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Theorem dalem54 29733
Description: Lemma for dath 29743. 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 29630 . . 3  |-  ( ph  ->  K  e.  HL )
323ad2ant1 976 . 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 29703 . . 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 29708 . . 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 29720 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  =/=  H )
18 eqid 2316 . . . 4  |-  ( LLines `  K )  =  (
LLines `  K )
195, 6, 18llni2 29519 . . 3  |-  ( ( ( K  e.  HL  /\  G  e.  A  /\  H  e.  A )  /\  G  =/=  H
)  ->  ( G  .\/  H )  e.  (
LLines `  K ) )
203, 13, 15, 17, 19syl31anc 1185 . 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 29732 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  e.  ( LLines `  K ) )
231dalemkelat 29631 . . . . . . 7  |-  ( ph  ->  K  e.  Lat )
24233ad2ant1 976 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  Lat )
25 eqid 2316 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
2625, 18llnbase 29516 . . . . . . . 8  |-  ( ( G  .\/  H )  e.  ( LLines `  K
)  ->  ( G  .\/  H )  e.  (
Base `  K )
)
2720, 26syl 15 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  .\/  H
)  e.  ( Base `  K ) )
281, 4, 5, 6, 7, 8, 9, 10, 11, 16dalem34 29713 . . . . . . . 8  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  e.  A )
2925, 6atbase 29297 . . . . . . . 8  |-  ( I  e.  A  ->  I  e.  ( Base `  K
) )
3028, 29syl 15 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  I  e.  ( Base `  K ) )
3125, 5latjcl 14205 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  I  e.  ( Base `  K )
)  ->  ( ( G  .\/  H )  .\/  I )  e.  (
Base `  K )
)
3224, 27, 30, 31syl3anc 1182 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .\/  I )  e.  ( Base `  K
) )
331, 9dalemyeb 29656 . . . . . . 7  |-  ( ph  ->  Y  e.  ( Base `  K ) )
34333ad2ant1 976 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  Y  e.  ( Base `  K ) )
3525, 4, 8latmle2 14232 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( ( G  .\/  H )  .\/  I )  e.  ( Base `  K
)  /\  Y  e.  ( Base `  K )
)  ->  ( (
( G  .\/  H
)  .\/  I )  ./\  Y )  .<_  Y )
3624, 32, 34, 35syl3anc 1182 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( ( G 
.\/  H )  .\/  I )  ./\  Y
)  .<_  Y )
3721, 36syl5eqbr 4093 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  .<_  Y )
381, 4, 5, 6, 7, 8, 9, 10, 11, 12dalem24 29704 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  G  .<_  Y )
3925, 6atbase 29297 . . . . . . . 8  |-  ( G  e.  A  ->  G  e.  ( Base `  K
) )
4013, 39syl 15 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  ( Base `  K ) )
4125, 6atbase 29297 . . . . . . . 8  |-  ( H  e.  A  ->  H  e.  ( Base `  K
) )
4215, 41syl 15 . . . . . . 7  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  H  e.  ( Base `  K ) )
4325, 4, 5latjle12 14217 . . . . . . 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 1184 . . . . . 6  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .<_  Y  /\  H  .<_  Y )  <-> 
( G  .\/  H
)  .<_  Y ) )
45 simpl 443 . . . . . 6  |-  ( ( G  .<_  Y  /\  H  .<_  Y )  ->  G  .<_  Y )
4644, 45syl6bir 220 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  .<_  Y  ->  G 
.<_  Y ) )
4738, 46mtod 168 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  ( G  .\/  H
)  .<_  Y )
48 nbrne2 4078 . . . 4  |-  ( ( B  .<_  Y  /\  -.  ( G  .\/  H
)  .<_  Y )  ->  B  =/=  ( G  .\/  H ) )
4937, 47, 48syl2anc 642 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  =/=  ( G  .\/  H ) )
5049necomd 2562 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  .\/  H
)  =/=  B )
51 hlatl 29368 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
523, 51syl 15 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  AtLat )
5325, 18llnbase 29516 . . . . 5  |-  ( B  e.  ( LLines `  K
)  ->  B  e.  ( Base `  K )
)
5422, 53syl 15 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  B  e.  ( Base `  K ) )
5525, 8latmcl 14206 . . . 4  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  B  e.  ( Base `  K )
)  ->  ( ( G  .\/  H )  ./\  B )  e.  ( Base `  K ) )
5624, 27, 54, 55syl3anc 1182 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  e.  ( Base `  K
) )
571, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16dalem52 29731 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  e.  A )
581, 5, 6dalempjqeb 29652 . . . . . 6  |-  ( ph  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
59583ad2ant1 976 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( P  .\/  Q
)  e.  ( Base `  K ) )
6025, 4, 8latmle1 14231 . . . . 5  |-  ( ( K  e.  Lat  /\  ( G  .\/  H )  e.  ( Base `  K
)  /\  ( P  .\/  Q )  e.  (
Base `  K )
)  ->  ( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( G  .\/  H ) )
6124, 27, 59, 60syl3anc 1182 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( G  .\/  H ) )
621, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16dalem51 29730 . . . . . 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 445 . . . . 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 29297 . . . . . . . 8  |-  ( c  e.  A  ->  c  e.  ( Base `  K
) )
6564anim2i 552 . . . . . . 7  |-  ( ( K  e.  HL  /\  c  e.  A )  ->  ( K  e.  HL  /\  c  e.  ( Base `  K ) ) )
66653anim1i 1138 . . . . . 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 227 . . . . . . 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 2316 . . . . . . 7  |-  ( ( G  .\/  H ) 
.\/  I )  =  ( ( G  .\/  H )  .\/  I )
69 eqid 2316 . . . . . . 7  |-  ( ( G  .\/  H ) 
./\  ( P  .\/  Q ) )  =  ( ( G  .\/  H
)  ./\  ( P  .\/  Q ) )
7067, 4, 5, 6, 8, 9, 68, 10, 21, 69dalem10 29680 . . . . . 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 1215 . . . . 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 15 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  B )
7325, 8latmcl 14206 . . . . . 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 1182 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  e.  ( Base `  K
) )
7525, 4, 8latlem12 14233 . . . . 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 1184 . . . 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 887 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  ( P  .\/  Q ) )  .<_  ( ( G  .\/  H )  ./\  B )
)
78 eqid 2316 . . . 4  |-  ( 0.
`  K )  =  ( 0. `  K
)
7925, 4, 78, 6atlen0 29318 . . 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 1185 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  =/=  ( 0. `  K
) )
818, 78, 6, 182llnmat 29531 . 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 1190 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( G  .\/  H )  ./\  B )  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701    =/= wne 2479   class class class wbr 4060   ` cfv 5292  (class class class)co 5900   Basecbs 13195   lecple 13262   joincjn 14127   meetcmee 14128   0.cp0 14192   Latclat 14200   Atomscatm 29271   AtLatcal 29272   HLchlt 29358   LLinesclln 29498   LPlanesclpl 29499
This theorem is referenced by:  dalem55  29734  dalem57  29736
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-undef 6340  df-riota 6346  df-poset 14129  df-plt 14141  df-lub 14157  df-glb 14158  df-join 14159  df-meet 14160  df-p0 14194  df-lat 14201  df-clat 14263  df-oposet 29184  df-ol 29186  df-oml 29187  df-covers 29274  df-ats 29275  df-atl 29306  df-cvlat 29330  df-hlat 29359  df-llines 29505  df-lplanes 29506  df-lvols 29507
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