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Theorem 5oalem2 24993
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 24705 . . . 4  |-  ( ( x  e.  A  /\  z  e.  C )  ->  ( x  -h  z
)  e.  ( A  +H  C ) )
43ad2ant2r 741 . . 3  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  ( z  e.  C  /\  w  e.  D ) )  -> 
( x  -h  z
)  e.  ( A  +H  C ) )
54adantr 462 . 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 24705 . . . . . . 7  |-  ( ( w  e.  D  /\  y  e.  B )  ->  ( w  -h  y
)  e.  ( D  +H  B ) )
98ancoms 450 . . . . . 6  |-  ( ( y  e.  B  /\  w  e.  D )  ->  ( w  -h  y
)  e.  ( D  +H  B ) )
107, 6shscomi 24701 . . . . . 6  |-  ( B  +H  D )  =  ( D  +H  B
)
119, 10syl6eleqr 2532 . . . . 5  |-  ( ( y  e.  B  /\  w  e.  D )  ->  ( w  -h  y
)  e.  ( B  +H  D ) )
1211ad2ant2l 740 . . . 4  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  ( z  e.  C  /\  w  e.  D ) )  -> 
( w  -h  y
)  e.  ( B  +H  D ) )
1312adantr 462 . . 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 24551 . . . . . 6  |-  ( x  e.  A  ->  x  e.  ~H )
157sheli 24551 . . . . . 6  |-  ( y  e.  B  ->  y  e.  ~H )
1614, 15anim12i 563 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( x  e.  ~H  /\  y  e.  ~H )
)
172sheli 24551 . . . . . 6  |-  ( z  e.  C  ->  z  e.  ~H )
186sheli 24551 . . . . . 6  |-  ( w  e.  D  ->  w  e.  ~H )
1917, 18anim12i 563 . . . . 5  |-  ( ( z  e.  C  /\  w  e.  D )  ->  ( z  e.  ~H  /\  w  e.  ~H )
)
2016, 19anim12i 563 . . . 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 6097 . . . . . . 7  |-  ( ( x  +h  y )  =  ( z  +h  w )  ->  (
( x  +h  y
)  -h  ( z  +h  y ) )  =  ( ( z  +h  w )  -h  ( z  +h  y
) ) )
2221adantl 463 . . . . . 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 458 . . . . . . . . . . . 12  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  y  e.  ~H )
2423anim2i 566 . . . . . . . . . . 11  |-  ( ( z  e.  ~H  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( z  e.  ~H  /\  y  e. 
~H ) )
2524ancoms 450 . . . . . . . . . 10  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  z  e.  ~H )  ->  ( z  e. 
~H  /\  y  e.  ~H ) )
26 hvsub4 24374 . . . . . . . . . 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 467 . . . . . . . . 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 24363 . . . . . . . . . . 11  |-  ( y  e.  ~H  ->  (
y  -h  y )  =  0h )
2928oveq2d 6106 . . . . . . . . . 10  |-  ( y  e.  ~H  ->  (
( x  -h  z
)  +h  ( y  -h  y ) )  =  ( ( x  -h  z )  +h 
0h ) )
3029ad2antlr 721 . . . . . . . . 9  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  z  e.  ~H )  ->  ( ( x  -h  z )  +h  ( y  -h  y
) )  =  ( ( x  -h  z
)  +h  0h )
)
31 hvsubcl 24354 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( x  -h  z
)  e.  ~H )
32 ax-hvaddid 24341 . . . . . . . . . . 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 709 . . . . . . . . 9  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  z  e.  ~H )  ->  ( ( x  -h  z )  +h 
0h )  =  ( x  -h  z ) )
3527, 30, 343eqtrd 2477 . . . . . . . 8  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  z  e.  ~H )  ->  ( ( x  +h  y )  -h  ( z  +h  y
) )  =  ( x  -h  z ) )
3635adantrr 711 . . . . . . 7  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
x  +h  y )  -h  ( z  +h  y ) )  =  ( x  -h  z
) )
3736adantr 462 . . . . . 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 458 . . . . . . . . . 10  |-  ( ( y  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( z  e.  ~H  /\  w  e. 
~H ) )
39 simpl 454 . . . . . . . . . . . 12  |-  ( ( z  e.  ~H  /\  w  e.  ~H )  ->  z  e.  ~H )
4039anim1i 565 . . . . . . . . . . 11  |-  ( ( ( z  e.  ~H  /\  w  e.  ~H )  /\  y  e.  ~H )  ->  ( z  e. 
~H  /\  y  e.  ~H ) )
4140ancoms 450 . . . . . . . . . 10  |-  ( ( y  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( z  e.  ~H  /\  y  e. 
~H ) )
42 hvsub4 24374 . . . . . . . . . 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 656 . . . . . . . . 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 24363 . . . . . . . . . . 11  |-  ( z  e.  ~H  ->  (
z  -h  z )  =  0h )
4544oveq1d 6105 . . . . . . . . . 10  |-  ( z  e.  ~H  ->  (
( z  -h  z
)  +h  ( w  -h  y ) )  =  ( 0h  +h  ( w  -h  y
) ) )
4645ad2antrl 722 . . . . . . . . 9  |-  ( ( y  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
z  -h  z )  +h  ( w  -h  y ) )  =  ( 0h  +h  (
w  -h  y ) ) )
47 hvsubcl 24354 . . . . . . . . . . . 12  |-  ( ( w  e.  ~H  /\  y  e.  ~H )  ->  ( w  -h  y
)  e.  ~H )
48 hvaddid2 24360 . . . . . . . . . . . 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 450 . . . . . . . . . 10  |-  ( ( y  e.  ~H  /\  w  e.  ~H )  ->  ( 0h  +h  (
w  -h  y ) )  =  ( w  -h  y ) )
5150adantrl 710 . . . . . . . . 9  |-  ( ( y  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( 0h  +h  ( w  -h  y
) )  =  ( w  -h  y ) )
5243, 46, 513eqtrd 2477 . . . . . . . 8  |-  ( ( y  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
z  +h  w )  -h  ( z  +h  y ) )  =  ( w  -h  y
) )
5352adantll 708 . . . . . . 7  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
z  +h  w )  -h  ( z  +h  y ) )  =  ( w  -h  y
) )
5453adantr 462 . . . . . 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 2481 . . . . 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 2507 . . . 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 468 . . 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 3537 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 1364    e. wcel 1761    i^i cin 3324  (class class class)co 6090   ~Hchil 24256    +h cva 24257   0hc0v 24261    -h cmv 24262   SHcsh 24265    +H cph 24268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-hilex 24336  ax-hfvadd 24337  ax-hvcom 24338  ax-hvass 24339  ax-hv0cl 24340  ax-hvaddid 24341  ax-hfvmul 24342  ax-hvmulid 24343  ax-hvdistr1 24345  ax-hvdistr2 24346  ax-hvmul0 24347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-recs 6828  df-rdg 6862  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-ltxr 9419  df-sub 9593  df-neg 9594  df-nn 10319  df-grpo 23613  df-ablo 23704  df-hvsub 24308  df-hlim 24309  df-sh 24544  df-ch 24559  df-shs 24646
This theorem is referenced by:  5oalem3  24994  5oalem4  24995
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