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Theorem 5oalem3 27309
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 838 . . . 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 27308 . . . . . 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 27308 . . . . . 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 570 . . . . 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 835 . . . 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 475 . . 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 26970 . . . . 5  |-  ( A  +H  F )  e.  SH
143, 5shscli 26970 . . . . 5  |-  ( B  +H  G )  e.  SH
1513, 14shincli 27015 . . . 4  |-  ( ( A  +H  F )  i^i  ( B  +H  G ) )  e.  SH
167, 4shscli 26970 . . . . 5  |-  ( C  +H  F )  e.  SH
178, 5shscli 26970 . . . . 5  |-  ( D  +H  G )  e.  SH
1816, 17shincli 27015 . . . 4  |-  ( ( C  +H  F )  i^i  ( D  +H  G ) )  e.  SH
1915, 18shsvsi 27020 . . 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 17 . 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 26867 . . . . . . 7  |-  ( x  e.  A  ->  x  e.  ~H )
2221adantr 467 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  B )  ->  x  e.  ~H )
237sheli 26867 . . . . . . 7  |-  ( z  e.  C  ->  z  e.  ~H )
2423adantr 467 . . . . . 6  |-  ( ( z  e.  C  /\  w  e.  D )  ->  z  e.  ~H )
2522, 24anim12i 570 . . . . 5  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  ( z  e.  C  /\  w  e.  D ) )  -> 
( x  e.  ~H  /\  z  e.  ~H )
)
264sheli 26867 . . . . . 6  |-  ( f  e.  F  ->  f  e.  ~H )
2726adantr 467 . . . . 5  |-  ( ( f  e.  F  /\  g  e.  G )  ->  f  e.  ~H )
28 hvsubsub4 26713 . . . . . . 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 840 . . . . . 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 26679 . . . . . . . 8  |-  ( f  e.  ~H  ->  (
f  -h  f )  =  0h )
3130oveq2d 6306 . . . . . . 7  |-  ( f  e.  ~H  ->  (
( x  -h  z
)  -h  ( f  -h  f ) )  =  ( ( x  -h  z )  -h 
0h ) )
32 hvsubcl 26670 . . . . . . . 8  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( x  -h  z
)  e.  ~H )
33 hvsub0 26729 . . . . . . . 8  |-  ( ( x  -h  z )  e.  ~H  ->  (
( x  -h  z
)  -h  0h )  =  ( x  -h  z ) )
3432, 33syl 17 . . . . . . 7  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( ( x  -h  z )  -h  0h )  =  ( x  -h  z ) )
3531, 34sylan9eqr 2507 . . . . . 6  |-  ( ( ( x  e.  ~H  /\  z  e.  ~H )  /\  f  e.  ~H )  ->  ( ( x  -h  z )  -h  ( f  -h  f
) )  =  ( x  -h  z ) )
3629, 35eqtrd 2485 . . . . 5  |-  ( ( ( x  e.  ~H  /\  z  e.  ~H )  /\  f  e.  ~H )  ->  ( ( x  -h  f )  -h  ( z  -h  f
) )  =  ( x  -h  z ) )
3725, 27, 36syl2an 480 . . . 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 2513 . . 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 467 . 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 214 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 setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887    i^i cin 3403  (class class class)co 6290   ~Hchil 26572    +h cva 26573   0hc0v 26577    -h cmv 26578   SHcsh 26581    +H cph 26584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-hilex 26652  ax-hfvadd 26653  ax-hvcom 26654  ax-hvass 26655  ax-hv0cl 26656  ax-hvaddid 26657  ax-hfvmul 26658  ax-hvmulid 26659  ax-hvmulass 26660  ax-hvdistr1 26661  ax-hvdistr2 26662  ax-hvmul0 26663
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-ltxr 9680  df-sub 9862  df-neg 9863  df-nn 10610  df-grpo 25919  df-ablo 26010  df-hvsub 26624  df-hlim 26625  df-sh 26860  df-ch 26874  df-shs 26961
This theorem is referenced by:  5oalem4  27310
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