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Theorem 3oalem3 22243
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 22038 . . . . . 6  |-  ( v  e.  ( B  +H  R )  <->  E. x  e.  B  E. y  e.  R  v  =  ( x  +h  y
) )
4 r2ex 2581 . . . . . 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 240 . . . . 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 22038 . . . . . 6  |-  ( v  e.  ( C  +H  S )  <->  E. z  e.  C  E. w  e.  S  v  =  ( z  +h  w
) )
9 r2ex 2581 . . . . . 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 240 . . . . 5  |-  ( v  e.  ( C  +H  S )  <->  E. z E. w ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )
115, 10anbi12i 678 . . . 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 3358 . . . 4  |-  ( v  e.  ( ( B  +H  R )  i^i  ( C  +H  S
) )  <->  ( v  e.  ( B  +H  R
)  /\  v  e.  ( C  +H  S
) ) )
13 ee4anv 1856 . . . 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 268 . . 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 22242 . . . . 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 1667 . . . 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 1667 . . 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 187 . 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 3184 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 set class
Syntax hints:    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   E.wrex 2544    i^i cin 3151    C_ wss 3152  (class class class)co 5858    +h cva 21500   CHcch 21509    +H cph 21511
This theorem is referenced by:  3oai  22247
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-hilex 21579  ax-hfvadd 21580  ax-hvcom 21581  ax-hvass 21582  ax-hv0cl 21583  ax-hvaddid 21584  ax-hfvmul 21585  ax-hvmulid 21586  ax-hvdistr1 21588  ax-hvdistr2 21589  ax-hvmul0 21590
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  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-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872  df-sub 9039  df-neg 9040  df-nn 9747  df-grpo 20858  df-ablo 20949  df-hvsub 21551  df-hlim 21552  df-sh 21786  df-ch 21801  df-shs 21887
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