HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  5oalem1 Unicode version

Theorem 5oalem1 22225
Description: Lemma for orthoarguesian law 5OA. (Contributed by NM, 1-Apr-2000.) (New usage is discouraged.)
Hypotheses
Ref Expression
5oalem1.1  |-  A  e.  SH
5oalem1.2  |-  B  e.  SH
5oalem1.3  |-  C  e.  SH
5oalem1.4  |-  R  e.  SH
Assertion
Ref Expression
5oalem1  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  v  =  ( x  +h  y ) )  /\  ( z  e.  C  /\  ( x  -h  z
)  e.  R ) )  ->  v  e.  ( B  +H  ( A  i^i  ( C  +H  R ) ) ) )

Proof of Theorem 5oalem1
StepHypRef Expression
1 simplll 736 . . . 4  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  v  =  ( x  +h  y ) )  /\  ( z  e.  C  /\  ( x  -h  z
)  e.  R ) )  ->  x  e.  A )
2 5oalem1.1 . . . . . . . 8  |-  A  e.  SH
32sheli 21785 . . . . . . 7  |-  ( x  e.  A  ->  x  e.  ~H )
43ad2antrr 708 . . . . . 6  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  v  =  ( x  +h  y
) )  ->  x  e.  ~H )
5 5oalem1.3 . . . . . . . 8  |-  C  e.  SH
65sheli 21785 . . . . . . 7  |-  ( z  e.  C  ->  z  e.  ~H )
76adantr 453 . . . . . 6  |-  ( ( z  e.  C  /\  ( x  -h  z
)  e.  R )  ->  z  e.  ~H )
8 hvaddsub12 21609 . . . . . . . 8  |-  ( ( x  e.  ~H  /\  z  e.  ~H  /\  z  e.  ~H )  ->  (
x  +h  ( z  -h  z ) )  =  ( z  +h  ( x  -h  z
) ) )
983anidm23 1243 . . . . . . 7  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( x  +h  (
z  -h  z ) )  =  ( z  +h  ( x  -h  z ) ) )
10 hvsubid 21597 . . . . . . . . 9  |-  ( z  e.  ~H  ->  (
z  -h  z )  =  0h )
1110oveq2d 5835 . . . . . . . 8  |-  ( z  e.  ~H  ->  (
x  +h  ( z  -h  z ) )  =  ( x  +h  0h ) )
12 ax-hvaddid 21576 . . . . . . . 8  |-  ( x  e.  ~H  ->  (
x  +h  0h )  =  x )
1311, 12sylan9eqr 2338 . . . . . . 7  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( x  +h  (
z  -h  z ) )  =  x )
149, 13eqtr3d 2318 . . . . . 6  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( z  +h  (
x  -h  z ) )  =  x )
154, 7, 14syl2an 465 . . . . 5  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  v  =  ( x  +h  y ) )  /\  ( z  e.  C  /\  ( x  -h  z
)  e.  R ) )  ->  ( z  +h  ( x  -h  z
) )  =  x )
16 5oalem1.4 . . . . . . 7  |-  R  e.  SH
175, 16shsvai 21935 . . . . . 6  |-  ( ( z  e.  C  /\  ( x  -h  z
)  e.  R )  ->  ( z  +h  ( x  -h  z
) )  e.  ( C  +H  R ) )
1817adantl 454 . . . . 5  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  v  =  ( x  +h  y ) )  /\  ( z  e.  C  /\  ( x  -h  z
)  e.  R ) )  ->  ( z  +h  ( x  -h  z
) )  e.  ( C  +H  R ) )
1915, 18eqeltrrd 2359 . . . 4  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  v  =  ( x  +h  y ) )  /\  ( z  e.  C  /\  ( x  -h  z
)  e.  R ) )  ->  x  e.  ( C  +H  R
) )
20 elin 3359 . . . 4  |-  ( x  e.  ( A  i^i  ( C  +H  R
) )  <->  ( x  e.  A  /\  x  e.  ( C  +H  R
) ) )
211, 19, 20sylanbrc 647 . . 3  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  v  =  ( x  +h  y ) )  /\  ( z  e.  C  /\  ( x  -h  z
)  e.  R ) )  ->  x  e.  ( A  i^i  ( C  +H  R ) ) )
22 simpllr 737 . . 3  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  v  =  ( x  +h  y ) )  /\  ( z  e.  C  /\  ( x  -h  z
)  e.  R ) )  ->  y  e.  B )
235, 16shscli 21888 . . . . . 6  |-  ( C  +H  R )  e.  SH
242, 23shincli 21933 . . . . 5  |-  ( A  i^i  ( C  +H  R ) )  e.  SH
25 5oalem1.2 . . . . 5  |-  B  e.  SH
2624, 25shsvai 21935 . . . 4  |-  ( ( x  e.  ( A  i^i  ( C  +H  R ) )  /\  y  e.  B )  ->  ( x  +h  y
)  e.  ( ( A  i^i  ( C  +H  R ) )  +H  B ) )
2724, 25shscomi 21934 . . . 4  |-  ( ( A  i^i  ( C  +H  R ) )  +H  B )  =  ( B  +H  ( A  i^i  ( C  +H  R ) ) )
2826, 27syl6eleq 2374 . . 3  |-  ( ( x  e.  ( A  i^i  ( C  +H  R ) )  /\  y  e.  B )  ->  ( x  +h  y
)  e.  ( B  +H  ( A  i^i  ( C  +H  R
) ) ) )
2921, 22, 28syl2anc 644 . 2  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  v  =  ( x  +h  y ) )  /\  ( z  e.  C  /\  ( x  -h  z
)  e.  R ) )  ->  ( x  +h  y )  e.  ( B  +H  ( A  i^i  ( C  +H  R ) ) ) )
30 eleq1 2344 . . 3  |-  ( v  =  ( x  +h  y )  ->  (
v  e.  ( B  +H  ( A  i^i  ( C  +H  R
) ) )  <->  ( x  +h  y )  e.  ( B  +H  ( A  i^i  ( C  +H  R ) ) ) ) )
3130ad2antlr 709 . 2  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  v  =  ( x  +h  y ) )  /\  ( z  e.  C  /\  ( x  -h  z
)  e.  R ) )  ->  ( v  e.  ( B  +H  ( A  i^i  ( C  +H  R ) ) )  <-> 
( x  +h  y
)  e.  ( B  +H  ( A  i^i  ( C  +H  R
) ) ) ) )
3229, 31mpbird 225 1  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  v  =  ( x  +h  y ) )  /\  ( z  e.  C  /\  ( x  -h  z
)  e.  R ) )  ->  v  e.  ( B  +H  ( A  i^i  ( C  +H  R ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1624    e. wcel 1685    i^i cin 3152  (class class class)co 5819   ~Hchil 21491    +h cva 21492   0hc0v 21496    -h cmv 21497   SHcsh 21500    +H cph 21503
This theorem is referenced by:  5oalem6  22230
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-hilex 21571  ax-hfvadd 21572  ax-hvcom 21573  ax-hvass 21574  ax-hv0cl 21575  ax-hvaddid 21576  ax-hfvmul 21577  ax-hvmulid 21578  ax-hvdistr1 21580  ax-hvdistr2 21581  ax-hvmul0 21582
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-po 4313  df-so 4314  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-iota 6252  df-riota 6299  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-pnf 8864  df-mnf 8865  df-ltxr 8867  df-sub 9034  df-neg 9035  df-grpo 20850  df-ablo 20941  df-hvsub 21543  df-sh 21778  df-shs 21879
  Copyright terms: Public domain W3C validator