MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ragflat3 Structured version   Unicode version

Theorem ragflat3 23791
Description: Right angle and colinearity. Theorem 8.9 of [Schwabhauser] p. 58. (Contributed by Thierry Arnoux, 4-Sep-2019.)
Hypotheses
Ref Expression
israg.p  |-  P  =  ( Base `  G
)
israg.d  |-  .-  =  ( dist `  G )
israg.i  |-  I  =  (Itv `  G )
israg.l  |-  L  =  (LineG `  G )
israg.s  |-  S  =  (pInvG `  G )
israg.g  |-  ( ph  ->  G  e. TarskiG )
israg.a  |-  ( ph  ->  A  e.  P )
israg.b  |-  ( ph  ->  B  e.  P )
israg.c  |-  ( ph  ->  C  e.  P )
ragflat3.1  |-  ( ph  ->  <" A B C ">  e.  (∟G `  G ) )
ragflat3.2  |-  ( ph  ->  ( C  e.  ( A L B )  \/  A  =  B ) )
Assertion
Ref Expression
ragflat3  |-  ( ph  ->  ( A  =  B  \/  C  =  B ) )

Proof of Theorem ragflat3
StepHypRef Expression
1 israg.p . . . 4  |-  P  =  ( Base `  G
)
2 israg.d . . . 4  |-  .-  =  ( dist `  G )
3 israg.i . . . 4  |-  I  =  (Itv `  G )
4 israg.l . . . 4  |-  L  =  (LineG `  G )
5 israg.s . . . 4  |-  S  =  (pInvG `  G )
6 israg.g . . . . 5  |-  ( ph  ->  G  e. TarskiG )
76adantr 465 . . . 4  |-  ( (
ph  /\  -.  A  =  B )  ->  G  e. TarskiG )
8 israg.c . . . . 5  |-  ( ph  ->  C  e.  P )
98adantr 465 . . . 4  |-  ( (
ph  /\  -.  A  =  B )  ->  C  e.  P )
10 israg.b . . . . 5  |-  ( ph  ->  B  e.  P )
1110adantr 465 . . . 4  |-  ( (
ph  /\  -.  A  =  B )  ->  B  e.  P )
12 israg.a . . . . 5  |-  ( ph  ->  A  e.  P )
1312adantr 465 . . . 4  |-  ( (
ph  /\  -.  A  =  B )  ->  A  e.  P )
14 ragflat3.1 . . . . . 6  |-  ( ph  ->  <" A B C ">  e.  (∟G `  G ) )
1514adantr 465 . . . . 5  |-  ( (
ph  /\  -.  A  =  B )  ->  <" A B C ">  e.  (∟G `  G ) )
16 simpr 461 . . . . . 6  |-  ( (
ph  /\  -.  A  =  B )  ->  -.  A  =  B )
1716neqned 2670 . . . . 5  |-  ( (
ph  /\  -.  A  =  B )  ->  A  =/=  B )
18 ragflat3.2 . . . . . . 7  |-  ( ph  ->  ( C  e.  ( A L B )  \/  A  =  B ) )
1918adantr 465 . . . . . 6  |-  ( (
ph  /\  -.  A  =  B )  ->  ( C  e.  ( A L B )  \/  A  =  B ) )
201, 4, 3, 7, 13, 11, 9, 19colrot1 23674 . . . . 5  |-  ( (
ph  /\  -.  A  =  B )  ->  ( A  e.  ( B L C )  \/  B  =  C ) )
211, 2, 3, 4, 5, 7, 13, 11, 9, 9, 15, 17, 20ragcol 23784 . . . 4  |-  ( (
ph  /\  -.  A  =  B )  ->  <" C B C ">  e.  (∟G `  G ) )
221, 2, 3, 4, 5, 7, 9, 11, 13, 21ragtriva 23790 . . 3  |-  ( (
ph  /\  -.  A  =  B )  ->  C  =  B )
2322ex 434 . 2  |-  ( ph  ->  ( -.  A  =  B  ->  C  =  B ) )
24 pm4.64 372 . 2  |-  ( ( -.  A  =  B  ->  C  =  B )  <->  ( A  =  B  \/  C  =  B ) )
2523, 24sylib 196 1  |-  ( ph  ->  ( A  =  B  \/  C  =  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767   ` cfv 5586  (class class class)co 6282   <"cs3 12766   Basecbs 14486   distcds 14560  TarskiGcstrkg 23553  Itvcitv 23560  LineGclng 23561  pInvGcmir 23746  ∟Gcrag 23778
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 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-hash 12370  df-word 12504  df-concat 12506  df-s1 12507  df-s2 12772  df-s3 12773  df-trkgc 23572  df-trkgb 23573  df-trkgcb 23574  df-trkg 23578  df-cgrg 23631  df-mir 23747  df-rag 23779
This theorem is referenced by:  ragncol  23794  mideulem  23813
  Copyright terms: Public domain W3C validator