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Theorem 5oalem6 27147
Description: Lemma for orthoarguesian law 5OA. (Contributed by NM, 4-May-2000.) (New usage is discouraged.)
Hypotheses
Ref Expression
5oalem5.1  |-  A  e.  SH
5oalem5.2  |-  B  e.  SH
5oalem5.3  |-  C  e.  SH
5oalem5.4  |-  D  e.  SH
5oalem5.5  |-  F  e.  SH
5oalem5.6  |-  G  e.  SH
5oalem5.7  |-  R  e.  SH
5oalem5.8  |-  S  e.  SH
Assertion
Ref Expression
5oalem6  |-  ( ( ( ( ( x  e.  A  /\  y  e.  B )  /\  h  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  D )  /\  h  =  ( z  +h  w ) ) )  /\  ( ( ( f  e.  F  /\  g  e.  G )  /\  h  =  (
f  +h  g ) )  /\  ( ( v  e.  R  /\  u  e.  S )  /\  h  =  (
v  +h  u ) ) ) )  ->  h  e.  ( B  +H  ( A  i^i  ( C  +H  ( ( ( ( A  +H  C
)  i^i  ( B  +H  D ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S
) )  +H  (
( C  +H  R
)  i^i  ( D  +H  S ) ) ) )  i^i  ( ( ( ( A  +H  F )  i^i  ( B  +H  G ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S ) ) ) )  +H  (
( ( C  +H  F )  i^i  ( D  +H  G ) )  i^i  ( ( ( C  +H  R )  i^i  ( D  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S ) ) ) ) ) ) ) ) ) )

Proof of Theorem 5oalem6
StepHypRef Expression
1 an4 831 . . . 4  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  h  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  D )  /\  h  =  ( z  +h  w ) ) )  <-> 
( ( ( x  e.  A  /\  y  e.  B )  /\  (
z  e.  C  /\  w  e.  D )
)  /\  ( h  =  ( x  +h  y )  /\  h  =  ( z  +h  w ) ) ) )
2 an4 831 . . . 4  |-  ( ( ( ( f  e.  F  /\  g  e.  G )  /\  h  =  ( f  +h  g ) )  /\  ( ( v  e.  R  /\  u  e.  S )  /\  h  =  ( v  +h  u ) ) )  <-> 
( ( ( f  e.  F  /\  g  e.  G )  /\  (
v  e.  R  /\  u  e.  S )
)  /\  ( h  =  ( f  +h  g )  /\  h  =  ( v  +h  u ) ) ) )
3 eqeq1 2433 . . . . . . . . . . 11  |-  ( h  =  ( x  +h  y )  ->  (
h  =  ( v  +h  u )  <->  ( x  +h  y )  =  ( v  +h  u ) ) )
43biimpcd 227 . . . . . . . . . 10  |-  ( h  =  ( v  +h  u )  ->  (
h  =  ( x  +h  y )  -> 
( x  +h  y
)  =  ( v  +h  u ) ) )
5 eqeq1 2433 . . . . . . . . . . 11  |-  ( h  =  ( z  +h  w )  ->  (
h  =  ( v  +h  u )  <->  ( z  +h  w )  =  ( v  +h  u ) ) )
65biimpcd 227 . . . . . . . . . 10  |-  ( h  =  ( v  +h  u )  ->  (
h  =  ( z  +h  w )  -> 
( z  +h  w
)  =  ( v  +h  u ) ) )
74, 6anim12d 565 . . . . . . . . 9  |-  ( h  =  ( v  +h  u )  ->  (
( h  =  ( x  +h  y )  /\  h  =  ( z  +h  w ) )  ->  ( (
x  +h  y )  =  ( v  +h  u )  /\  (
z  +h  w )  =  ( v  +h  u ) ) ) )
8 eqeq1 2433 . . . . . . . . . 10  |-  ( h  =  ( f  +h  g )  ->  (
h  =  ( v  +h  u )  <->  ( f  +h  g )  =  ( v  +h  u ) ) )
98biimpcd 227 . . . . . . . . 9  |-  ( h  =  ( v  +h  u )  ->  (
h  =  ( f  +h  g )  -> 
( f  +h  g
)  =  ( v  +h  u ) ) )
107, 9anim12d 565 . . . . . . . 8  |-  ( h  =  ( v  +h  u )  ->  (
( ( h  =  ( x  +h  y
)  /\  h  =  ( z  +h  w
) )  /\  h  =  ( f  +h  g ) )  -> 
( ( ( x  +h  y )  =  ( v  +h  u
)  /\  ( z  +h  w )  =  ( v  +h  u ) )  /\  ( f  +h  g )  =  ( v  +h  u
) ) ) )
1110expdcom 440 . . . . . . 7  |-  ( ( h  =  ( x  +h  y )  /\  h  =  ( z  +h  w ) )  -> 
( h  =  ( f  +h  g )  ->  ( h  =  ( v  +h  u
)  ->  ( (
( x  +h  y
)  =  ( v  +h  u )  /\  ( z  +h  w
)  =  ( v  +h  u ) )  /\  ( f  +h  g )  =  ( v  +h  u ) ) ) ) )
1211imp32 434 . . . . . 6  |-  ( ( ( h  =  ( x  +h  y )  /\  h  =  ( z  +h  w ) )  /\  ( h  =  ( f  +h  g )  /\  h  =  ( v  +h  u ) ) )  ->  ( ( ( x  +h  y )  =  ( v  +h  u )  /\  (
z  +h  w )  =  ( v  +h  u ) )  /\  ( f  +h  g
)  =  ( v  +h  u ) ) )
1312anim2i 571 . . . . 5  |-  ( ( ( ( ( x  e.  A  /\  y  e.  B )  /\  (
z  e.  C  /\  w  e.  D )
)  /\  ( (
f  e.  F  /\  g  e.  G )  /\  ( v  e.  R  /\  u  e.  S
) ) )  /\  ( ( h  =  ( x  +h  y
)  /\  h  =  ( z  +h  w
) )  /\  (
h  =  ( f  +h  g )  /\  h  =  ( v  +h  u ) ) ) )  ->  ( (
( ( x  e.  A  /\  y  e.  B )  /\  (
z  e.  C  /\  w  e.  D )
)  /\  ( (
f  e.  F  /\  g  e.  G )  /\  ( v  e.  R  /\  u  e.  S
) ) )  /\  ( ( ( x  +h  y )  =  ( v  +h  u
)  /\  ( z  +h  w )  =  ( v  +h  u ) )  /\  ( f  +h  g )  =  ( v  +h  u
) ) ) )
1413an4s 833 . . . 4  |-  ( ( ( ( ( x  e.  A  /\  y  e.  B )  /\  (
z  e.  C  /\  w  e.  D )
)  /\  ( h  =  ( x  +h  y )  /\  h  =  ( z  +h  w ) ) )  /\  ( ( ( f  e.  F  /\  g  e.  G )  /\  ( v  e.  R  /\  u  e.  S
) )  /\  (
h  =  ( f  +h  g )  /\  h  =  ( v  +h  u ) ) ) )  ->  ( (
( ( x  e.  A  /\  y  e.  B )  /\  (
z  e.  C  /\  w  e.  D )
)  /\  ( (
f  e.  F  /\  g  e.  G )  /\  ( v  e.  R  /\  u  e.  S
) ) )  /\  ( ( ( x  +h  y )  =  ( v  +h  u
)  /\  ( z  +h  w )  =  ( v  +h  u ) )  /\  ( f  +h  g )  =  ( v  +h  u
) ) ) )
151, 2, 14syl2anb 481 . . 3  |-  ( ( ( ( ( x  e.  A  /\  y  e.  B )  /\  h  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  D )  /\  h  =  ( z  +h  w ) ) )  /\  ( ( ( f  e.  F  /\  g  e.  G )  /\  h  =  (
f  +h  g ) )  /\  ( ( v  e.  R  /\  u  e.  S )  /\  h  =  (
v  +h  u ) ) ) )  -> 
( ( ( ( x  e.  A  /\  y  e.  B )  /\  ( z  e.  C  /\  w  e.  D
) )  /\  (
( f  e.  F  /\  g  e.  G
)  /\  ( v  e.  R  /\  u  e.  S ) ) )  /\  ( ( ( x  +h  y )  =  ( v  +h  u )  /\  (
z  +h  w )  =  ( v  +h  u ) )  /\  ( f  +h  g
)  =  ( v  +h  u ) ) ) )
16 5oalem5.1 . . . 4  |-  A  e.  SH
17 5oalem5.2 . . . 4  |-  B  e.  SH
18 5oalem5.3 . . . 4  |-  C  e.  SH
19 5oalem5.4 . . . 4  |-  D  e.  SH
20 5oalem5.5 . . . 4  |-  F  e.  SH
21 5oalem5.6 . . . 4  |-  G  e.  SH
22 5oalem5.7 . . . 4  |-  R  e.  SH
23 5oalem5.8 . . . 4  |-  S  e.  SH
2416, 17, 18, 19, 20, 21, 22, 235oalem5 27146 . . 3  |-  ( ( ( ( ( x  e.  A  /\  y  e.  B )  /\  (
z  e.  C  /\  w  e.  D )
)  /\  ( (
f  e.  F  /\  g  e.  G )  /\  ( v  e.  R  /\  u  e.  S
) ) )  /\  ( ( ( x  +h  y )  =  ( v  +h  u
)  /\  ( z  +h  w )  =  ( v  +h  u ) )  /\  ( f  +h  g )  =  ( v  +h  u
) ) )  -> 
( x  -h  z
)  e.  ( ( ( ( A  +H  C )  i^i  ( B  +H  D ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S ) )  +H  ( ( C  +H  R )  i^i  ( D  +H  S ) ) ) )  i^i  (
( ( ( A  +H  F )  i^i  ( B  +H  G
) )  i^i  (
( ( A  +H  R )  i^i  ( B  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S
) ) ) )  +H  ( ( ( C  +H  F )  i^i  ( D  +H  G ) )  i^i  ( ( ( C  +H  R )  i^i  ( D  +H  S
) )  +H  (
( F  +H  R
)  i^i  ( G  +H  S ) ) ) ) ) ) )
2515, 24syl 17 . 2  |-  ( ( ( ( ( x  e.  A  /\  y  e.  B )  /\  h  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  D )  /\  h  =  ( z  +h  w ) ) )  /\  ( ( ( f  e.  F  /\  g  e.  G )  /\  h  =  (
f  +h  g ) )  /\  ( ( v  e.  R  /\  u  e.  S )  /\  h  =  (
v  +h  u ) ) ) )  -> 
( x  -h  z
)  e.  ( ( ( ( A  +H  C )  i^i  ( B  +H  D ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S ) )  +H  ( ( C  +H  R )  i^i  ( D  +H  S ) ) ) )  i^i  (
( ( ( A  +H  F )  i^i  ( B  +H  G
) )  i^i  (
( ( A  +H  R )  i^i  ( B  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S
) ) ) )  +H  ( ( ( C  +H  F )  i^i  ( D  +H  G ) )  i^i  ( ( ( C  +H  R )  i^i  ( D  +H  S
) )  +H  (
( F  +H  R
)  i^i  ( G  +H  S ) ) ) ) ) ) )
2616, 18shscli 26805 . . . . . . . . . 10  |-  ( A  +H  C )  e.  SH
2717, 19shscli 26805 . . . . . . . . . 10  |-  ( B  +H  D )  e.  SH
2826, 27shincli 26850 . . . . . . . . 9  |-  ( ( A  +H  C )  i^i  ( B  +H  D ) )  e.  SH
2916, 22shscli 26805 . . . . . . . . . . 11  |-  ( A  +H  R )  e.  SH
3017, 23shscli 26805 . . . . . . . . . . 11  |-  ( B  +H  S )  e.  SH
3129, 30shincli 26850 . . . . . . . . . 10  |-  ( ( A  +H  R )  i^i  ( B  +H  S ) )  e.  SH
3218, 22shscli 26805 . . . . . . . . . . 11  |-  ( C  +H  R )  e.  SH
3319, 23shscli 26805 . . . . . . . . . . 11  |-  ( D  +H  S )  e.  SH
3432, 33shincli 26850 . . . . . . . . . 10  |-  ( ( C  +H  R )  i^i  ( D  +H  S ) )  e.  SH
3531, 34shscli 26805 . . . . . . . . 9  |-  ( ( ( A  +H  R
)  i^i  ( B  +H  S ) )  +H  ( ( C  +H  R )  i^i  ( D  +H  S ) ) )  e.  SH
3628, 35shincli 26850 . . . . . . . 8  |-  ( ( ( A  +H  C
)  i^i  ( B  +H  D ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S
) )  +H  (
( C  +H  R
)  i^i  ( D  +H  S ) ) ) )  e.  SH
3716, 20shscli 26805 . . . . . . . . . . 11  |-  ( A  +H  F )  e.  SH
3817, 21shscli 26805 . . . . . . . . . . 11  |-  ( B  +H  G )  e.  SH
3937, 38shincli 26850 . . . . . . . . . 10  |-  ( ( A  +H  F )  i^i  ( B  +H  G ) )  e.  SH
4020, 22shscli 26805 . . . . . . . . . . . 12  |-  ( F  +H  R )  e.  SH
4121, 23shscli 26805 . . . . . . . . . . . 12  |-  ( G  +H  S )  e.  SH
4240, 41shincli 26850 . . . . . . . . . . 11  |-  ( ( F  +H  R )  i^i  ( G  +H  S ) )  e.  SH
4331, 42shscli 26805 . . . . . . . . . 10  |-  ( ( ( A  +H  R
)  i^i  ( B  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S ) ) )  e.  SH
4439, 43shincli 26850 . . . . . . . . 9  |-  ( ( ( A  +H  F
)  i^i  ( B  +H  G ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S
) )  +H  (
( F  +H  R
)  i^i  ( G  +H  S ) ) ) )  e.  SH
4518, 20shscli 26805 . . . . . . . . . . 11  |-  ( C  +H  F )  e.  SH
4619, 21shscli 26805 . . . . . . . . . . 11  |-  ( D  +H  G )  e.  SH
4745, 46shincli 26850 . . . . . . . . . 10  |-  ( ( C  +H  F )  i^i  ( D  +H  G ) )  e.  SH
4834, 42shscli 26805 . . . . . . . . . 10  |-  ( ( ( C  +H  R
)  i^i  ( D  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S ) ) )  e.  SH
4947, 48shincli 26850 . . . . . . . . 9  |-  ( ( ( C  +H  F
)  i^i  ( D  +H  G ) )  i^i  ( ( ( C  +H  R )  i^i  ( D  +H  S
) )  +H  (
( F  +H  R
)  i^i  ( G  +H  S ) ) ) )  e.  SH
5044, 49shscli 26805 . . . . . . . 8  |-  ( ( ( ( A  +H  F )  i^i  ( B  +H  G ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S ) ) ) )  +H  (
( ( C  +H  F )  i^i  ( D  +H  G ) )  i^i  ( ( ( C  +H  R )  i^i  ( D  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S ) ) ) ) )  e.  SH
5136, 50shincli 26850 . . . . . . 7  |-  ( ( ( ( A  +H  C )  i^i  ( B  +H  D ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S ) )  +H  ( ( C  +H  R )  i^i  ( D  +H  S ) ) ) )  i^i  (
( ( ( A  +H  F )  i^i  ( B  +H  G
) )  i^i  (
( ( A  +H  R )  i^i  ( B  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S
) ) ) )  +H  ( ( ( C  +H  F )  i^i  ( D  +H  G ) )  i^i  ( ( ( C  +H  R )  i^i  ( D  +H  S
) )  +H  (
( F  +H  R
)  i^i  ( G  +H  S ) ) ) ) ) )  e.  SH
5216, 17, 18, 515oalem1 27142 . . . . . 6  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  h  =  ( x  +h  y ) )  /\  ( z  e.  C  /\  ( x  -h  z
)  e.  ( ( ( ( A  +H  C )  i^i  ( B  +H  D ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S ) )  +H  ( ( C  +H  R )  i^i  ( D  +H  S ) ) ) )  i^i  (
( ( ( A  +H  F )  i^i  ( B  +H  G
) )  i^i  (
( ( A  +H  R )  i^i  ( B  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S
) ) ) )  +H  ( ( ( C  +H  F )  i^i  ( D  +H  G ) )  i^i  ( ( ( C  +H  R )  i^i  ( D  +H  S
) )  +H  (
( F  +H  R
)  i^i  ( G  +H  S ) ) ) ) ) ) ) )  ->  h  e.  ( B  +H  ( A  i^i  ( C  +H  ( ( ( ( A  +H  C )  i^i  ( B  +H  D ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S
) )  +H  (
( C  +H  R
)  i^i  ( D  +H  S ) ) ) )  i^i  ( ( ( ( A  +H  F )  i^i  ( B  +H  G ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S ) ) ) )  +H  (
( ( C  +H  F )  i^i  ( D  +H  G ) )  i^i  ( ( ( C  +H  R )  i^i  ( D  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S ) ) ) ) ) ) ) ) ) )
5352expr 618 . . . . 5  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  h  =  ( x  +h  y ) )  /\  z  e.  C )  ->  ( ( x  -h  z )  e.  ( ( ( ( A  +H  C )  i^i  ( B  +H  D
) )  i^i  (
( ( A  +H  R )  i^i  ( B  +H  S ) )  +H  ( ( C  +H  R )  i^i  ( D  +H  S
) ) ) )  i^i  ( ( ( ( A  +H  F
)  i^i  ( B  +H  G ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S
) )  +H  (
( F  +H  R
)  i^i  ( G  +H  S ) ) ) )  +H  ( ( ( C  +H  F
)  i^i  ( D  +H  G ) )  i^i  ( ( ( C  +H  R )  i^i  ( D  +H  S
) )  +H  (
( F  +H  R
)  i^i  ( G  +H  S ) ) ) ) ) )  ->  h  e.  ( B  +H  ( A  i^i  ( C  +H  ( ( ( ( A  +H  C
)  i^i  ( B  +H  D ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S
) )  +H  (
( C  +H  R
)  i^i  ( D  +H  S ) ) ) )  i^i  ( ( ( ( A  +H  F )  i^i  ( B  +H  G ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S ) ) ) )  +H  (
( ( C  +H  F )  i^i  ( D  +H  G ) )  i^i  ( ( ( C  +H  R )  i^i  ( D  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S ) ) ) ) ) ) ) ) ) ) )
5453adantrr 721 . . . 4  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  h  =  ( x  +h  y ) )  /\  ( z  e.  C  /\  w  e.  D
) )  ->  (
( x  -h  z
)  e.  ( ( ( ( A  +H  C )  i^i  ( B  +H  D ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S ) )  +H  ( ( C  +H  R )  i^i  ( D  +H  S ) ) ) )  i^i  (
( ( ( A  +H  F )  i^i  ( B  +H  G
) )  i^i  (
( ( A  +H  R )  i^i  ( B  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S
) ) ) )  +H  ( ( ( C  +H  F )  i^i  ( D  +H  G ) )  i^i  ( ( ( C  +H  R )  i^i  ( D  +H  S
) )  +H  (
( F  +H  R
)  i^i  ( G  +H  S ) ) ) ) ) )  ->  h  e.  ( B  +H  ( A  i^i  ( C  +H  ( ( ( ( A  +H  C
)  i^i  ( B  +H  D ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S
) )  +H  (
( C  +H  R
)  i^i  ( D  +H  S ) ) ) )  i^i  ( ( ( ( A  +H  F )  i^i  ( B  +H  G ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S ) ) ) )  +H  (
( ( C  +H  F )  i^i  ( D  +H  G ) )  i^i  ( ( ( C  +H  R )  i^i  ( D  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S ) ) ) ) ) ) ) ) ) ) )
5554adantrr 721 . . 3  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  h  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  D )  /\  h  =  ( z  +h  w ) ) )  ->  ( ( x  -h  z )  e.  ( ( ( ( A  +H  C )  i^i  ( B  +H  D ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S
) )  +H  (
( C  +H  R
)  i^i  ( D  +H  S ) ) ) )  i^i  ( ( ( ( A  +H  F )  i^i  ( B  +H  G ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S ) ) ) )  +H  (
( ( C  +H  F )  i^i  ( D  +H  G ) )  i^i  ( ( ( C  +H  R )  i^i  ( D  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S ) ) ) ) ) )  ->  h  e.  ( B  +H  ( A  i^i  ( C  +H  ( ( ( ( A  +H  C )  i^i  ( B  +H  D ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S
) )  +H  (
( C  +H  R
)  i^i  ( D  +H  S ) ) ) )  i^i  ( ( ( ( A  +H  F )  i^i  ( B  +H  G ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S ) ) ) )  +H  (
( ( C  +H  F )  i^i  ( D  +H  G ) )  i^i  ( ( ( C  +H  R )  i^i  ( D  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S ) ) ) ) ) ) ) ) ) ) )
5655adantr 466 . 2  |-  ( ( ( ( ( x  e.  A  /\  y  e.  B )  /\  h  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  D )  /\  h  =  ( z  +h  w ) ) )  /\  ( ( ( f  e.  F  /\  g  e.  G )  /\  h  =  (
f  +h  g ) )  /\  ( ( v  e.  R  /\  u  e.  S )  /\  h  =  (
v  +h  u ) ) ) )  -> 
( ( x  -h  z )  e.  ( ( ( ( A  +H  C )  i^i  ( B  +H  D
) )  i^i  (
( ( A  +H  R )  i^i  ( B  +H  S ) )  +H  ( ( C  +H  R )  i^i  ( D  +H  S
) ) ) )  i^i  ( ( ( ( A  +H  F
)  i^i  ( B  +H  G ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S
) )  +H  (
( F  +H  R
)  i^i  ( G  +H  S ) ) ) )  +H  ( ( ( C  +H  F
)  i^i  ( D  +H  G ) )  i^i  ( ( ( C  +H  R )  i^i  ( D  +H  S
) )  +H  (
( F  +H  R
)  i^i  ( G  +H  S ) ) ) ) ) )  ->  h  e.  ( B  +H  ( A  i^i  ( C  +H  ( ( ( ( A  +H  C
)  i^i  ( B  +H  D ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S
) )  +H  (
( C  +H  R
)  i^i  ( D  +H  S ) ) ) )  i^i  ( ( ( ( A  +H  F )  i^i  ( B  +H  G ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S ) ) ) )  +H  (
( ( C  +H  F )  i^i  ( D  +H  G ) )  i^i  ( ( ( C  +H  R )  i^i  ( D  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S ) ) ) ) ) ) ) ) ) ) )
5725, 56mpd 15 1  |-  ( ( ( ( ( x  e.  A  /\  y  e.  B )  /\  h  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  D )  /\  h  =  ( z  +h  w ) ) )  /\  ( ( ( f  e.  F  /\  g  e.  G )  /\  h  =  (
f  +h  g ) )  /\  ( ( v  e.  R  /\  u  e.  S )  /\  h  =  (
v  +h  u ) ) ) )  ->  h  e.  ( B  +H  ( A  i^i  ( C  +H  ( ( ( ( A  +H  C
)  i^i  ( B  +H  D ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S
) )  +H  (
( C  +H  R
)  i^i  ( D  +H  S ) ) ) )  i^i  ( ( ( ( A  +H  F )  i^i  ( B  +H  G ) )  i^i  ( ( ( A  +H  R )  i^i  ( B  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S ) ) ) )  +H  (
( ( C  +H  F )  i^i  ( D  +H  G ) )  i^i  ( ( ( C  +H  R )  i^i  ( D  +H  S ) )  +H  ( ( F  +H  R )  i^i  ( G  +H  S ) ) ) ) ) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870    i^i cin 3441  (class class class)co 6305    +h cva 26408    -h cmv 26413   SHcsh 26416    +H cph 26419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-hilex 26487  ax-hfvadd 26488  ax-hvcom 26489  ax-hvass 26490  ax-hv0cl 26491  ax-hvaddid 26492  ax-hfvmul 26493  ax-hvmulid 26494  ax-hvmulass 26495  ax-hvdistr1 26496  ax-hvdistr2 26497  ax-hvmul0 26498
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-ltxr 9679  df-sub 9861  df-neg 9862  df-nn 10610  df-grpo 25764  df-ablo 25855  df-hvsub 26459  df-hlim 26460  df-sh 26695  df-ch 26709  df-shs 26796
This theorem is referenced by:  5oalem7  27148
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