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Theorem 3oalem1 22296
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 21867 . . . 4  |-  ( x  e.  B  ->  x  e.  ~H )
3 3oalem1.3 . . . . 5  |-  R  e. 
CH
43cheli 21867 . . . 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 21647 . . . . 5  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  +h  y
)  e.  ~H )
7 eleq1 2376 . . . . 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 21867 . . . 4  |-  ( z  e.  C  ->  z  e.  ~H )
13 3oalem1.4 . . . . 5  |-  S  e. 
CH
1413cheli 21867 . . . 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 1633    e. wcel 1701  (class class class)co 5900   ~Hchil 21554    +h cva 21555   CHcch 21564
This theorem is referenced by:  3oalem2  22297
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251  ax-hilex 21634  ax-hfvadd 21635
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-fv 5300  df-ov 5903  df-sh 21841  df-ch 21856
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