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Theorem 3oalem3 27329
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
3oalem3  |-  ( ( B  +H  R )  i^i  ( C  +H  S ) )  C_  ( B  +H  ( R  i^i  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S ) ) ) ) )

Proof of Theorem 3oalem3
Dummy variables  x  y  z  w  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 3oalem1.1 . . . . . . 7  |-  B  e. 
CH
2 3oalem1.3 . . . . . . 7  |-  R  e. 
CH
31, 2chseli 27124 . . . . . 6  |-  ( v  e.  ( B  +H  R )  <->  E. x  e.  B  E. y  e.  R  v  =  ( x  +h  y
) )
4 r2ex 2915 . . . . . 6  |-  ( E. x  e.  B  E. y  e.  R  v  =  ( x  +h  y )  <->  E. x E. y ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) ) )
53, 4bitri 253 . . . . 5  |-  ( v  e.  ( B  +H  R )  <->  E. x E. y ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) ) )
6 3oalem1.2 . . . . . . 7  |-  C  e. 
CH
7 3oalem1.4 . . . . . . 7  |-  S  e. 
CH
86, 7chseli 27124 . . . . . 6  |-  ( v  e.  ( C  +H  S )  <->  E. z  e.  C  E. w  e.  S  v  =  ( z  +h  w
) )
9 r2ex 2915 . . . . . 6  |-  ( E. z  e.  C  E. w  e.  S  v  =  ( z  +h  w )  <->  E. z E. w ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )
108, 9bitri 253 . . . . 5  |-  ( v  e.  ( C  +H  S )  <->  E. z E. w ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )
115, 10anbi12i 704 . . . 4  |-  ( ( v  e.  ( B  +H  R )  /\  v  e.  ( C  +H  S ) )  <->  ( E. x E. y ( ( x  e.  B  /\  y  e.  R )  /\  v  =  (
x  +h  y ) )  /\  E. z E. w ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) ) )
12 elin 3619 . . . 4  |-  ( v  e.  ( ( B  +H  R )  i^i  ( C  +H  S
) )  <->  ( v  e.  ( B  +H  R
)  /\  v  e.  ( C  +H  S
) ) )
13 ee4anv 2082 . . . 4  |-  ( E. x E. y E. z E. w ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  <-> 
( E. x E. y ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  E. z E. w ( ( z  e.  C  /\  w  e.  S
)  /\  v  =  ( z  +h  w
) ) ) )
1411, 12, 133bitr4i 281 . . 3  |-  ( v  e.  ( ( B  +H  R )  i^i  ( C  +H  S
) )  <->  E. x E. y E. z E. w ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  (
x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  (
z  +h  w ) ) ) )
151, 6, 2, 73oalem2 27328 . . . . 5  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  v  e.  ( B  +H  ( R  i^i  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S ) ) ) ) ) )
1615exlimivv 1780 . . . 4  |-  ( E. z E. w ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  v  e.  ( B  +H  ( R  i^i  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S ) ) ) ) ) )
1716exlimivv 1780 . . 3  |-  ( E. x E. y E. z E. w ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  v  e.  ( B  +H  ( R  i^i  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S ) ) ) ) ) )
1814, 17sylbi 199 . 2  |-  ( v  e.  ( ( B  +H  R )  i^i  ( C  +H  S
) )  ->  v  e.  ( B  +H  ( R  i^i  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S ) ) ) ) ) )
1918ssriv 3438 1  |-  ( ( B  +H  R )  i^i  ( C  +H  S ) )  C_  ( B  +H  ( R  i^i  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 371    = wceq 1446   E.wex 1665    e. wcel 1889   E.wrex 2740    i^i cin 3405    C_ wss 3406  (class class class)co 6295    +h cva 26585   CHcch 26594    +H cph 26596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-hilex 26664  ax-hfvadd 26665  ax-hvcom 26666  ax-hvass 26667  ax-hv0cl 26668  ax-hvaddid 26669  ax-hfvmul 26670  ax-hvmulid 26671  ax-hvdistr1 26673  ax-hvdistr2 26674  ax-hvmul0 26675
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-er 7368  df-map 7479  df-en 7575  df-dom 7576  df-sdom 7577  df-pnf 9682  df-mnf 9683  df-ltxr 9685  df-sub 9867  df-neg 9868  df-nn 10617  df-grpo 25931  df-ablo 26022  df-hvsub 26636  df-hlim 26637  df-sh 26872  df-ch 26886  df-shs 26973
This theorem is referenced by:  3oai  27333
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