HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  5oalem4 Structured version   Unicode version

Theorem 5oalem4 23159
Description: Lemma for orthoarguesian law 5OA. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
Hypotheses
Ref Expression
5oalem3.1  |-  A  e.  SH
5oalem3.2  |-  B  e.  SH
5oalem3.3  |-  C  e.  SH
5oalem3.4  |-  D  e.  SH
5oalem3.5  |-  F  e.  SH
5oalem3.6  |-  G  e.  SH
Assertion
Ref Expression
5oalem4  |-  ( ( ( ( ( x  e.  A  /\  y  e.  B )  /\  (
z  e.  C  /\  w  e.  D )
)  /\  ( f  e.  F  /\  g  e.  G ) )  /\  ( ( x  +h  y )  =  ( f  +h  g )  /\  ( z  +h  w )  =  ( f  +h  g ) ) )  ->  (
x  -h  z )  e.  ( ( ( A  +H  C )  i^i  ( B  +H  D ) )  i^i  ( ( ( A  +H  F )  i^i  ( B  +H  G
) )  +H  (
( C  +H  F
)  i^i  ( D  +H  G ) ) ) ) )

Proof of Theorem 5oalem4
StepHypRef Expression
1 eqtr3 2455 . . . 4  |-  ( ( ( x  +h  y
)  =  ( f  +h  g )  /\  ( z  +h  w
)  =  ( f  +h  g ) )  ->  ( x  +h  y )  =  ( z  +h  w ) )
2 5oalem3.1 . . . . 5  |-  A  e.  SH
3 5oalem3.2 . . . . 5  |-  B  e.  SH
4 5oalem3.3 . . . . 5  |-  C  e.  SH
5 5oalem3.4 . . . . 5  |-  D  e.  SH
62, 3, 4, 55oalem2 23157 . . . 4  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  (
z  e.  C  /\  w  e.  D )
)  /\  ( x  +h  y )  =  ( z  +h  w ) )  ->  ( x  -h  z )  e.  ( ( A  +H  C
)  i^i  ( B  +H  D ) ) )
71, 6sylan2 461 . . 3  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  (
z  e.  C  /\  w  e.  D )
)  /\  ( (
x  +h  y )  =  ( f  +h  g )  /\  (
z  +h  w )  =  ( f  +h  g ) ) )  ->  ( x  -h  z )  e.  ( ( A  +H  C
)  i^i  ( B  +H  D ) ) )
87adantlr 696 . 2  |-  ( ( ( ( ( x  e.  A  /\  y  e.  B )  /\  (
z  e.  C  /\  w  e.  D )
)  /\  ( f  e.  F  /\  g  e.  G ) )  /\  ( ( x  +h  y )  =  ( f  +h  g )  /\  ( z  +h  w )  =  ( f  +h  g ) ) )  ->  (
x  -h  z )  e.  ( ( A  +H  C )  i^i  ( B  +H  D
) ) )
9 5oalem3.5 . . 3  |-  F  e.  SH
10 5oalem3.6 . . 3  |-  G  e.  SH
112, 3, 4, 5, 9, 105oalem3 23158 . 2  |-  ( ( ( ( ( x  e.  A  /\  y  e.  B )  /\  (
z  e.  C  /\  w  e.  D )
)  /\  ( f  e.  F  /\  g  e.  G ) )  /\  ( ( x  +h  y )  =  ( f  +h  g )  /\  ( z  +h  w )  =  ( f  +h  g ) ) )  ->  (
x  -h  z )  e.  ( ( ( A  +H  F )  i^i  ( B  +H  G ) )  +H  ( ( C  +H  F )  i^i  ( D  +H  G ) ) ) )
12 elin 3530 . 2  |-  ( ( x  -h  z )  e.  ( ( ( A  +H  C )  i^i  ( B  +H  D ) )  i^i  ( ( ( A  +H  F )  i^i  ( B  +H  G
) )  +H  (
( C  +H  F
)  i^i  ( D  +H  G ) ) ) )  <->  ( ( x  -h  z )  e.  ( ( A  +H  C )  i^i  ( B  +H  D ) )  /\  ( x  -h  z )  e.  ( ( ( A  +H  F )  i^i  ( B  +H  G ) )  +H  ( ( C  +H  F )  i^i  ( D  +H  G
) ) ) ) )
138, 11, 12sylanbrc 646 1  |-  ( ( ( ( ( x  e.  A  /\  y  e.  B )  /\  (
z  e.  C  /\  w  e.  D )
)  /\  ( f  e.  F  /\  g  e.  G ) )  /\  ( ( x  +h  y )  =  ( f  +h  g )  /\  ( z  +h  w )  =  ( f  +h  g ) ) )  ->  (
x  -h  z )  e.  ( ( ( A  +H  C )  i^i  ( B  +H  D ) )  i^i  ( ( ( A  +H  F )  i^i  ( B  +H  G
) )  +H  (
( C  +H  F
)  i^i  ( D  +H  G ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    i^i cin 3319  (class class class)co 6081    +h cva 22423    -h cmv 22428   SHcsh 22431    +H cph 22434
This theorem is referenced by:  5oalem5  23160
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-hilex 22502  ax-hfvadd 22503  ax-hvcom 22504  ax-hvass 22505  ax-hv0cl 22506  ax-hvaddid 22507  ax-hfvmul 22508  ax-hvmulid 22509  ax-hvmulass 22510  ax-hvdistr1 22511  ax-hvdistr2 22512  ax-hvmul0 22513
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-ltxr 9125  df-sub 9293  df-neg 9294  df-nn 10001  df-grpo 21779  df-ablo 21870  df-hvsub 22474  df-hlim 22475  df-sh 22709  df-ch 22724  df-shs 22810
  Copyright terms: Public domain W3C validator