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Theorem 3oalem1 22241
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 21812 . . . 4  |-  ( x  e.  B  ->  x  e.  ~H )
3 3oalem1.3 . . . . 5  |-  R  e. 
CH
43cheli 21812 . . . 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 21592 . . . . 5  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  +h  y
)  e.  ~H )
7 eleq1 2343 . . . . 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 21812 . . . 4  |-  ( z  e.  C  ->  z  e.  ~H )
13 3oalem1.4 . . . . 5  |-  S  e. 
CH
1413cheli 21812 . . . 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 1623    e. wcel 1684  (class class class)co 5858   ~Hchil 21499    +h cva 21500   CHcch 21509
This theorem is referenced by:  3oalem2  22242
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-hilex 21579  ax-hfvadd 21580
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-sh 21786  df-ch 21801
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