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Theorem 5oalem1 22081
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 737 . . . 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 21623 . . . . . . 7  |-  ( x  e.  A  ->  x  e.  ~H )
43ad2antrr 709 . . . . . 6  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  v  =  ( x  +h  y
) )  ->  x  e.  ~H )
5 5oalem1.3 . . . . . . . 8  |-  C  e.  SH
65sheli 21623 . . . . . . 7  |-  ( z  e.  C  ->  z  e.  ~H )
76adantr 453 . . . . . 6  |-  ( ( z  e.  C  /\  ( x  -h  z
)  e.  R )  ->  z  e.  ~H )
8 hvaddsub12 21447 . . . . . . . 8  |-  ( ( x  e.  ~H  /\  z  e.  ~H  /\  z  e.  ~H )  ->  (
x  +h  ( z  -h  z ) )  =  ( z  +h  ( x  -h  z
) ) )
983anidm23 1246 . . . . . . 7  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( x  +h  (
z  -h  z ) )  =  ( z  +h  ( x  -h  z ) ) )
10 hvsubid 21435 . . . . . . . . 9  |-  ( z  e.  ~H  ->  (
z  -h  z )  =  0h )
1110oveq2d 5726 . . . . . . . 8  |-  ( z  e.  ~H  ->  (
x  +h  ( z  -h  z ) )  =  ( x  +h  0h ) )
12 ax-hvaddid 21414 . . . . . . . 8  |-  ( x  e.  ~H  ->  (
x  +h  0h )  =  x )
1311, 12sylan9eqr 2307 . . . . . . 7  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( x  +h  (
z  -h  z ) )  =  x )
149, 13eqtr3d 2287 . . . . . 6  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( z  +h  (
x  -h  z ) )  =  x )
154, 7, 14syl2an 465 . . . . 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 21773 . . . . . 6  |-  ( ( z  e.  C  /\  ( x  -h  z
)  e.  R )  ->  ( z  +h  ( x  -h  z
) )  e.  ( C  +H  R ) )
1817adantl 454 . . . . 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 2328 . . . 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 3266 . . . 4  |-  ( x  e.  ( A  i^i  ( C  +H  R
) )  <->  ( x  e.  A  /\  x  e.  ( C  +H  R
) ) )
211, 19, 20sylanbrc 648 . . 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 738 . . 3  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  v  =  ( x  +h  y ) )  /\  ( z  e.  C  /\  ( x  -h  z
)  e.  R ) )  ->  y  e.  B )
235, 16shscli 21726 . . . . . 6  |-  ( C  +H  R )  e.  SH
242, 23shincli 21771 . . . . 5  |-  ( A  i^i  ( C  +H  R ) )  e.  SH
25 5oalem1.2 . . . . 5  |-  B  e.  SH
2624, 25shsvai 21773 . . . 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 21772 . . . 4  |-  ( ( A  i^i  ( C  +H  R ) )  +H  B )  =  ( B  +H  ( A  i^i  ( C  +H  R ) ) )
2826, 27syl6eleq 2343 . . 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 645 . 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 2313 . . 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 710 . 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 225 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 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621    i^i cin 3077  (class class class)co 5710   ~Hchil 21329    +h cva 21330   0hc0v 21334    -h cmv 21335   SHcsh 21338    +H cph 21341
This theorem is referenced by:  5oalem6  22086
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-hilex 21409  ax-hfvadd 21410  ax-hvcom 21411  ax-hvass 21412  ax-hv0cl 21413  ax-hvaddid 21414  ax-hfvmul 21415  ax-hvmulid 21416  ax-hvdistr1 21418  ax-hvdistr2 21419  ax-hvmul0 21420
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-po 4207  df-so 4208  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-iota 6143  df-riota 6190  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-pnf 8749  df-mnf 8750  df-ltxr 8752  df-sub 8919  df-neg 8920  df-grpo 20688  df-ablo 20779  df-hvsub 21381  df-sh 21616  df-shs 21717
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