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Theorem 5oalem2 23110
Description: Lemma for orthoarguesian law 5OA. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
Hypotheses
Ref Expression
5oalem2.1  |-  A  e.  SH
5oalem2.2  |-  B  e.  SH
5oalem2.3  |-  C  e.  SH
5oalem2.4  |-  D  e.  SH
Assertion
Ref Expression
5oalem2  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  (
z  e.  C  /\  w  e.  D )
)  /\  ( x  +h  y )  =  ( z  +h  w ) )  ->  ( x  -h  z )  e.  ( ( A  +H  C
)  i^i  ( B  +H  D ) ) )

Proof of Theorem 5oalem2
StepHypRef Expression
1 5oalem2.1 . . . . 5  |-  A  e.  SH
2 5oalem2.3 . . . . 5  |-  C  e.  SH
31, 2shsvsi 22822 . . . 4  |-  ( ( x  e.  A  /\  z  e.  C )  ->  ( x  -h  z
)  e.  ( A  +H  C ) )
43ad2ant2r 728 . . 3  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  ( z  e.  C  /\  w  e.  D ) )  -> 
( x  -h  z
)  e.  ( A  +H  C ) )
54adantr 452 . 2  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  (
z  e.  C  /\  w  e.  D )
)  /\  ( x  +h  y )  =  ( z  +h  w ) )  ->  ( x  -h  z )  e.  ( A  +H  C ) )
6 5oalem2.4 . . . . . . . 8  |-  D  e.  SH
7 5oalem2.2 . . . . . . . 8  |-  B  e.  SH
86, 7shsvsi 22822 . . . . . . 7  |-  ( ( w  e.  D  /\  y  e.  B )  ->  ( w  -h  y
)  e.  ( D  +H  B ) )
98ancoms 440 . . . . . 6  |-  ( ( y  e.  B  /\  w  e.  D )  ->  ( w  -h  y
)  e.  ( D  +H  B ) )
107, 6shscomi 22818 . . . . . 6  |-  ( B  +H  D )  =  ( D  +H  B
)
119, 10syl6eleqr 2495 . . . . 5  |-  ( ( y  e.  B  /\  w  e.  D )  ->  ( w  -h  y
)  e.  ( B  +H  D ) )
1211ad2ant2l 727 . . . 4  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  ( z  e.  C  /\  w  e.  D ) )  -> 
( w  -h  y
)  e.  ( B  +H  D ) )
1312adantr 452 . . 3  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  (
z  e.  C  /\  w  e.  D )
)  /\  ( x  +h  y )  =  ( z  +h  w ) )  ->  ( w  -h  y )  e.  ( B  +H  D ) )
141sheli 22669 . . . . . 6  |-  ( x  e.  A  ->  x  e.  ~H )
157sheli 22669 . . . . . 6  |-  ( y  e.  B  ->  y  e.  ~H )
1614, 15anim12i 550 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( x  e.  ~H  /\  y  e.  ~H )
)
172sheli 22669 . . . . . 6  |-  ( z  e.  C  ->  z  e.  ~H )
186sheli 22669 . . . . . 6  |-  ( w  e.  D  ->  w  e.  ~H )
1917, 18anim12i 550 . . . . 5  |-  ( ( z  e.  C  /\  w  e.  D )  ->  ( z  e.  ~H  /\  w  e.  ~H )
)
2016, 19anim12i 550 . . . 4  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  ( z  e.  C  /\  w  e.  D ) )  -> 
( ( x  e. 
~H  /\  y  e.  ~H )  /\  (
z  e.  ~H  /\  w  e.  ~H )
) )
21 oveq1 6047 . . . . . . 7  |-  ( ( x  +h  y )  =  ( z  +h  w )  ->  (
( x  +h  y
)  -h  ( z  +h  y ) )  =  ( ( z  +h  w )  -h  ( z  +h  y
) ) )
2221adantl 453 . . . . . 6  |-  ( ( ( ( x  e. 
~H  /\  y  e.  ~H )  /\  (
z  e.  ~H  /\  w  e.  ~H )
)  /\  ( x  +h  y )  =  ( z  +h  w ) )  ->  ( (
x  +h  y )  -h  ( z  +h  y ) )  =  ( ( z  +h  w )  -h  (
z  +h  y ) ) )
23 simpr 448 . . . . . . . . . . . 12  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  y  e.  ~H )
2423anim2i 553 . . . . . . . . . . 11  |-  ( ( z  e.  ~H  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( z  e.  ~H  /\  y  e. 
~H ) )
2524ancoms 440 . . . . . . . . . 10  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  z  e.  ~H )  ->  ( z  e. 
~H  /\  y  e.  ~H ) )
26 hvsub4 22492 . . . . . . . . . 10  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  ( z  e.  ~H  /\  y  e.  ~H )
)  ->  ( (
x  +h  y )  -h  ( z  +h  y ) )  =  ( ( x  -h  z )  +h  (
y  -h  y ) ) )
2725, 26syldan 457 . . . . . . . . 9  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  z  e.  ~H )  ->  ( ( x  +h  y )  -h  ( z  +h  y
) )  =  ( ( x  -h  z
)  +h  ( y  -h  y ) ) )
28 hvsubid 22481 . . . . . . . . . . 11  |-  ( y  e.  ~H  ->  (
y  -h  y )  =  0h )
2928oveq2d 6056 . . . . . . . . . 10  |-  ( y  e.  ~H  ->  (
( x  -h  z
)  +h  ( y  -h  y ) )  =  ( ( x  -h  z )  +h 
0h ) )
3029ad2antlr 708 . . . . . . . . 9  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  z  e.  ~H )  ->  ( ( x  -h  z )  +h  ( y  -h  y
) )  =  ( ( x  -h  z
)  +h  0h )
)
31 hvsubcl 22473 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( x  -h  z
)  e.  ~H )
32 ax-hvaddid 22460 . . . . . . . . . . 11  |-  ( ( x  -h  z )  e.  ~H  ->  (
( x  -h  z
)  +h  0h )  =  ( x  -h  z ) )
3331, 32syl 16 . . . . . . . . . 10  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( ( x  -h  z )  +h  0h )  =  ( x  -h  z ) )
3433adantlr 696 . . . . . . . . 9  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  z  e.  ~H )  ->  ( ( x  -h  z )  +h 
0h )  =  ( x  -h  z ) )
3527, 30, 343eqtrd 2440 . . . . . . . 8  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  z  e.  ~H )  ->  ( ( x  +h  y )  -h  ( z  +h  y
) )  =  ( x  -h  z ) )
3635adantrr 698 . . . . . . 7  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
x  +h  y )  -h  ( z  +h  y ) )  =  ( x  -h  z
) )
3736adantr 452 . . . . . 6  |-  ( ( ( ( x  e. 
~H  /\  y  e.  ~H )  /\  (
z  e.  ~H  /\  w  e.  ~H )
)  /\  ( x  +h  y )  =  ( z  +h  w ) )  ->  ( (
x  +h  y )  -h  ( z  +h  y ) )  =  ( x  -h  z
) )
38 simpr 448 . . . . . . . . . 10  |-  ( ( y  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( z  e.  ~H  /\  w  e. 
~H ) )
39 simpl 444 . . . . . . . . . . . 12  |-  ( ( z  e.  ~H  /\  w  e.  ~H )  ->  z  e.  ~H )
4039anim1i 552 . . . . . . . . . . 11  |-  ( ( ( z  e.  ~H  /\  w  e.  ~H )  /\  y  e.  ~H )  ->  ( z  e. 
~H  /\  y  e.  ~H ) )
4140ancoms 440 . . . . . . . . . 10  |-  ( ( y  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( z  e.  ~H  /\  y  e. 
~H ) )
42 hvsub4 22492 . . . . . . . . . 10  |-  ( ( ( z  e.  ~H  /\  w  e.  ~H )  /\  ( z  e.  ~H  /\  y  e.  ~H )
)  ->  ( (
z  +h  w )  -h  ( z  +h  y ) )  =  ( ( z  -h  z )  +h  (
w  -h  y ) ) )
4338, 41, 42syl2anc 643 . . . . . . . . 9  |-  ( ( y  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
z  +h  w )  -h  ( z  +h  y ) )  =  ( ( z  -h  z )  +h  (
w  -h  y ) ) )
44 hvsubid 22481 . . . . . . . . . . 11  |-  ( z  e.  ~H  ->  (
z  -h  z )  =  0h )
4544oveq1d 6055 . . . . . . . . . 10  |-  ( z  e.  ~H  ->  (
( z  -h  z
)  +h  ( w  -h  y ) )  =  ( 0h  +h  ( w  -h  y
) ) )
4645ad2antrl 709 . . . . . . . . 9  |-  ( ( y  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
z  -h  z )  +h  ( w  -h  y ) )  =  ( 0h  +h  (
w  -h  y ) ) )
47 hvsubcl 22473 . . . . . . . . . . . 12  |-  ( ( w  e.  ~H  /\  y  e.  ~H )  ->  ( w  -h  y
)  e.  ~H )
48 hvaddid2 22478 . . . . . . . . . . . 12  |-  ( ( w  -h  y )  e.  ~H  ->  ( 0h  +h  ( w  -h  y ) )  =  ( w  -h  y
) )
4947, 48syl 16 . . . . . . . . . . 11  |-  ( ( w  e.  ~H  /\  y  e.  ~H )  ->  ( 0h  +h  (
w  -h  y ) )  =  ( w  -h  y ) )
5049ancoms 440 . . . . . . . . . 10  |-  ( ( y  e.  ~H  /\  w  e.  ~H )  ->  ( 0h  +h  (
w  -h  y ) )  =  ( w  -h  y ) )
5150adantrl 697 . . . . . . . . 9  |-  ( ( y  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( 0h  +h  ( w  -h  y
) )  =  ( w  -h  y ) )
5243, 46, 513eqtrd 2440 . . . . . . . 8  |-  ( ( y  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
z  +h  w )  -h  ( z  +h  y ) )  =  ( w  -h  y
) )
5352adantll 695 . . . . . . 7  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
z  +h  w )  -h  ( z  +h  y ) )  =  ( w  -h  y
) )
5453adantr 452 . . . . . 6  |-  ( ( ( ( x  e. 
~H  /\  y  e.  ~H )  /\  (
z  e.  ~H  /\  w  e.  ~H )
)  /\  ( x  +h  y )  =  ( z  +h  w ) )  ->  ( (
z  +h  w )  -h  ( z  +h  y ) )  =  ( w  -h  y
) )
5522, 37, 543eqtr3d 2444 . . . . 5  |-  ( ( ( ( x  e. 
~H  /\  y  e.  ~H )  /\  (
z  e.  ~H  /\  w  e.  ~H )
)  /\  ( x  +h  y )  =  ( z  +h  w ) )  ->  ( x  -h  z )  =  ( w  -h  y ) )
5655eleq1d 2470 . . . 4  |-  ( ( ( ( x  e. 
~H  /\  y  e.  ~H )  /\  (
z  e.  ~H  /\  w  e.  ~H )
)  /\  ( x  +h  y )  =  ( z  +h  w ) )  ->  ( (
x  -h  z )  e.  ( B  +H  D )  <->  ( w  -h  y )  e.  ( B  +H  D ) ) )
5720, 56sylan 458 . . 3  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  (
z  e.  C  /\  w  e.  D )
)  /\  ( x  +h  y )  =  ( z  +h  w ) )  ->  ( (
x  -h  z )  e.  ( B  +H  D )  <->  ( w  -h  y )  e.  ( B  +H  D ) ) )
5813, 57mpbird 224 . 2  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  (
z  e.  C  /\  w  e.  D )
)  /\  ( x  +h  y )  =  ( z  +h  w ) )  ->  ( x  -h  z )  e.  ( B  +H  D ) )
59 elin 3490 . 2  |-  ( ( x  -h  z )  e.  ( ( A  +H  C )  i^i  ( B  +H  D
) )  <->  ( (
x  -h  z )  e.  ( A  +H  C )  /\  (
x  -h  z )  e.  ( B  +H  D ) ) )
605, 58, 59sylanbrc 646 1  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  (
z  e.  C  /\  w  e.  D )
)  /\  ( x  +h  y )  =  ( z  +h  w ) )  ->  ( x  -h  z )  e.  ( ( A  +H  C
)  i^i  ( B  +H  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    i^i cin 3279  (class class class)co 6040   ~Hchil 22375    +h cva 22376   0hc0v 22380    -h cmv 22381   SHcsh 22384    +H cph 22387
This theorem is referenced by:  5oalem3  23111  5oalem4  23112
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-hilex 22455  ax-hfvadd 22456  ax-hvcom 22457  ax-hvass 22458  ax-hv0cl 22459  ax-hvaddid 22460  ax-hfvmul 22461  ax-hvmulid 22462  ax-hvdistr1 22464  ax-hvdistr2 22465  ax-hvmul0 22466
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-ltxr 9081  df-sub 9249  df-neg 9250  df-nn 9957  df-grpo 21732  df-ablo 21823  df-hvsub 22427  df-hlim 22428  df-sh 22662  df-ch 22677  df-shs 22763
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