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Theorem 5oalem3 25010
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 825 . . . 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 25009 . . . . . 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 25009 . . . . . 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 566 . . . . 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 822 . . . 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 472 . . 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 24671 . . . . 5  |-  ( A  +H  F )  e.  SH
143, 5shscli 24671 . . . . 5  |-  ( B  +H  G )  e.  SH
1513, 14shincli 24716 . . . 4  |-  ( ( A  +H  F )  i^i  ( B  +H  G ) )  e.  SH
167, 4shscli 24671 . . . . 5  |-  ( C  +H  F )  e.  SH
178, 5shscli 24671 . . . . 5  |-  ( D  +H  G )  e.  SH
1816, 17shincli 24716 . . . 4  |-  ( ( C  +H  F )  i^i  ( D  +H  G ) )  e.  SH
1915, 18shsvsi 24721 . . 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 24567 . . . . . . 7  |-  ( x  e.  A  ->  x  e.  ~H )
2221adantr 465 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  B )  ->  x  e.  ~H )
237sheli 24567 . . . . . . 7  |-  ( z  e.  C  ->  z  e.  ~H )
2423adantr 465 . . . . . 6  |-  ( ( z  e.  C  /\  w  e.  D )  ->  z  e.  ~H )
2522, 24anim12i 566 . . . . 5  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  ( z  e.  C  /\  w  e.  D ) )  -> 
( x  e.  ~H  /\  z  e.  ~H )
)
264sheli 24567 . . . . . 6  |-  ( f  e.  F  ->  f  e.  ~H )
2726adantr 465 . . . . 5  |-  ( ( f  e.  F  /\  g  e.  G )  ->  f  e.  ~H )
28 hvsubsub4 24413 . . . . . . 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 827 . . . . . 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 24379 . . . . . . . 8  |-  ( f  e.  ~H  ->  (
f  -h  f )  =  0h )
3130oveq2d 6102 . . . . . . 7  |-  ( f  e.  ~H  ->  (
( x  -h  z
)  -h  ( f  -h  f ) )  =  ( ( x  -h  z )  -h 
0h ) )
32 hvsubcl 24370 . . . . . . . 8  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( x  -h  z
)  e.  ~H )
33 hvsub0 24429 . . . . . . . 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 2492 . . . . . 6  |-  ( ( ( x  e.  ~H  /\  z  e.  ~H )  /\  f  e.  ~H )  ->  ( ( x  -h  z )  -h  ( f  -h  f
) )  =  ( x  -h  z ) )
3629, 35eqtrd 2470 . . . . 5  |-  ( ( ( x  e.  ~H  /\  z  e.  ~H )  /\  f  e.  ~H )  ->  ( ( x  -h  f )  -h  ( z  -h  f
) )  =  ( x  -h  z ) )
3725, 27, 36syl2an 477 . . . 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 2504 . . 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 465 . 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 210 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 184    /\ wa 369    = wceq 1369    e. wcel 1756    i^i cin 3322  (class class class)co 6086   ~Hchil 24272    +h cva 24273   0hc0v 24277    -h cmv 24278   SHcsh 24281    +H cph 24284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-hilex 24352  ax-hfvadd 24353  ax-hvcom 24354  ax-hvass 24355  ax-hv0cl 24356  ax-hvaddid 24357  ax-hfvmul 24358  ax-hvmulid 24359  ax-hvmulass 24360  ax-hvdistr1 24361  ax-hvdistr2 24362  ax-hvmul0 24363
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-recs 6824  df-rdg 6858  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-ltxr 9415  df-sub 9589  df-neg 9590  df-nn 10315  df-grpo 23629  df-ablo 23720  df-hvsub 24324  df-hlim 24325  df-sh 24560  df-ch 24575  df-shs 24662
This theorem is referenced by:  5oalem4  25011
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