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Theorem 3oalem1 22257
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 21828 . . . 4  |-  ( x  e.  B  ->  x  e.  ~H )
3 3oalem1.3 . . . . 5  |-  R  e. 
CH
43cheli 21828 . . . 4  |-  ( y  e.  R  ->  y  e.  ~H )
52, 4anim12i 549 . . 3  |-  ( ( x  e.  B  /\  y  e.  R )  ->  ( x  e.  ~H  /\  y  e.  ~H )
)
6 hvaddcl 21608 . . . . 5  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  +h  y
)  e.  ~H )
7 eleq1 2356 . . . . 5  |-  ( v  =  ( x  +h  y )  ->  (
v  e.  ~H  <->  ( x  +h  y )  e.  ~H ) )
86, 7syl5ibrcom 213 . . . 4  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( v  =  ( x  +h  y )  ->  v  e.  ~H ) )
98imdistani 671 . . 3  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  v  =  (
x  +h  y ) )  ->  ( (
x  e.  ~H  /\  y  e.  ~H )  /\  v  e.  ~H ) )
105, 9sylan 457 . 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 21828 . . . 4  |-  ( z  e.  C  ->  z  e.  ~H )
13 3oalem1.4 . . . . 5  |-  S  e. 
CH
1413cheli 21828 . . . 4  |-  ( w  e.  S  ->  w  e.  ~H )
1512, 14anim12i 549 . . 3  |-  ( ( z  e.  C  /\  w  e.  S )  ->  ( z  e.  ~H  /\  w  e.  ~H )
)
1615adantr 451 . 2  |-  ( ( ( z  e.  C  /\  w  e.  S
)  /\  v  =  ( z  +h  w
) )  ->  (
z  e.  ~H  /\  w  e.  ~H )
)
1710, 16anim12i 549 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 358    = wceq 1632    e. wcel 1696  (class class class)co 5874   ~Hchil 21515    +h cva 21516   CHcch 21525
This theorem is referenced by:  3oalem2  22258
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-hilex 21595  ax-hfvadd 21596
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-sh 21802  df-ch 21817
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