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Theorem 5oalem1 23109
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 22669 . . . . . . 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 22669 . . . . . . 7  |-  ( z  e.  C  ->  z  e.  ~H )
76adantr 452 . . . . . 6  |-  ( ( z  e.  C  /\  ( x  -h  z
)  e.  R )  ->  z  e.  ~H )
8 hvaddsub12 22493 . . . . . . . 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 22481 . . . . . . . . 9  |-  ( z  e.  ~H  ->  (
z  -h  z )  =  0h )
1110oveq2d 6056 . . . . . . . 8  |-  ( z  e.  ~H  ->  (
x  +h  ( z  -h  z ) )  =  ( x  +h  0h ) )
12 ax-hvaddid 22460 . . . . . . . 8  |-  ( x  e.  ~H  ->  (
x  +h  0h )  =  x )
1311, 12sylan9eqr 2458 . . . . . . 7  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( x  +h  (
z  -h  z ) )  =  x )
149, 13eqtr3d 2438 . . . . . 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 22819 . . . . . 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 2479 . . . 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 3490 . . . 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 22772 . . . . . 6  |-  ( C  +H  R )  e.  SH
242, 23shincli 22817 . . . . 5  |-  ( A  i^i  ( C  +H  R ) )  e.  SH
25 5oalem1.2 . . . . 5  |-  B  e.  SH
2624, 25shsvai 22819 . . . 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 22818 . . . 4  |-  ( ( A  i^i  ( C  +H  R ) )  +H  B )  =  ( B  +H  ( A  i^i  ( C  +H  R ) ) )
2826, 27syl6eleq 2494 . . 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 2464 . . 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 1649    e. wcel 1721    i^i cin 3279  (class class class)co 6040   ~Hchil 22375    +h cva 22376   0hc0v 22380    -h cmv 22381   SHcsh 22384    +H cph 22387
This theorem is referenced by:  5oalem6  23114
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-hilex 22455  ax-hfvadd 22456  ax-hvcom 22457  ax-hvass 22458  ax-hv0cl 22459  ax-hvaddid 22460  ax-hfvmul 22461  ax-hvmulid 22462  ax-hvdistr1 22464  ax-hvdistr2 22465  ax-hvmul0 22466
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  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-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-ltxr 9081  df-sub 9249  df-neg 9250  df-grpo 21732  df-ablo 21823  df-hvsub 22427  df-sh 22662  df-shs 22763
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