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Theorem 5oalem3 23150
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 803 . . . 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 23149 . . . . . 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 23149 . . . . . 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 550 . . . . 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 800 . . . 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 459 . . 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 22811 . . . . 5  |-  ( A  +H  F )  e.  SH
143, 5shscli 22811 . . . . 5  |-  ( B  +H  G )  e.  SH
1513, 14shincli 22856 . . . 4  |-  ( ( A  +H  F )  i^i  ( B  +H  G ) )  e.  SH
167, 4shscli 22811 . . . . 5  |-  ( C  +H  F )  e.  SH
178, 5shscli 22811 . . . . 5  |-  ( D  +H  G )  e.  SH
1816, 17shincli 22856 . . . 4  |-  ( ( C  +H  F )  i^i  ( D  +H  G ) )  e.  SH
1915, 18shsvsi 22861 . . 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 16 . 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 22708 . . . . . . 7  |-  ( x  e.  A  ->  x  e.  ~H )
2221adantr 452 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  B )  ->  x  e.  ~H )
237sheli 22708 . . . . . . 7  |-  ( z  e.  C  ->  z  e.  ~H )
2423adantr 452 . . . . . 6  |-  ( ( z  e.  C  /\  w  e.  D )  ->  z  e.  ~H )
2522, 24anim12i 550 . . . . 5  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  ( z  e.  C  /\  w  e.  D ) )  -> 
( x  e.  ~H  /\  z  e.  ~H )
)
264sheli 22708 . . . . . 6  |-  ( f  e.  F  ->  f  e.  ~H )
2726adantr 452 . . . . 5  |-  ( ( f  e.  F  /\  g  e.  G )  ->  f  e.  ~H )
28 hvsubsub4 22554 . . . . . . 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 805 . . . . . 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 22520 . . . . . . . 8  |-  ( f  e.  ~H  ->  (
f  -h  f )  =  0h )
3130oveq2d 6089 . . . . . . 7  |-  ( f  e.  ~H  ->  (
( x  -h  z
)  -h  ( f  -h  f ) )  =  ( ( x  -h  z )  -h 
0h ) )
32 hvsubcl 22512 . . . . . . . 8  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( x  -h  z
)  e.  ~H )
33 hvsub0 22570 . . . . . . . 8  |-  ( ( x  -h  z )  e.  ~H  ->  (
( x  -h  z
)  -h  0h )  =  ( x  -h  z ) )
3432, 33syl 16 . . . . . . 7  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( ( x  -h  z )  -h  0h )  =  ( x  -h  z ) )
3531, 34sylan9eqr 2489 . . . . . 6  |-  ( ( ( x  e.  ~H  /\  z  e.  ~H )  /\  f  e.  ~H )  ->  ( ( x  -h  z )  -h  ( f  -h  f
) )  =  ( x  -h  z ) )
3629, 35eqtrd 2467 . . . . 5  |-  ( ( ( x  e.  ~H  /\  z  e.  ~H )  /\  f  e.  ~H )  ->  ( ( x  -h  f )  -h  ( z  -h  f
) )  =  ( x  -h  z ) )
3725, 27, 36syl2an 464 . . . 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 2501 . . 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 452 . 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 202 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 177    /\ wa 359    = wceq 1652    e. wcel 1725    i^i cin 3311  (class class class)co 6073   ~Hchil 22414    +h cva 22415   0hc0v 22419    -h cmv 22420   SHcsh 22423    +H cph 22426
This theorem is referenced by:  5oalem4  23151
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-hilex 22494  ax-hfvadd 22495  ax-hvcom 22496  ax-hvass 22497  ax-hv0cl 22498  ax-hvaddid 22499  ax-hfvmul 22500  ax-hvmulid 22501  ax-hvmulass 22502  ax-hvdistr1 22503  ax-hvdistr2 22504  ax-hvmul0 22505
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-ltxr 9117  df-sub 9285  df-neg 9286  df-nn 9993  df-grpo 21771  df-ablo 21862  df-hvsub 22466  df-hlim 22467  df-sh 22701  df-ch 22716  df-shs 22802
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