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Theorem 3oalem1 23164
Description: Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
3oalem1.1  |-  B  e. 
CH
3oalem1.2  |-  C  e. 
CH
3oalem1.3  |-  R  e. 
CH
3oalem1.4  |-  S  e. 
CH
Assertion
Ref Expression
3oalem1  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  v  e.  ~H )  /\  ( z  e. 
~H  /\  w  e.  ~H ) ) )
Distinct variable groups:    x, y,
z, w, v, B   
x, C, y, z, w, v    x, R, y, z, w, v   
x, S, y, z, w, v

Proof of Theorem 3oalem1
StepHypRef Expression
1 3oalem1.1 . . . . 5  |-  B  e. 
CH
21cheli 22735 . . . 4  |-  ( x  e.  B  ->  x  e.  ~H )
3 3oalem1.3 . . . . 5  |-  R  e. 
CH
43cheli 22735 . . . 4  |-  ( y  e.  R  ->  y  e.  ~H )
52, 4anim12i 550 . . 3  |-  ( ( x  e.  B  /\  y  e.  R )  ->  ( x  e.  ~H  /\  y  e.  ~H )
)
6 hvaddcl 22515 . . . . 5  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  +h  y
)  e.  ~H )
7 eleq1 2496 . . . . 5  |-  ( v  =  ( x  +h  y )  ->  (
v  e.  ~H  <->  ( x  +h  y )  e.  ~H ) )
86, 7syl5ibrcom 214 . . . 4  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( v  =  ( x  +h  y )  ->  v  e.  ~H ) )
98imdistani 672 . . 3  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  v  =  (
x  +h  y ) )  ->  ( (
x  e.  ~H  /\  y  e.  ~H )  /\  v  e.  ~H ) )
105, 9sylan 458 . 2  |-  ( ( ( x  e.  B  /\  y  e.  R
)  /\  v  =  ( x  +h  y
) )  ->  (
( x  e.  ~H  /\  y  e.  ~H )  /\  v  e.  ~H ) )
11 3oalem1.2 . . . . 5  |-  C  e. 
CH
1211cheli 22735 . . . 4  |-  ( z  e.  C  ->  z  e.  ~H )
13 3oalem1.4 . . . . 5  |-  S  e. 
CH
1413cheli 22735 . . . 4  |-  ( w  e.  S  ->  w  e.  ~H )
1512, 14anim12i 550 . . 3  |-  ( ( z  e.  C  /\  w  e.  S )  ->  ( z  e.  ~H  /\  w  e.  ~H )
)
1615adantr 452 . 2  |-  ( ( ( z  e.  C  /\  w  e.  S
)  /\  v  =  ( z  +h  w
) )  ->  (
z  e.  ~H  /\  w  e.  ~H )
)
1710, 16anim12i 550 1  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  v  e.  ~H )  /\  ( z  e. 
~H  /\  w  e.  ~H ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725  (class class class)co 6081   ~Hchil 22422    +h cva 22423   CHcch 22432
This theorem is referenced by:  3oalem2  23165
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-hilex 22502  ax-hfvadd 22503
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-sh 22709  df-ch 22724
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