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Theorem 5oalem1 23156
Description: Lemma for orthoarguesian law 5OA. (Contributed by NM, 1-Apr-2000.) (New usage is discouraged.)
Hypotheses
Ref Expression
5oalem1.1  |-  A  e.  SH
5oalem1.2  |-  B  e.  SH
5oalem1.3  |-  C  e.  SH
5oalem1.4  |-  R  e.  SH
Assertion
Ref Expression
5oalem1  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  v  =  ( x  +h  y ) )  /\  ( z  e.  C  /\  ( x  -h  z
)  e.  R ) )  ->  v  e.  ( B  +H  ( A  i^i  ( C  +H  R ) ) ) )

Proof of Theorem 5oalem1
StepHypRef Expression
1 simplll 735 . . . 4  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  v  =  ( x  +h  y ) )  /\  ( z  e.  C  /\  ( x  -h  z
)  e.  R ) )  ->  x  e.  A )
2 5oalem1.1 . . . . . . . 8  |-  A  e.  SH
32sheli 22716 . . . . . . 7  |-  ( x  e.  A  ->  x  e.  ~H )
43ad2antrr 707 . . . . . 6  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  v  =  ( x  +h  y
) )  ->  x  e.  ~H )
5 5oalem1.3 . . . . . . . 8  |-  C  e.  SH
65sheli 22716 . . . . . . 7  |-  ( z  e.  C  ->  z  e.  ~H )
76adantr 452 . . . . . 6  |-  ( ( z  e.  C  /\  ( x  -h  z
)  e.  R )  ->  z  e.  ~H )
8 hvaddsub12 22540 . . . . . . . 8  |-  ( ( x  e.  ~H  /\  z  e.  ~H  /\  z  e.  ~H )  ->  (
x  +h  ( z  -h  z ) )  =  ( z  +h  ( x  -h  z
) ) )
983anidm23 1243 . . . . . . 7  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( x  +h  (
z  -h  z ) )  =  ( z  +h  ( x  -h  z ) ) )
10 hvsubid 22528 . . . . . . . . 9  |-  ( z  e.  ~H  ->  (
z  -h  z )  =  0h )
1110oveq2d 6097 . . . . . . . 8  |-  ( z  e.  ~H  ->  (
x  +h  ( z  -h  z ) )  =  ( x  +h  0h ) )
12 ax-hvaddid 22507 . . . . . . . 8  |-  ( x  e.  ~H  ->  (
x  +h  0h )  =  x )
1311, 12sylan9eqr 2490 . . . . . . 7  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( x  +h  (
z  -h  z ) )  =  x )
149, 13eqtr3d 2470 . . . . . 6  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( z  +h  (
x  -h  z ) )  =  x )
154, 7, 14syl2an 464 . . . . 5  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  v  =  ( x  +h  y ) )  /\  ( z  e.  C  /\  ( x  -h  z
)  e.  R ) )  ->  ( z  +h  ( x  -h  z
) )  =  x )
16 5oalem1.4 . . . . . . 7  |-  R  e.  SH
175, 16shsvai 22866 . . . . . 6  |-  ( ( z  e.  C  /\  ( x  -h  z
)  e.  R )  ->  ( z  +h  ( x  -h  z
) )  e.  ( C  +H  R ) )
1817adantl 453 . . . . 5  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  v  =  ( x  +h  y ) )  /\  ( z  e.  C  /\  ( x  -h  z
)  e.  R ) )  ->  ( z  +h  ( x  -h  z
) )  e.  ( C  +H  R ) )
1915, 18eqeltrrd 2511 . . . 4  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  v  =  ( x  +h  y ) )  /\  ( z  e.  C  /\  ( x  -h  z
)  e.  R ) )  ->  x  e.  ( C  +H  R
) )
20 elin 3530 . . . 4  |-  ( x  e.  ( A  i^i  ( C  +H  R
) )  <->  ( x  e.  A  /\  x  e.  ( C  +H  R
) ) )
211, 19, 20sylanbrc 646 . . 3  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  v  =  ( x  +h  y ) )  /\  ( z  e.  C  /\  ( x  -h  z
)  e.  R ) )  ->  x  e.  ( A  i^i  ( C  +H  R ) ) )
22 simpllr 736 . . 3  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  v  =  ( x  +h  y ) )  /\  ( z  e.  C  /\  ( x  -h  z
)  e.  R ) )  ->  y  e.  B )
235, 16shscli 22819 . . . . . 6  |-  ( C  +H  R )  e.  SH
242, 23shincli 22864 . . . . 5  |-  ( A  i^i  ( C  +H  R ) )  e.  SH
25 5oalem1.2 . . . . 5  |-  B  e.  SH
2624, 25shsvai 22866 . . . 4  |-  ( ( x  e.  ( A  i^i  ( C  +H  R ) )  /\  y  e.  B )  ->  ( x  +h  y
)  e.  ( ( A  i^i  ( C  +H  R ) )  +H  B ) )
2724, 25shscomi 22865 . . . 4  |-  ( ( A  i^i  ( C  +H  R ) )  +H  B )  =  ( B  +H  ( A  i^i  ( C  +H  R ) ) )
2826, 27syl6eleq 2526 . . 3  |-  ( ( x  e.  ( A  i^i  ( C  +H  R ) )  /\  y  e.  B )  ->  ( x  +h  y
)  e.  ( B  +H  ( A  i^i  ( C  +H  R
) ) ) )
2921, 22, 28syl2anc 643 . 2  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  v  =  ( x  +h  y ) )  /\  ( z  e.  C  /\  ( x  -h  z
)  e.  R ) )  ->  ( x  +h  y )  e.  ( B  +H  ( A  i^i  ( C  +H  R ) ) ) )
30 eleq1 2496 . . 3  |-  ( v  =  ( x  +h  y )  ->  (
v  e.  ( B  +H  ( A  i^i  ( C  +H  R
) ) )  <->  ( x  +h  y )  e.  ( B  +H  ( A  i^i  ( C  +H  R ) ) ) ) )
3130ad2antlr 708 . 2  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  v  =  ( x  +h  y ) )  /\  ( z  e.  C  /\  ( x  -h  z
)  e.  R ) )  ->  ( v  e.  ( B  +H  ( A  i^i  ( C  +H  R ) ) )  <-> 
( x  +h  y
)  e.  ( B  +H  ( A  i^i  ( C  +H  R
) ) ) ) )
3229, 31mpbird 224 1  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  v  =  ( x  +h  y ) )  /\  ( z  e.  C  /\  ( x  -h  z
)  e.  R ) )  ->  v  e.  ( B  +H  ( A  i^i  ( C  +H  R ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    i^i cin 3319  (class class class)co 6081   ~Hchil 22422    +h cva 22423   0hc0v 22427    -h cmv 22428   SHcsh 22431    +H cph 22434
This theorem is referenced by:  5oalem6  23161
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-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-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-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  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-er 6905  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-grpo 21779  df-ablo 21870  df-hvsub 22474  df-sh 22709  df-shs 22810
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