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Theorem 5oalem1 26386
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 757 . . . 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 25945 . . . . . . 7  |-  ( x  e.  A  ->  x  e.  ~H )
43ad2antrr 725 . . . . . 6  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  v  =  ( x  +h  y
) )  ->  x  e.  ~H )
5 5oalem1.3 . . . . . . . 8  |-  C  e.  SH
65sheli 25945 . . . . . . 7  |-  ( z  e.  C  ->  z  e.  ~H )
76adantr 465 . . . . . 6  |-  ( ( z  e.  C  /\  ( x  -h  z
)  e.  R )  ->  z  e.  ~H )
8 hvaddsub12 25769 . . . . . . . 8  |-  ( ( x  e.  ~H  /\  z  e.  ~H  /\  z  e.  ~H )  ->  (
x  +h  ( z  -h  z ) )  =  ( z  +h  ( x  -h  z
) ) )
983anidm23 1287 . . . . . . 7  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( x  +h  (
z  -h  z ) )  =  ( z  +h  ( x  -h  z ) ) )
10 hvsubid 25757 . . . . . . . . 9  |-  ( z  e.  ~H  ->  (
z  -h  z )  =  0h )
1110oveq2d 6311 . . . . . . . 8  |-  ( z  e.  ~H  ->  (
x  +h  ( z  -h  z ) )  =  ( x  +h  0h ) )
12 ax-hvaddid 25735 . . . . . . . 8  |-  ( x  e.  ~H  ->  (
x  +h  0h )  =  x )
1311, 12sylan9eqr 2530 . . . . . . 7  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( x  +h  (
z  -h  z ) )  =  x )
149, 13eqtr3d 2510 . . . . . 6  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( z  +h  (
x  -h  z ) )  =  x )
154, 7, 14syl2an 477 . . . . 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 26096 . . . . . 6  |-  ( ( z  e.  C  /\  ( x  -h  z
)  e.  R )  ->  ( z  +h  ( x  -h  z
) )  e.  ( C  +H  R ) )
1817adantl 466 . . . . 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 2556 . . . 4  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  v  =  ( x  +h  y ) )  /\  ( z  e.  C  /\  ( x  -h  z
)  e.  R ) )  ->  x  e.  ( C  +H  R
) )
201, 19elind 3693 . . 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 ) ) )
21 simpllr 758 . . 3  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  v  =  ( x  +h  y ) )  /\  ( z  e.  C  /\  ( x  -h  z
)  e.  R ) )  ->  y  e.  B )
225, 16shscli 26049 . . . . . 6  |-  ( C  +H  R )  e.  SH
232, 22shincli 26094 . . . . 5  |-  ( A  i^i  ( C  +H  R ) )  e.  SH
24 5oalem1.2 . . . . 5  |-  B  e.  SH
2523, 24shsvai 26096 . . . 4  |-  ( ( x  e.  ( A  i^i  ( C  +H  R ) )  /\  y  e.  B )  ->  ( x  +h  y
)  e.  ( ( A  i^i  ( C  +H  R ) )  +H  B ) )
2623, 24shscomi 26095 . . . 4  |-  ( ( A  i^i  ( C  +H  R ) )  +H  B )  =  ( B  +H  ( A  i^i  ( C  +H  R ) ) )
2725, 26syl6eleq 2565 . . 3  |-  ( ( x  e.  ( A  i^i  ( C  +H  R ) )  /\  y  e.  B )  ->  ( x  +h  y
)  e.  ( B  +H  ( A  i^i  ( C  +H  R
) ) ) )
2820, 21, 27syl2anc 661 . 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 ) ) ) )
29 eleq1 2539 . . 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 ) ) ) ) )
3029ad2antlr 726 . 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
) ) ) ) )
3128, 30mpbird 232 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    i^i cin 3480  (class class class)co 6295   ~Hchil 25650    +h cva 25651   0hc0v 25655    -h cmv 25656   SHcsh 25659    +H cph 25662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-hilex 25730  ax-hfvadd 25731  ax-hvcom 25732  ax-hvass 25733  ax-hv0cl 25734  ax-hvaddid 25735  ax-hfvmul 25736  ax-hvmulid 25737  ax-hvdistr1 25739  ax-hvdistr2 25740  ax-hvmul0 25741
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-ltxr 9645  df-sub 9819  df-neg 9820  df-grpo 25007  df-ablo 25098  df-hvsub 25702  df-sh 25938  df-shs 26040
This theorem is referenced by:  5oalem6  26391
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