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Theorem dalem54 30523
Description: Lemma for dath 30533. 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 30420 . . 3  |-  ( ph  ->  K  e.  HL )
323ad2ant1 978 . 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 30493 . . 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 30498 . . 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 30510 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  =/=  H )
18 eqid 2436 . . . 4  |-  ( LLines `  K )  =  (
LLines `  K )
195, 6, 18llni2 30309 . . 3  |-  ( ( ( K  e.  HL  /\  G  e.  A  /\  H  e.  A )  /\  G  =/=  H
)  ->  ( G  .\/  H )  e.  (
LLines `  K ) )
203, 13, 15, 17, 19syl31anc 1187 . 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 30522 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  e.  ( LLines `  K ) )
231dalemkelat 30421 . . . . . . 7  |-  ( ph  ->  K  e.  Lat )
24233ad2ant1 978 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  Lat )
25 eqid 2436 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
2625, 18llnbase 30306 . . . . . . . 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 30503 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  e.  A )
2925, 6atbase 30087 . . . . . . . 8  |-  ( I  e.  A  ->  I  e.  ( Base `  K
) )
3028, 29syl 16 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  e.  ( Base `  K ) )
3125, 5latjcl 14479 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  I  e.  ( Base `  K )
)  ->  ( ( G  .\/  H )  .\/  I )  e.  (
Base `  K )
)
3224, 27, 30, 31syl3anc 1184 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  I )  e.  ( Base `  K
) )
331, 9dalemyeb 30446 . . . . . . 7  |-  ( ph  ->  Y  e.  ( Base `  K ) )
34333ad2ant1 978 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  e.  ( Base `  K ) )
3525, 4, 8latmle2 14506 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( ( G  .\/  H )  .\/  I )  e.  ( Base `  K
)  /\  Y  e.  ( Base `  K )
)  ->  ( (
( G  .\/  H
)  .\/  I )  ./\  Y )  .<_  Y )
3624, 32, 34, 35syl3anc 1184 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( G 
.\/  H )  .\/  I )  ./\  Y
)  .<_  Y )
3721, 36syl5eqbr 4245 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  .<_  Y )
381, 4, 5, 6, 7, 8, 9, 10, 11, 12dalem24 30494 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  G  .<_  Y )
3925, 6atbase 30087 . . . . . . . 8  |-  ( G  e.  A  ->  G  e.  ( Base `  K
) )
4013, 39syl 16 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  ( Base `  K ) )
4125, 6atbase 30087 . . . . . . . 8  |-  ( H  e.  A  ->  H  e.  ( Base `  K
) )
4215, 41syl 16 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  H  e.  ( Base `  K ) )
4325, 4, 5latjle12 14491 . . . . . . 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 1186 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .<_  Y  /\  H  .<_  Y )  <-> 
( G  .\/  H
)  .<_  Y ) )
45 simpl 444 . . . . . 6  |-  ( ( G  .<_  Y  /\  H  .<_  Y )  ->  G  .<_  Y )
4644, 45syl6bir 221 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .<_  Y  ->  G 
.<_  Y ) )
4738, 46mtod 170 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  ( G  .\/  H
)  .<_  Y )
48 nbrne2 4230 . . . 4  |-  ( ( B  .<_  Y  /\  -.  ( G  .\/  H
)  .<_  Y )  ->  B  =/=  ( G  .\/  H ) )
4937, 47, 48syl2anc 643 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  =/=  ( G  .\/  H ) )
5049necomd 2687 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  .\/  H
)  =/=  B )
51 hlatl 30158 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
523, 51syl 16 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  AtLat )
5325, 18llnbase 30306 . . . . 5  |-  ( B  e.  ( LLines `  K
)  ->  B  e.  ( Base `  K )
)
5422, 53syl 16 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  e.  ( Base `  K ) )
5525, 8latmcl 14480 . . . 4  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  B  e.  ( Base `  K )
)  ->  ( ( G  .\/  H )  ./\  B )  e.  ( Base `  K ) )
5624, 27, 54, 55syl3anc 1184 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  e.  ( Base `  K
) )
571, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16dalem52 30521 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  e.  A )
581, 5, 6dalempjqeb 30442 . . . . . 6  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
59583ad2ant1 978 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  .\/  Q
)  e.  ( Base `  K ) )
6025, 4, 8latmle1 14505 . . . . 5  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  ( P  .\/  Q )  e.  (
Base `  K )
)  ->  ( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( G  .\/  H ) )
6124, 27, 59, 60syl3anc 1184 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( G  .\/  H ) )
621, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16dalem51 30520 . . . . . 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 446 . . . . 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 30087 . . . . . . . 8  |-  ( c  e.  A  ->  c  e.  ( Base `  K
) )
6564anim2i 553 . . . . . . 7  |-  ( ( K  e.  HL  /\  c  e.  A )  ->  ( K  e.  HL  /\  c  e.  ( Base `  K ) ) )
66653anim1i 1140 . . . . . 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 228 . . . . . . 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 2436 . . . . . . 7  |-  ( ( G  .\/  H ) 
.\/  I )  =  ( ( G  .\/  H )  .\/  I )
69 eqid 2436 . . . . . . 7  |-  ( ( G  .\/  H ) 
./\  ( P  .\/  Q ) )  =  ( ( G  .\/  H
)  ./\  ( P  .\/  Q ) )
7067, 4, 5, 6, 8, 9, 68, 10, 21, 69dalem10 30470 . . . . . 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 1217 . . . . 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 14480 . . . . . 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 1184 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  e.  ( Base `  K
) )
7525, 4, 8latlem12 14507 . . . . 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 1186 . . . 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 888 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( ( G  .\/  H )  ./\  B )
)
78 eqid 2436 . . . 4  |-  ( 0.
`  K )  =  ( 0. `  K
)
7925, 4, 78, 6atlen0 30108 . . 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 1187 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  =/=  ( 0. `  K
) )
818, 78, 6, 182llnmat 30321 . 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 1192 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   lecple 13536   joincjn 14401   meetcmee 14402   0.cp0 14466   Latclat 14474   Atomscatm 30061   AtLatcal 30062   HLchlt 30148   LLinesclln 30288   LPlanesclpl 30289
This theorem is referenced by:  dalem55  30524  dalem57  30526
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-llines 30295  df-lplanes 30296  df-lvols 30297
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