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Theorem 5oalem2 26396
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 26108 . . . 4  |-  ( ( x  e.  A  /\  z  e.  C )  ->  ( x  -h  z
)  e.  ( A  +H  C ) )
43ad2ant2r 746 . . 3  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  ( z  e.  C  /\  w  e.  D ) )  -> 
( x  -h  z
)  e.  ( A  +H  C ) )
54adantr 465 . 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 26108 . . . . . . 7  |-  ( ( w  e.  D  /\  y  e.  B )  ->  ( w  -h  y
)  e.  ( D  +H  B ) )
98ancoms 453 . . . . . 6  |-  ( ( y  e.  B  /\  w  e.  D )  ->  ( w  -h  y
)  e.  ( D  +H  B ) )
107, 6shscomi 26104 . . . . . 6  |-  ( B  +H  D )  =  ( D  +H  B
)
119, 10syl6eleqr 2566 . . . . 5  |-  ( ( y  e.  B  /\  w  e.  D )  ->  ( w  -h  y
)  e.  ( B  +H  D ) )
1211ad2ant2l 745 . . . 4  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  ( z  e.  C  /\  w  e.  D ) )  -> 
( w  -h  y
)  e.  ( B  +H  D ) )
1312adantr 465 . . 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 25954 . . . . . 6  |-  ( x  e.  A  ->  x  e.  ~H )
157sheli 25954 . . . . . 6  |-  ( y  e.  B  ->  y  e.  ~H )
1614, 15anim12i 566 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( x  e.  ~H  /\  y  e.  ~H )
)
172sheli 25954 . . . . . 6  |-  ( z  e.  C  ->  z  e.  ~H )
186sheli 25954 . . . . . 6  |-  ( w  e.  D  ->  w  e.  ~H )
1917, 18anim12i 566 . . . . 5  |-  ( ( z  e.  C  /\  w  e.  D )  ->  ( z  e.  ~H  /\  w  e.  ~H )
)
2016, 19anim12i 566 . . . 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 6302 . . . . . . 7  |-  ( ( x  +h  y )  =  ( z  +h  w )  ->  (
( x  +h  y
)  -h  ( z  +h  y ) )  =  ( ( z  +h  w )  -h  ( z  +h  y
) ) )
2221adantl 466 . . . . . 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 461 . . . . . . . . . . . 12  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  y  e.  ~H )
2423anim2i 569 . . . . . . . . . . 11  |-  ( ( z  e.  ~H  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( z  e.  ~H  /\  y  e. 
~H ) )
2524ancoms 453 . . . . . . . . . 10  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  z  e.  ~H )  ->  ( z  e. 
~H  /\  y  e.  ~H ) )
26 hvsub4 25777 . . . . . . . . . 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 470 . . . . . . . . 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 25766 . . . . . . . . . . 11  |-  ( y  e.  ~H  ->  (
y  -h  y )  =  0h )
2928oveq2d 6311 . . . . . . . . . 10  |-  ( y  e.  ~H  ->  (
( x  -h  z
)  +h  ( y  -h  y ) )  =  ( ( x  -h  z )  +h 
0h ) )
3029ad2antlr 726 . . . . . . . . 9  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  z  e.  ~H )  ->  ( ( x  -h  z )  +h  ( y  -h  y
) )  =  ( ( x  -h  z
)  +h  0h )
)
31 hvsubcl 25757 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( x  -h  z
)  e.  ~H )
32 ax-hvaddid 25744 . . . . . . . . . . 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 714 . . . . . . . . 9  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  z  e.  ~H )  ->  ( ( x  -h  z )  +h 
0h )  =  ( x  -h  z ) )
3527, 30, 343eqtrd 2512 . . . . . . . 8  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  z  e.  ~H )  ->  ( ( x  +h  y )  -h  ( z  +h  y
) )  =  ( x  -h  z ) )
3635adantrr 716 . . . . . . 7  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
x  +h  y )  -h  ( z  +h  y ) )  =  ( x  -h  z
) )
3736adantr 465 . . . . . 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 461 . . . . . . . . . 10  |-  ( ( y  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( z  e.  ~H  /\  w  e. 
~H ) )
39 simpl 457 . . . . . . . . . . . 12  |-  ( ( z  e.  ~H  /\  w  e.  ~H )  ->  z  e.  ~H )
4039anim1i 568 . . . . . . . . . . 11  |-  ( ( ( z  e.  ~H  /\  w  e.  ~H )  /\  y  e.  ~H )  ->  ( z  e. 
~H  /\  y  e.  ~H ) )
4140ancoms 453 . . . . . . . . . 10  |-  ( ( y  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( z  e.  ~H  /\  y  e. 
~H ) )
42 hvsub4 25777 . . . . . . . . . 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 661 . . . . . . . . 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 25766 . . . . . . . . . . 11  |-  ( z  e.  ~H  ->  (
z  -h  z )  =  0h )
4544oveq1d 6310 . . . . . . . . . 10  |-  ( z  e.  ~H  ->  (
( z  -h  z
)  +h  ( w  -h  y ) )  =  ( 0h  +h  ( w  -h  y
) ) )
4645ad2antrl 727 . . . . . . . . 9  |-  ( ( y  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
z  -h  z )  +h  ( w  -h  y ) )  =  ( 0h  +h  (
w  -h  y ) ) )
47 hvsubcl 25757 . . . . . . . . . . . 12  |-  ( ( w  e.  ~H  /\  y  e.  ~H )  ->  ( w  -h  y
)  e.  ~H )
48 hvaddid2 25763 . . . . . . . . . . . 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 453 . . . . . . . . . 10  |-  ( ( y  e.  ~H  /\  w  e.  ~H )  ->  ( 0h  +h  (
w  -h  y ) )  =  ( w  -h  y ) )
5150adantrl 715 . . . . . . . . 9  |-  ( ( y  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( 0h  +h  ( w  -h  y
) )  =  ( w  -h  y ) )
5243, 46, 513eqtrd 2512 . . . . . . . 8  |-  ( ( y  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
z  +h  w )  -h  ( z  +h  y ) )  =  ( w  -h  y
) )
5352adantll 713 . . . . . . 7  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
z  +h  w )  -h  ( z  +h  y ) )  =  ( w  -h  y
) )
5453adantr 465 . . . . . 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 2516 . . . . 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 2536 . . . 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 471 . . 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 232 . 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 ) )
595, 58elind 3693 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    i^i cin 3480  (class class class)co 6295   ~Hchil 25659    +h cva 25660   0hc0v 25664    -h cmv 25665   SHcsh 25668    +H cph 25671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-hilex 25739  ax-hfvadd 25740  ax-hvcom 25741  ax-hvass 25742  ax-hv0cl 25743  ax-hvaddid 25744  ax-hfvmul 25745  ax-hvmulid 25746  ax-hvdistr1 25748  ax-hvdistr2 25749  ax-hvmul0 25750
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-recs 7054  df-rdg 7088  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-ltxr 9645  df-sub 9819  df-neg 9820  df-nn 10549  df-grpo 25016  df-ablo 25107  df-hvsub 25711  df-hlim 25712  df-sh 25947  df-ch 25962  df-shs 26049
This theorem is referenced by:  5oalem3  26397  5oalem4  26398
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