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Theorem 5oalem3 26974
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 830 . . . 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 26973 . . . . . 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 26973 . . . . . 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 564 . . . . 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 827 . . . 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 470 . . 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 26635 . . . . 5  |-  ( A  +H  F )  e.  SH
143, 5shscli 26635 . . . . 5  |-  ( B  +H  G )  e.  SH
1513, 14shincli 26680 . . . 4  |-  ( ( A  +H  F )  i^i  ( B  +H  G ) )  e.  SH
167, 4shscli 26635 . . . . 5  |-  ( C  +H  F )  e.  SH
178, 5shscli 26635 . . . . 5  |-  ( D  +H  G )  e.  SH
1816, 17shincli 26680 . . . 4  |-  ( ( C  +H  F )  i^i  ( D  +H  G ) )  e.  SH
1915, 18shsvsi 26685 . . 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 26531 . . . . . . 7  |-  ( x  e.  A  ->  x  e.  ~H )
2221adantr 463 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  B )  ->  x  e.  ~H )
237sheli 26531 . . . . . . 7  |-  ( z  e.  C  ->  z  e.  ~H )
2423adantr 463 . . . . . 6  |-  ( ( z  e.  C  /\  w  e.  D )  ->  z  e.  ~H )
2522, 24anim12i 564 . . . . 5  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  ( z  e.  C  /\  w  e.  D ) )  -> 
( x  e.  ~H  /\  z  e.  ~H )
)
264sheli 26531 . . . . . 6  |-  ( f  e.  F  ->  f  e.  ~H )
2726adantr 463 . . . . 5  |-  ( ( f  e.  F  /\  g  e.  G )  ->  f  e.  ~H )
28 hvsubsub4 26377 . . . . . . 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 832 . . . . . 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 26343 . . . . . . . 8  |-  ( f  e.  ~H  ->  (
f  -h  f )  =  0h )
3130oveq2d 6293 . . . . . . 7  |-  ( f  e.  ~H  ->  (
( x  -h  z
)  -h  ( f  -h  f ) )  =  ( ( x  -h  z )  -h 
0h ) )
32 hvsubcl 26334 . . . . . . . 8  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( x  -h  z
)  e.  ~H )
33 hvsub0 26393 . . . . . . . 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 2465 . . . . . 6  |-  ( ( ( x  e.  ~H  /\  z  e.  ~H )  /\  f  e.  ~H )  ->  ( ( x  -h  z )  -h  ( f  -h  f
) )  =  ( x  -h  z ) )
3629, 35eqtrd 2443 . . . . 5  |-  ( ( ( x  e.  ~H  /\  z  e.  ~H )  /\  f  e.  ~H )  ->  ( ( x  -h  f )  -h  ( z  -h  f
) )  =  ( x  -h  z ) )
3725, 27, 36syl2an 475 . . . 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 2471 . . 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 463 . 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 367    = wceq 1405    e. wcel 1842    i^i cin 3412  (class class class)co 6277   ~Hchil 26236    +h cva 26237   0hc0v 26241    -h cmv 26242   SHcsh 26245    +H cph 26248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-hilex 26316  ax-hfvadd 26317  ax-hvcom 26318  ax-hvass 26319  ax-hv0cl 26320  ax-hvaddid 26321  ax-hfvmul 26322  ax-hvmulid 26323  ax-hvmulass 26324  ax-hvdistr1 26325  ax-hvdistr2 26326  ax-hvmul0 26327
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-er 7347  df-map 7458  df-en 7554  df-dom 7555  df-sdom 7556  df-pnf 9659  df-mnf 9660  df-ltxr 9662  df-sub 9842  df-neg 9843  df-nn 10576  df-grpo 25593  df-ablo 25684  df-hvsub 26288  df-hlim 26289  df-sh 26524  df-ch 26539  df-shs 26626
This theorem is referenced by:  5oalem4  26975
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