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Theorem 5oalem3 22251
Description: Lemma for orthoarguesian law 5OA. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
Hypotheses
Ref Expression
5oalem3.1  |-  A  e.  SH
5oalem3.2  |-  B  e.  SH
5oalem3.3  |-  C  e.  SH
5oalem3.4  |-  D  e.  SH
5oalem3.5  |-  F  e.  SH
5oalem3.6  |-  G  e.  SH
Assertion
Ref Expression
5oalem3  |-  ( ( ( ( ( x  e.  A  /\  y  e.  B )  /\  (
z  e.  C  /\  w  e.  D )
)  /\  ( f  e.  F  /\  g  e.  G ) )  /\  ( ( x  +h  y )  =  ( f  +h  g )  /\  ( z  +h  w )  =  ( f  +h  g ) ) )  ->  (
x  -h  z )  e.  ( ( ( A  +H  F )  i^i  ( B  +H  G ) )  +H  ( ( C  +H  F )  i^i  ( D  +H  G ) ) ) )

Proof of Theorem 5oalem3
StepHypRef Expression
1 anandir 802 . . . 4  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  (
z  e.  C  /\  w  e.  D )
)  /\  ( f  e.  F  /\  g  e.  G ) )  <->  ( (
( x  e.  A  /\  y  e.  B
)  /\  ( f  e.  F  /\  g  e.  G ) )  /\  ( ( z  e.  C  /\  w  e.  D )  /\  (
f  e.  F  /\  g  e.  G )
) ) )
2 5oalem3.1 . . . . . . 7  |-  A  e.  SH
3 5oalem3.2 . . . . . . 7  |-  B  e.  SH
4 5oalem3.5 . . . . . . 7  |-  F  e.  SH
5 5oalem3.6 . . . . . . 7  |-  G  e.  SH
62, 3, 4, 55oalem2 22250 . . . . . 6  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  (
f  e.  F  /\  g  e.  G )
)  /\  ( x  +h  y )  =  ( f  +h  g ) )  ->  ( x  -h  f )  e.  ( ( A  +H  F
)  i^i  ( B  +H  G ) ) )
7 5oalem3.3 . . . . . . 7  |-  C  e.  SH
8 5oalem3.4 . . . . . . 7  |-  D  e.  SH
97, 8, 4, 55oalem2 22250 . . . . . 6  |-  ( ( ( ( z  e.  C  /\  w  e.  D )  /\  (
f  e.  F  /\  g  e.  G )
)  /\  ( z  +h  w )  =  ( f  +h  g ) )  ->  ( z  -h  f )  e.  ( ( C  +H  F
)  i^i  ( D  +H  G ) ) )
106, 9anim12i 549 . . . . 5  |-  ( ( ( ( ( x  e.  A  /\  y  e.  B )  /\  (
f  e.  F  /\  g  e.  G )
)  /\  ( x  +h  y )  =  ( f  +h  g ) )  /\  ( ( ( z  e.  C  /\  w  e.  D
)  /\  ( f  e.  F  /\  g  e.  G ) )  /\  ( z  +h  w
)  =  ( f  +h  g ) ) )  ->  ( (
x  -h  f )  e.  ( ( A  +H  F )  i^i  ( B  +H  G
) )  /\  (
z  -h  f )  e.  ( ( C  +H  F )  i^i  ( D  +H  G
) ) ) )
1110an4s 799 . . . 4  |-  ( ( ( ( ( x  e.  A  /\  y  e.  B )  /\  (
f  e.  F  /\  g  e.  G )
)  /\  ( (
z  e.  C  /\  w  e.  D )  /\  ( f  e.  F  /\  g  e.  G
) ) )  /\  ( ( x  +h  y )  =  ( f  +h  g )  /\  ( z  +h  w )  =  ( f  +h  g ) ) )  ->  (
( x  -h  f
)  e.  ( ( A  +H  F )  i^i  ( B  +H  G ) )  /\  ( z  -h  f
)  e.  ( ( C  +H  F )  i^i  ( D  +H  G ) ) ) )
121, 11sylanb 458 . . 3  |-  ( ( ( ( ( x  e.  A  /\  y  e.  B )  /\  (
z  e.  C  /\  w  e.  D )
)  /\  ( f  e.  F  /\  g  e.  G ) )  /\  ( ( x  +h  y )  =  ( f  +h  g )  /\  ( z  +h  w )  =  ( f  +h  g ) ) )  ->  (
( x  -h  f
)  e.  ( ( A  +H  F )  i^i  ( B  +H  G ) )  /\  ( z  -h  f
)  e.  ( ( C  +H  F )  i^i  ( D  +H  G ) ) ) )
132, 4shscli 21912 . . . . 5  |-  ( A  +H  F )  e.  SH
143, 5shscli 21912 . . . . 5  |-  ( B  +H  G )  e.  SH
1513, 14shincli 21957 . . . 4  |-  ( ( A  +H  F )  i^i  ( B  +H  G ) )  e.  SH
167, 4shscli 21912 . . . . 5  |-  ( C  +H  F )  e.  SH
178, 5shscli 21912 . . . . 5  |-  ( D  +H  G )  e.  SH
1816, 17shincli 21957 . . . 4  |-  ( ( C  +H  F )  i^i  ( D  +H  G ) )  e.  SH
1915, 18shsvsi 21962 . . 3  |-  ( ( ( x  -h  f
)  e.  ( ( A  +H  F )  i^i  ( B  +H  G ) )  /\  ( z  -h  f
)  e.  ( ( C  +H  F )  i^i  ( D  +H  G ) ) )  ->  ( ( x  -h  f )  -h  ( z  -h  f
) )  e.  ( ( ( A  +H  F )  i^i  ( B  +H  G ) )  +H  ( ( C  +H  F )  i^i  ( D  +H  G
) ) ) )
2012, 19syl 15 . 2  |-  ( ( ( ( ( x  e.  A  /\  y  e.  B )  /\  (
z  e.  C  /\  w  e.  D )
)  /\  ( f  e.  F  /\  g  e.  G ) )  /\  ( ( x  +h  y )  =  ( f  +h  g )  /\  ( z  +h  w )  =  ( f  +h  g ) ) )  ->  (
( x  -h  f
)  -h  ( z  -h  f ) )  e.  ( ( ( A  +H  F )  i^i  ( B  +H  G ) )  +H  ( ( C  +H  F )  i^i  ( D  +H  G ) ) ) )
212sheli 21809 . . . . . . 7  |-  ( x  e.  A  ->  x  e.  ~H )
2221adantr 451 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  B )  ->  x  e.  ~H )
237sheli 21809 . . . . . . 7  |-  ( z  e.  C  ->  z  e.  ~H )
2423adantr 451 . . . . . 6  |-  ( ( z  e.  C  /\  w  e.  D )  ->  z  e.  ~H )
2522, 24anim12i 549 . . . . 5  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  ( z  e.  C  /\  w  e.  D ) )  -> 
( x  e.  ~H  /\  z  e.  ~H )
)
264sheli 21809 . . . . . 6  |-  ( f  e.  F  ->  f  e.  ~H )
2726adantr 451 . . . . 5  |-  ( ( f  e.  F  /\  g  e.  G )  ->  f  e.  ~H )
28 hvsubsub4 21655 . . . . . . 7  |-  ( ( ( x  e.  ~H  /\  f  e.  ~H )  /\  ( z  e.  ~H  /\  f  e.  ~H )
)  ->  ( (
x  -h  f )  -h  ( z  -h  f ) )  =  ( ( x  -h  z )  -h  (
f  -h  f ) ) )
2928anandirs 804 . . . . . 6  |-  ( ( ( x  e.  ~H  /\  z  e.  ~H )  /\  f  e.  ~H )  ->  ( ( x  -h  f )  -h  ( z  -h  f
) )  =  ( ( x  -h  z
)  -h  ( f  -h  f ) ) )
30 hvsubid 21621 . . . . . . . 8  |-  ( f  e.  ~H  ->  (
f  -h  f )  =  0h )
3130oveq2d 5890 . . . . . . 7  |-  ( f  e.  ~H  ->  (
( x  -h  z
)  -h  ( f  -h  f ) )  =  ( ( x  -h  z )  -h 
0h ) )
32 hvsubcl 21613 . . . . . . . 8  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( x  -h  z
)  e.  ~H )
33 hvsub0 21671 . . . . . . . 8  |-  ( ( x  -h  z )  e.  ~H  ->  (
( x  -h  z
)  -h  0h )  =  ( x  -h  z ) )
3432, 33syl 15 . . . . . . 7  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( ( x  -h  z )  -h  0h )  =  ( x  -h  z ) )
3531, 34sylan9eqr 2350 . . . . . 6  |-  ( ( ( x  e.  ~H  /\  z  e.  ~H )  /\  f  e.  ~H )  ->  ( ( x  -h  z )  -h  ( f  -h  f
) )  =  ( x  -h  z ) )
3629, 35eqtrd 2328 . . . . 5  |-  ( ( ( x  e.  ~H  /\  z  e.  ~H )  /\  f  e.  ~H )  ->  ( ( x  -h  f )  -h  ( z  -h  f
) )  =  ( x  -h  z ) )
3725, 27, 36syl2an 463 . . . 4  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  (
z  e.  C  /\  w  e.  D )
)  /\  ( f  e.  F  /\  g  e.  G ) )  -> 
( ( x  -h  f )  -h  (
z  -h  f ) )  =  ( x  -h  z ) )
3837eleq1d 2362 . . 3  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  (
z  e.  C  /\  w  e.  D )
)  /\  ( f  e.  F  /\  g  e.  G ) )  -> 
( ( ( x  -h  f )  -h  ( z  -h  f
) )  e.  ( ( ( A  +H  F )  i^i  ( B  +H  G ) )  +H  ( ( C  +H  F )  i^i  ( D  +H  G
) ) )  <->  ( x  -h  z )  e.  ( ( ( A  +H  F )  i^i  ( B  +H  G ) )  +H  ( ( C  +H  F )  i^i  ( D  +H  G
) ) ) ) )
3938adantr 451 . 2  |-  ( ( ( ( ( x  e.  A  /\  y  e.  B )  /\  (
z  e.  C  /\  w  e.  D )
)  /\  ( f  e.  F  /\  g  e.  G ) )  /\  ( ( x  +h  y )  =  ( f  +h  g )  /\  ( z  +h  w )  =  ( f  +h  g ) ) )  ->  (
( ( x  -h  f )  -h  (
z  -h  f ) )  e.  ( ( ( A  +H  F
)  i^i  ( B  +H  G ) )  +H  ( ( C  +H  F )  i^i  ( D  +H  G ) ) )  <->  ( x  -h  z )  e.  ( ( ( A  +H  F )  i^i  ( B  +H  G ) )  +H  ( ( C  +H  F )  i^i  ( D  +H  G
) ) ) ) )
4020, 39mpbid 201 1  |-  ( ( ( ( ( x  e.  A  /\  y  e.  B )  /\  (
z  e.  C  /\  w  e.  D )
)  /\  ( f  e.  F  /\  g  e.  G ) )  /\  ( ( x  +h  y )  =  ( f  +h  g )  /\  ( z  +h  w )  =  ( f  +h  g ) ) )  ->  (
x  -h  z )  e.  ( ( ( A  +H  F )  i^i  ( B  +H  G ) )  +H  ( ( C  +H  F )  i^i  ( D  +H  G ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    i^i cin 3164  (class class class)co 5874   ~Hchil 21515    +h cva 21516   0hc0v 21520    -h cmv 21521   SHcsh 21524    +H cph 21527
This theorem is referenced by:  5oalem4  22252
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-hilex 21595  ax-hfvadd 21596  ax-hvcom 21597  ax-hvass 21598  ax-hv0cl 21599  ax-hvaddid 21600  ax-hfvmul 21601  ax-hvmulid 21602  ax-hvmulass 21603  ax-hvdistr1 21604  ax-hvdistr2 21605  ax-hvmul0 21606
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-ltxr 8888  df-sub 9055  df-neg 9056  df-nn 9763  df-grpo 20874  df-ablo 20965  df-hvsub 21567  df-hlim 21568  df-sh 21802  df-ch 21817  df-shs 21903
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