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Theorem 5oalem2 25058
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 24770 . . . 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 24770 . . . . . . 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 24766 . . . . . 6  |-  ( B  +H  D )  =  ( D  +H  B
)
119, 10syl6eleqr 2534 . . . . 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 24616 . . . . . 6  |-  ( x  e.  A  ->  x  e.  ~H )
157sheli 24616 . . . . . 6  |-  ( y  e.  B  ->  y  e.  ~H )
1614, 15anim12i 566 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( x  e.  ~H  /\  y  e.  ~H )
)
172sheli 24616 . . . . . 6  |-  ( z  e.  C  ->  z  e.  ~H )
186sheli 24616 . . . . . 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 6098 . . . . . . 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 24439 . . . . . . . . . 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 24428 . . . . . . . . . . 11  |-  ( y  e.  ~H  ->  (
y  -h  y )  =  0h )
2928oveq2d 6107 . . . . . . . . . 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 24419 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( x  -h  z
)  e.  ~H )
32 ax-hvaddid 24406 . . . . . . . . . . 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 2479 . . . . . . . 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 24439 . . . . . . . . . 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 24428 . . . . . . . . . . 11  |-  ( z  e.  ~H  ->  (
z  -h  z )  =  0h )
4544oveq1d 6106 . . . . . . . . . 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 24419 . . . . . . . . . . . 12  |-  ( ( w  e.  ~H  /\  y  e.  ~H )  ->  ( w  -h  y
)  e.  ~H )
48 hvaddid2 24425 . . . . . . . . . . . 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 2479 . . . . . . . 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 2483 . . . . 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 2509 . . . 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 3540 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 1369    e. wcel 1756    i^i cin 3327  (class class class)co 6091   ~Hchil 24321    +h cva 24322   0hc0v 24326    -h cmv 24327   SHcsh 24330    +H cph 24333
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 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-hilex 24401  ax-hfvadd 24402  ax-hvcom 24403  ax-hvass 24404  ax-hv0cl 24405  ax-hvaddid 24406  ax-hfvmul 24407  ax-hvmulid 24408  ax-hvdistr1 24410  ax-hvdistr2 24411  ax-hvmul0 24412
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-recs 6832  df-rdg 6866  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-ltxr 9423  df-sub 9597  df-neg 9598  df-nn 10323  df-grpo 23678  df-ablo 23769  df-hvsub 24373  df-hlim 24374  df-sh 24609  df-ch 24624  df-shs 24711
This theorem is referenced by:  5oalem3  25059  5oalem4  25060
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