MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ragperp Structured version   Unicode version

Theorem ragperp 23830
Description: Deduce that two lines are perpendicular from a right angle statement. One direction of theorem 8.13 of [Schwabhauser] p. 59. (Contributed by Thierry Arnoux, 20-Oct-2019.)
Hypotheses
Ref Expression
isperp.p  |-  P  =  ( Base `  G
)
isperp.d  |-  .-  =  ( dist `  G )
isperp.i  |-  I  =  (Itv `  G )
isperp.l  |-  L  =  (LineG `  G )
isperp.g  |-  ( ph  ->  G  e. TarskiG )
isperp.a  |-  ( ph  ->  A  e.  ran  L
)
ragperp.b  |-  ( ph  ->  B  e.  ran  L
)
ragperp.x  |-  ( ph  ->  X  e.  ( A  i^i  B ) )
ragperp.u  |-  ( ph  ->  U  e.  A )
ragperp.v  |-  ( ph  ->  V  e.  B )
ragperp.1  |-  ( ph  ->  U  =/=  X )
ragperp.2  |-  ( ph  ->  V  =/=  X )
ragperp.r  |-  ( ph  ->  <" U X V ">  e.  (∟G `  G ) )
Assertion
Ref Expression
ragperp  |-  ( ph  ->  A (⟂G `  G
) B )

Proof of Theorem ragperp
Dummy variables  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isperp.p . . . 4  |-  P  =  ( Base `  G
)
2 isperp.d . . . 4  |-  .-  =  ( dist `  G )
3 isperp.i . . . 4  |-  I  =  (Itv `  G )
4 isperp.l . . . 4  |-  L  =  (LineG `  G )
5 eqid 2467 . . . 4  |-  (pInvG `  G )  =  (pInvG `  G )
6 isperp.g . . . . 5  |-  ( ph  ->  G  e. TarskiG )
76adantr 465 . . . 4  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  G  e. TarskiG )
8 ragperp.b . . . . . 6  |-  ( ph  ->  B  e.  ran  L
)
98adantr 465 . . . . 5  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  B  e.  ran  L )
10 simprr 756 . . . . 5  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  -> 
v  e.  B )
111, 4, 3, 7, 9, 10tglnpt 23692 . . . 4  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  -> 
v  e.  P )
12 isperp.a . . . . . 6  |-  ( ph  ->  A  e.  ran  L
)
1312adantr 465 . . . . 5  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  A  e.  ran  L )
14 inss1 3718 . . . . . . 7  |-  ( A  i^i  B )  C_  A
15 ragperp.x . . . . . . 7  |-  ( ph  ->  X  e.  ( A  i^i  B ) )
1614, 15sseldi 3502 . . . . . 6  |-  ( ph  ->  X  e.  A )
1716adantr 465 . . . . 5  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  X  e.  A )
181, 4, 3, 7, 13, 17tglnpt 23692 . . . 4  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  X  e.  P )
19 simprl 755 . . . . 5  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  u  e.  A )
201, 4, 3, 7, 13, 19tglnpt 23692 . . . 4  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  u  e.  P )
21 ragperp.v . . . . . . 7  |-  ( ph  ->  V  e.  B )
2221adantr 465 . . . . . 6  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  V  e.  B )
231, 4, 3, 7, 9, 22tglnpt 23692 . . . . 5  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  V  e.  P )
24 ragperp.u . . . . . . . . 9  |-  ( ph  ->  U  e.  A )
2524adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  U  e.  A )
261, 4, 3, 7, 13, 25tglnpt 23692 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  U  e.  P )
27 ragperp.r . . . . . . . 8  |-  ( ph  ->  <" U X V ">  e.  (∟G `  G ) )
2827adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  <" U X V ">  e.  (∟G `  G ) )
29 ragperp.1 . . . . . . . 8  |-  ( ph  ->  U  =/=  X )
3029adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  U  =/=  X )
3125adantr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  U  e.  A )
327adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  G  e. TarskiG )
3318adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  X  e.  P )
3420adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  u  e.  P )
35 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  -.  X  =  u )
3635neqned 2670 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  X  =/=  u )
3713adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  A  e.  ran  L )
3817adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  X  e.  A )
3919adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  u  e.  A )
401, 3, 4, 32, 33, 34, 36, 36, 37, 38, 39tglinethru 23758 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  A  =  ( X L u ) )
4131, 40eleqtrd 2557 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  u )  ->  U  e.  ( X L u ) )
4241ex 434 . . . . . . . . 9  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  -> 
( -.  X  =  u  ->  U  e.  ( X L u ) ) )
4342orrd 378 . . . . . . . 8  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  -> 
( X  =  u  \/  U  e.  ( X L u ) ) )
4443orcomd 388 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  -> 
( U  e.  ( X L u )  \/  X  =  u ) )
451, 2, 3, 4, 5, 7, 26, 18, 23, 20, 28, 30, 44ragcol 23812 . . . . . 6  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  <" u X V ">  e.  (∟G `  G ) )
461, 2, 3, 4, 5, 7, 20, 18, 23, 45ragcom 23811 . . . . 5  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  <" V X u ">  e.  (∟G `  G ) )
47 ragperp.2 . . . . . 6  |-  ( ph  ->  V  =/=  X )
4847adantr 465 . . . . 5  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  V  =/=  X )
4922adantr 465 . . . . . . . . 9  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  ->  V  e.  B )
507adantr 465 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  ->  G  e. TarskiG )
5118adantr 465 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  ->  X  e.  P )
5211adantr 465 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  -> 
v  e.  P )
53 simpr 461 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  ->  -.  X  =  v
)
5453neqned 2670 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  ->  X  =/=  v )
558ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  ->  B  e.  ran  L )
56 inss2 3719 . . . . . . . . . . . 12  |-  ( A  i^i  B )  C_  B
5756, 15sseldi 3502 . . . . . . . . . . 11  |-  ( ph  ->  X  e.  B )
5857ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  ->  X  e.  B )
5910adantr 465 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  -> 
v  e.  B )
601, 3, 4, 50, 51, 52, 54, 54, 55, 58, 59tglinethru 23758 . . . . . . . . 9  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  ->  B  =  ( X L v ) )
6149, 60eleqtrd 2557 . . . . . . . 8  |-  ( ( ( ph  /\  (
u  e.  A  /\  v  e.  B )
)  /\  -.  X  =  v )  ->  V  e.  ( X L v ) )
6261ex 434 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  -> 
( -.  X  =  v  ->  V  e.  ( X L v ) ) )
6362orrd 378 . . . . . 6  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  -> 
( X  =  v  \/  V  e.  ( X L v ) ) )
6463orcomd 388 . . . . 5  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  -> 
( V  e.  ( X L v )  \/  X  =  v ) )
651, 2, 3, 4, 5, 7, 23, 18, 20, 11, 46, 48, 64ragcol 23812 . . . 4  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  <" v X u ">  e.  (∟G `  G ) )
661, 2, 3, 4, 5, 7, 11, 18, 20, 65ragcom 23811 . . 3  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  B ) )  ->  <" u X v ">  e.  (∟G `  G ) )
6766ralrimivva 2885 . 2  |-  ( ph  ->  A. u  e.  A  A. v  e.  B  <" u X v ">  e.  (∟G `  G ) )
681, 2, 3, 4, 6, 12, 8, 15isperp2 23828 . 2  |-  ( ph  ->  ( A (⟂G `  G
) B  <->  A. u  e.  A  A. v  e.  B  <" u X v ">  e.  (∟G `  G )
) )
6967, 68mpbird 232 1  |-  ( ph  ->  A (⟂G `  G
) B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814    i^i cin 3475   class class class wbr 4447   ran crn 5000   ` cfv 5588  (class class class)co 6284   <"cs3 12770   Basecbs 14490   distcds 14564  TarskiGcstrkg 23581  Itvcitv 23588  LineGclng 23589  pInvGcmir 23774  ∟Gcrag 23806  ⟂Gcperpg 23808
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
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 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-fzo 11793  df-hash 12374  df-word 12508  df-concat 12510  df-s1 12511  df-s2 12776  df-s3 12777  df-trkgc 23600  df-trkgb 23601  df-trkgcb 23602  df-trkg 23606  df-cgrg 23659  df-mir 23775  df-rag 23807  df-perpg 23809
This theorem is referenced by:  footex  23831  colperpexlem3  23839  mideulem  23841  lmimid  23864  hypcgrlem1  23869  hypcgrlem2  23870
  Copyright terms: Public domain W3C validator