 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( G 
.\/  H )  .\/  I )  ./\  Y
)  .<_  Y )
3721, 36syl5eqbr 4489 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  .<_  Y )
381, 4, 5, 6, 7, 8, 9, 10, 11, 12dalem24 35564 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  G  .<_  Y )
3925, 6atbase 35157 . . . . . . . 8  |-  ( G  e.  A  ->  G  e.  ( Base `  K
) )
4013, 39syl 16 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  ( Base `  K ) )
4125, 6atbase 35157 . . . . . . . 8  |-  ( H  e.  A  ->  H  e.  ( Base `  K
) )
4215, 41syl 16 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  H  e.  ( Base `  K ) )
4325, 4, 5latjle12 15819 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( G  e.  ( Base `  K )  /\  H  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) ) )  -> 
( ( G  .<_  Y  /\  H  .<_  Y )  <-> 
( G  .\/  H
)  .<_  Y ) )
4424, 40, 42, 34, 43syl13anc 1230 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .<_  Y  /\  H  .<_  Y )  <-> 
( G  .\/  H
)  .<_  Y ) )
45 simpl 457 . . . . . 6  |-  ( ( G  .<_  Y  /\  H  .<_  Y )  ->  G  .<_  Y )
4644, 45syl6bir 229 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .<_  Y  ->  G 
.<_  Y ) )
4738, 46mtod 177 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  ( G  .\/  H
)  .<_  Y )
48 nbrne2 4474 . . . 4  |-  ( ( B  .<_  Y  /\  -.  ( G  .\/  H
)  .<_  Y )  ->  B  =/=  ( G  .\/  H ) )
4937, 47, 48syl2anc 661 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  =/=  ( G  .\/  H ) )
5049necomd 2728 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  .\/  H
)  =/=  B )
51 hlatl 35228 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
523, 51syl 16 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  AtLat )
5325, 18llnbase 35376 . . . . 5  |-  ( B  e.  ( LLines `  K
)  ->  B  e.  ( Base `  K )
)
5422, 53syl 16 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  e.  ( Base `  K ) )
5525, 8latmcl 15809 . . . 4  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  B  e.  ( Base `  K )
)  ->  ( ( G  .\/  H )  ./\  B )  e.  ( Base `  K ) )
5624, 27, 54, 55syl3anc 1228 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  e.  ( Base `  K
) )
571, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16dalem52 35591 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  e.  A )
581, 5, 6dalempjqeb 35512 . . . . . 6  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
59583ad2ant1 1017 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  .\/  Q
)  e.  ( Base `  K ) )
6025, 4, 8latmle1 15833 . . . . 5  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  ( P  .\/  Q )  e.  (
Base `  K )
)  ->  ( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( G  .\/  H ) )
6124, 27, 59, 60syl3anc 1228 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( G  .\/  H ) )
621, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16dalem51 35590 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( ( K  e.  HL  /\  c  e.  A )  /\  ( G  e.  A  /\  H  e.  A  /\  I  e.  A
)  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  /\  ( ( ( G 
.\/  H )  .\/  I )  e.  O  /\  Y  e.  O
)  /\  ( ( -.  c  .<_  ( G 
.\/  H )  /\  -.  c  .<_  ( H 
.\/  I )  /\  -.  c  .<_  ( I 
.\/  G ) )  /\  ( -.  c  .<_  ( P  .\/  Q
)  /\  -.  c  .<_  ( Q  .\/  R
)  /\  -.  c  .<_  ( R  .\/  P
) )  /\  (
c  .<_  ( G  .\/  P )  /\  c  .<_  ( H  .\/  Q )  /\  c  .<_  ( I 
.\/  R ) ) ) )  /\  (
( G  .\/  H
)  .\/  I )  =/=  Y ) )
6362simpld 459 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( K  e.  HL  /\  c  e.  A )  /\  ( G  e.  A  /\  H  e.  A  /\  I  e.  A )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  (
( ( G  .\/  H )  .\/  I )  e.  O  /\  Y  e.  O )  /\  (
( -.  c  .<_  ( G  .\/  H )  /\  -.  c  .<_  ( H  .\/  I )  /\  -.  c  .<_  ( I  .\/  G ) )  /\  ( -.  c  .<_  ( P  .\/  Q )  /\  -.  c  .<_  ( Q  .\/  R )  /\  -.  c  .<_  ( R  .\/  P
) )  /\  (
c  .<_  ( G  .\/  P )  /\  c  .<_  ( H  .\/  Q )  /\  c  .<_  ( I 
.\/  R ) ) ) ) )
6425, 6atbase 35157 . . . . . . . 8  |-  ( c  e.  A  ->  c  e.  ( Base `  K
) )
6564anim2i 569 . . . . . . 7  |-  ( ( K  e.  HL  /\  c  e.  A )  ->  ( K  e.  HL  /\  c  e.  ( Base `  K ) ) )
66653anim1i 1182 . . . . . 6  |-  ( ( ( K  e.  HL  /\  c  e.  A )  /\  ( G  e.  A  /\  H  e.  A  /\  I  e.  A )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )
)  ->  ( ( K  e.  HL  /\  c  e.  ( Base `  K
) )  /\  ( G  e.  A  /\  H  e.  A  /\  I  e.  A )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) ) )
67 biid 236 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  c  e.  ( Base `  K
) )  /\  ( G  e.  A  /\  H  e.  A  /\  I  e.  A )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  (
( ( G  .\/  H )  .\/  I )  e.  O  /\  Y  e.  O )  /\  (
( -.  c  .<_  ( G  .\/  H )  /\  -.  c  .<_  ( H  .\/  I )  /\  -.  c  .<_  ( I  .\/  G ) )  /\  ( -.  c  .<_  ( P  .\/  Q )  /\  -.  c  .<_  ( Q  .\/  R )  /\  -.  c  .<_  ( R  .\/  P
) )  /\  (
c  .<_  ( G  .\/  P )  /\  c  .<_  ( H  .\/  Q )  /\  c  .<_  ( I 
.\/  R ) ) ) )  <->  ( (
( K  e.  HL  /\  c  e.  ( Base `  K ) )  /\  ( G  e.  A  /\  H  e.  A  /\  I  e.  A
)  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  /\  ( ( ( G 
.\/  H )  .\/  I )  e.  O  /\  Y  e.  O
)  /\  ( ( -.  c  .<_  ( G 
.\/  H )  /\  -.  c  .<_  ( H 
.\/  I )  /\  -.  c  .<_  ( I 
.\/  G ) )  /\  ( -.  c  .<_  ( P  .\/  Q
)  /\  -.  c  .<_  ( Q  .\/  R
)  /\  -.  c  .<_  ( R  .\/  P
) )  /\  (
c  .<_  ( G  .\/  P )  /\  c  .<_  ( H  .\/  Q )  /\  c  .<_  ( I 
.\/  R ) ) ) ) )
68 eqid 2457 . . . . . . 7  |-  ( ( G  .\/  H ) 
.\/  I )  =  ( ( G  .\/  H )  .\/  I )
69 eqid 2457 . . . . . . 7  |-  ( ( G  .\/  H ) 
./\  ( P  .\/  Q ) )  =  ( ( G  .\/  H
)  ./\  ( P  .\/  Q ) )
7067, 4, 5, 6, 8, 9, 68, 10, 21, 69dalem10 35540 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  c  e.  ( Base `  K
) )  /\  ( G  e.  A  /\  H  e.  A  /\  I  e.  A )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  (
( ( G  .\/  H )  .\/  I )  e.  O  /\  Y  e.  O )  /\  (
( -.  c  .<_  ( G  .\/  H )  /\  -.  c  .<_  ( H  .\/  I )  /\  -.  c  .<_  ( I  .\/  G ) )  /\  ( -.  c  .<_  ( P  .\/  Q )  /\  -.  c  .<_  ( Q  .\/  R )  /\  -.  c  .<_  ( R  .\/  P
) )  /\  (
c  .<_  ( G  .\/  P )  /\  c  .<_  ( H  .\/  Q )  /\  c  .<_  ( I 
.\/  R ) ) ) )  ->  (
( G  .\/  H
)  ./\  ( P  .\/  Q ) )  .<_  B )
7166, 70syl3an1 1261 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  c  e.  A )  /\  ( G  e.  A  /\  H  e.  A  /\  I  e.  A )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  /\  (
( ( G  .\/  H )  .\/  I )  e.  O  /\  Y  e.  O )  /\  (
( -.  c  .<_  ( G  .\/  H )  /\  -.  c  .<_  ( H  .\/  I )  /\  -.  c  .<_  ( I  .\/  G ) )  /\  ( -.  c  .<_  ( P  .\/  Q )  /\  -.  c  .<_  ( Q  .\/  R )  /\  -.  c  .<_  ( R  .\/  P
) )  /\  (
c  .<_  ( G  .\/  P )  /\  c  .<_  ( H  .\/  Q )  /\  c  .<_  ( I 
.\/  R ) ) ) )  ->  (
( G  .\/  H
)  ./\  ( P  .\/  Q ) )  .<_  B )
7263, 71syl 16 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  B )
7325, 8latmcl 15809 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  ( P  .\/  Q )  e.  (
Base `  K )
)  ->  ( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  e.  ( Base `  K ) )
7424, 27, 59, 73syl3anc 1228 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  e.  ( Base `  K
) )
7525, 4, 8latlem12 15835 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( ( G 
.\/  H )  ./\  ( P  .\/  Q ) )  e.  ( Base `  K )  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  B  e.  ( Base `  K )
) )  ->  (
( ( ( G 
.\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( G  .\/  H )  /\  (
( G  .\/  H
)  ./\  ( P  .\/  Q ) )  .<_  B )  <->  ( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( ( G  .\/  H )  ./\  B ) ) )
7624, 74, 27, 54, 75syl13anc 1230 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( ( G  .\/  H ) 
./\  ( P  .\/  Q ) )  .<_  ( G 
.\/  H )  /\  ( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  B )  <->  ( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( ( G  .\/  H )  ./\  B ) ) )
7761, 72, 76mpbi2and 921 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( ( G  .\/  H )  ./\  B )
)
78 eqid 2457 . . . 4  |-  ( 0.
`  K )  =  ( 0. `  K
)
7925, 4, 78, 6atlen0 35178 . . 3  |-  ( ( ( K  e.  AtLat  /\  ( ( G  .\/  H )  ./\  B )  e.  ( Base `  K
)  /\  ( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  e.  A )  /\  ( ( G 
.\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( ( G  .\/  H )  ./\  B ) )  ->  (
( G  .\/  H
)  ./\  B )  =/=  ( 0. `  K
) )
8052, 56, 57, 77, 79syl31anc 1231 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  =/=  ( 0. `  K
) )
818, 78, 6, 182llnmat 35391 . 2  |-  ( ( ( K  e.  HL  /\  ( G  .\/  H
)  e.  ( LLines `  K )  /\  B  e.  ( LLines `  K )
)  /\  ( ( G  .\/  H )  =/= 
B  /\  ( ( G  .\/  H )  ./\  B )  =/=  ( 0.
`  K ) ) )  ->  ( ( G  .\/  H )  ./\  B )  e.  A )
823, 20, 22, 50, 80, 81syl32anc 1236 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   Basecbs 14644   lecple 14719   joincjn 15700   meetcmee 15701   0.cp0 15794   Latclat 15802   Atomscatm 35131   AtLatcal 35132   HLchlt 35218   LLinesclln 35358   LPlanesclpl 35359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-preset 15684  df-poset 15702  df-plt 15715  df-lub 15731  df-glb 15732  df-join 15733  df-meet 15734  df-p0 15796  df-lat 15803  df-clat 15865  df-oposet 35044  df-ol 35046  df-oml 35047  df-covers 35134  df-ats 35135  df-atl 35166  df-cvlat 35190  df-hlat 35219  df-llines 35365  df-lplanes 35366  df-lvols 35367
This theorem is referenced by:  dalem55  35594  dalem57  35596
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