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Theorem ragcom 23204
Description: Commutative rule for right angles. Theorem 8.2 of [Schwabhauser] p. 57. (Contributed by Thierry Arnoux, 25-Aug-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 )
ragcom.1  |-  ( ph  ->  <" A B C ">  e.  (∟G `  G ) )
Assertion
Ref Expression
ragcom  |-  ( ph  ->  <" C B A ">  e.  (∟G `  G ) )

Proof of Theorem ragcom
StepHypRef Expression
1 israg.p . . . 4  |-  P  =  ( Base `  G
)
2 israg.d . . . 4  |-  .-  =  ( dist `  G )
3 israg.i . . . 4  |-  I  =  (Itv `  G )
4 israg.g . . . 4  |-  ( ph  ->  G  e. TarskiG )
5 israg.a . . . 4  |-  ( ph  ->  A  e.  P )
6 israg.c . . . 4  |-  ( ph  ->  C  e.  P )
7 israg.l . . . . 5  |-  L  =  (LineG `  G )
8 israg.s . . . . 5  |-  S  =  (pInvG `  G )
9 israg.b . . . . 5  |-  ( ph  ->  B  e.  P )
10 eqid 2450 . . . . 5  |-  ( S `
 B )  =  ( S `  B
)
111, 2, 3, 7, 8, 4, 9, 10, 6mircl 23177 . . . 4  |-  ( ph  ->  ( ( S `  B ) `  C
)  e.  P )
12 ragcom.1 . . . . 5  |-  ( ph  ->  <" A B C ">  e.  (∟G `  G ) )
131, 2, 3, 7, 8, 4, 5, 9, 6israg 23203 . . . . 5  |-  ( ph  ->  ( <" A B C ">  e.  (∟G `  G )  <->  ( A  .-  C )  =  ( A  .-  ( ( S `  B ) `
 C ) ) ) )
1412, 13mpbid 210 . . . 4  |-  ( ph  ->  ( A  .-  C
)  =  ( A 
.-  ( ( S `
 B ) `  C ) ) )
151, 2, 3, 4, 5, 6, 5, 11, 14tgcgrcomlr 23037 . . 3  |-  ( ph  ->  ( C  .-  A
)  =  ( ( ( S `  B
) `  C )  .-  A ) )
161, 2, 3, 7, 8, 4, 9, 10, 11, 5miriso 23185 . . 3  |-  ( ph  ->  ( ( ( S `
 B ) `  ( ( S `  B ) `  C
) )  .-  (
( S `  B
) `  A )
)  =  ( ( ( S `  B
) `  C )  .-  A ) )
171, 2, 3, 7, 8, 4, 9, 10, 6mirmir 23178 . . . 4  |-  ( ph  ->  ( ( S `  B ) `  (
( S `  B
) `  C )
)  =  C )
1817oveq1d 6191 . . 3  |-  ( ph  ->  ( ( ( S `
 B ) `  ( ( S `  B ) `  C
) )  .-  (
( S `  B
) `  A )
)  =  ( C 
.-  ( ( S `
 B ) `  A ) ) )
1915, 16, 183eqtr2d 2496 . 2  |-  ( ph  ->  ( C  .-  A
)  =  ( C 
.-  ( ( S `
 B ) `  A ) ) )
201, 2, 3, 7, 8, 4, 6, 9, 5israg 23203 . 2  |-  ( ph  ->  ( <" C B A ">  e.  (∟G `  G )  <->  ( C  .-  A )  =  ( C  .-  ( ( S `  B ) `
 A ) ) ) )
2119, 20mpbird 232 1  |-  ( ph  ->  <" C B A ">  e.  (∟G `  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1757   ` cfv 5502  (class class class)co 6176   <"cs3 12557   Basecbs 14262   distcds 14335  TarskiGcstrkg 22991  Itvcitv 22998  LineGclng 22999  pInvGcmir 23167  ∟Gcrag 23199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-cnex 9425  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-addrcl 9430  ax-mulcl 9431  ax-mulrcl 9432  ax-mulcom 9433  ax-addass 9434  ax-mulass 9435  ax-distr 9436  ax-i2m1 9437  ax-1ne0 9438  ax-1rid 9439  ax-rnegex 9440  ax-rrecex 9441  ax-cnre 9442  ax-pre-lttri 9443  ax-pre-lttrn 9444  ax-pre-ltadd 9445  ax-pre-mulgt0 9446
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rmo 2800  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-int 4213  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-om 6563  df-1st 6663  df-2nd 6664  df-recs 6918  df-rdg 6952  df-1o 7006  df-oadd 7010  df-er 7187  df-map 7302  df-pm 7303  df-en 7397  df-dom 7398  df-sdom 7399  df-fin 7400  df-card 8196  df-cda 8424  df-pnf 9507  df-mnf 9508  df-xr 9509  df-ltxr 9510  df-le 9511  df-sub 9684  df-neg 9685  df-nn 10410  df-2 10467  df-3 10468  df-n0 10667  df-z 10734  df-uz 10949  df-fz 11525  df-fzo 11636  df-hash 12191  df-word 12317  df-concat 12319  df-s1 12320  df-s2 12563  df-s3 12564  df-trkgc 23010  df-trkgb 23011  df-trkgcb 23012  df-trkg 23016  df-mir 23168  df-rag 23200
This theorem is referenced by:  ragflat  23210  ragtriva  23211  perpcom  23218  ragperp  23222  footex  23223  colperplem3  23228  mideulem  23230
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