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Theorem 3oalem1 25064
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 24634 . . . 4  |-  ( x  e.  B  ->  x  e.  ~H )
3 3oalem1.3 . . . . 5  |-  R  e. 
CH
43cheli 24634 . . . 4  |-  ( y  e.  R  ->  y  e.  ~H )
52, 4anim12i 566 . . 3  |-  ( ( x  e.  B  /\  y  e.  R )  ->  ( x  e.  ~H  /\  y  e.  ~H )
)
6 hvaddcl 24413 . . . . 5  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  +h  y
)  e.  ~H )
7 eleq1 2502 . . . . 5  |-  ( v  =  ( x  +h  y )  ->  (
v  e.  ~H  <->  ( x  +h  y )  e.  ~H ) )
86, 7syl5ibrcom 222 . . . 4  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( v  =  ( x  +h  y )  ->  v  e.  ~H ) )
98imdistani 690 . . 3  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  v  =  (
x  +h  y ) )  ->  ( (
x  e.  ~H  /\  y  e.  ~H )  /\  v  e.  ~H ) )
105, 9sylan 471 . 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 24634 . . . 4  |-  ( z  e.  C  ->  z  e.  ~H )
13 3oalem1.4 . . . . 5  |-  S  e. 
CH
1413cheli 24634 . . . 4  |-  ( w  e.  S  ->  w  e.  ~H )
1512, 14anim12i 566 . . 3  |-  ( ( z  e.  C  /\  w  e.  S )  ->  ( z  e.  ~H  /\  w  e.  ~H )
)
1615adantr 465 . 2  |-  ( ( ( z  e.  C  /\  w  e.  S
)  /\  v  =  ( z  +h  w
) )  ->  (
z  e.  ~H  /\  w  e.  ~H )
)
1710, 16anim12i 566 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756  (class class class)co 6090   ~Hchil 24320    +h cva 24321   CHcch 24330
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-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pr 4530  ax-hilex 24400  ax-hfvadd 24401
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-fv 5425  df-ov 6093  df-sh 24608  df-ch 24623
This theorem is referenced by:  3oalem2  25065
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