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Theorem israg 23091
Description: Property for 3 points A, B, C to form a right angle. Definition 8.1 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 )
Assertion
Ref Expression
israg  |-  ( ph  ->  ( <" A B C ">  e.  (∟G `  G )  <->  ( A  .-  C )  =  ( A  .-  ( ( S `  B ) `
 C ) ) ) )

Proof of Theorem israg
Dummy variables  g  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 israg.a . . . 4  |-  ( ph  ->  A  e.  P )
2 israg.b . . . 4  |-  ( ph  ->  B  e.  P )
3 israg.c . . . 4  |-  ( ph  ->  C  e.  P )
41, 2, 3s3cld 12497 . . 3  |-  ( ph  ->  <" A B C ">  e. Word  P )
5 fveq2 5691 . . . . . 6  |-  ( w  =  <" A B C ">  ->  (
# `  w )  =  ( # `  <" A B C "> ) )
65eqeq1d 2451 . . . . 5  |-  ( w  =  <" A B C ">  ->  ( ( # `  w
)  =  3  <->  ( # `
 <" A B C "> )  =  3 ) )
7 fveq1 5690 . . . . . . 7  |-  ( w  =  <" A B C ">  ->  ( w `  0 )  =  ( <" A B C "> `  0
) )
8 fveq1 5690 . . . . . . 7  |-  ( w  =  <" A B C ">  ->  ( w `  2 )  =  ( <" A B C "> `  2
) )
97, 8oveq12d 6109 . . . . . 6  |-  ( w  =  <" A B C ">  ->  ( ( w `  0
)  .-  ( w `  2 ) )  =  ( ( <" A B C "> `  0
)  .-  ( <" A B C "> `  2 ) ) )
10 fveq1 5690 . . . . . . . . 9  |-  ( w  =  <" A B C ">  ->  ( w `  1 )  =  ( <" A B C "> `  1
) )
1110fveq2d 5695 . . . . . . . 8  |-  ( w  =  <" A B C ">  ->  ( S `  ( w `
 1 ) )  =  ( S `  ( <" A B C "> `  1
) ) )
1211, 8fveq12d 5697 . . . . . . 7  |-  ( w  =  <" A B C ">  ->  ( ( S `  (
w `  1 )
) `  ( w `  2 ) )  =  ( ( S `
 ( <" A B C "> `  1
) ) `  ( <" A B C "> `  2
) ) )
137, 12oveq12d 6109 . . . . . 6  |-  ( w  =  <" A B C ">  ->  ( ( w `  0
)  .-  ( ( S `  ( w `  1 ) ) `
 ( w ` 
2 ) ) )  =  ( ( <" A B C "> `  0
)  .-  ( ( S `  ( <" A B C "> `  1 ) ) `
 ( <" A B C "> `  2
) ) ) )
149, 13eqeq12d 2457 . . . . 5  |-  ( w  =  <" A B C ">  ->  ( ( ( w ` 
0 )  .-  (
w `  2 )
)  =  ( ( w `  0 ) 
.-  ( ( S `
 ( w ` 
1 ) ) `  ( w `  2
) ) )  <->  ( ( <" A B C "> `  0
)  .-  ( <" A B C "> `  2 ) )  =  ( ( <" A B C "> `  0
)  .-  ( ( S `  ( <" A B C "> `  1 ) ) `
 ( <" A B C "> `  2
) ) ) ) )
156, 14anbi12d 710 . . . 4  |-  ( w  =  <" A B C ">  ->  ( ( ( # `  w
)  =  3  /\  ( ( w ` 
0 )  .-  (
w `  2 )
)  =  ( ( w `  0 ) 
.-  ( ( S `
 ( w ` 
1 ) ) `  ( w `  2
) ) ) )  <-> 
( ( # `  <" A B C "> )  =  3  /\  ( ( <" A B C "> `  0
)  .-  ( <" A B C "> `  2 ) )  =  ( ( <" A B C "> `  0
)  .-  ( ( S `  ( <" A B C "> `  1 ) ) `
 ( <" A B C "> `  2
) ) ) ) ) )
1615elrab3 3118 . . 3  |-  ( <" A B C ">  e. Word  P  ->  ( <" A B C ">  e.  { w  e. Word  P  | 
( ( # `  w
)  =  3  /\  ( ( w ` 
0 )  .-  (
w `  2 )
)  =  ( ( w `  0 ) 
.-  ( ( S `
 ( w ` 
1 ) ) `  ( w `  2
) ) ) ) }  <->  ( ( # `  <" A B C "> )  =  3  /\  (
( <" A B C "> `  0
)  .-  ( <" A B C "> `  2 ) )  =  ( ( <" A B C "> `  0
)  .-  ( ( S `  ( <" A B C "> `  1 ) ) `
 ( <" A B C "> `  2
) ) ) ) ) )
174, 16syl 16 . 2  |-  ( ph  ->  ( <" A B C ">  e.  { w  e. Word  P  | 
( ( # `  w
)  =  3  /\  ( ( w ` 
0 )  .-  (
w `  2 )
)  =  ( ( w `  0 ) 
.-  ( ( S `
 ( w ` 
1 ) ) `  ( w `  2
) ) ) ) }  <->  ( ( # `  <" A B C "> )  =  3  /\  (
( <" A B C "> `  0
)  .-  ( <" A B C "> `  2 ) )  =  ( ( <" A B C "> `  0
)  .-  ( ( S `  ( <" A B C "> `  1 ) ) `
 ( <" A B C "> `  2
) ) ) ) ) )
18 df-rag 23088 . . . . 5  |- ∟G  =  ( g  e.  _V  |->  { w  e. Word  ( Base `  g )  |  ( ( # `  w
)  =  3  /\  ( ( w ` 
0 ) ( dist `  g ) ( w `
 2 ) )  =  ( ( w `
 0 ) (
dist `  g )
( ( (pInvG `  g ) `  (
w `  1 )
) `  ( w `  2 ) ) ) ) } )
1918a1i 11 . . . 4  |-  ( ph  -> ∟G 
=  ( g  e. 
_V  |->  { w  e. Word 
( Base `  g )  |  ( ( # `  w )  =  3  /\  ( ( w `
 0 ) (
dist `  g )
( w `  2
) )  =  ( ( w `  0
) ( dist `  g
) ( ( (pInvG `  g ) `  (
w `  1 )
) `  ( w `  2 ) ) ) ) } ) )
20 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  g  =  G )  ->  g  =  G )
2120fveq2d 5695 . . . . . . 7  |-  ( (
ph  /\  g  =  G )  ->  ( Base `  g )  =  ( Base `  G
) )
22 israg.p . . . . . . 7  |-  P  =  ( Base `  G
)
2321, 22syl6eqr 2493 . . . . . 6  |-  ( (
ph  /\  g  =  G )  ->  ( Base `  g )  =  P )
24 wrdeq 12251 . . . . . 6  |-  ( (
Base `  g )  =  P  -> Word  ( Base `  g )  = Word  P
)
2523, 24syl 16 . . . . 5  |-  ( (
ph  /\  g  =  G )  -> Word  ( Base `  g )  = Word  P
)
26 biidd 237 . . . . . 6  |-  ( (
ph  /\  g  =  G )  ->  (
( # `  w )  =  3  <->  ( # `  w
)  =  3 ) )
2720fveq2d 5695 . . . . . . . . 9  |-  ( (
ph  /\  g  =  G )  ->  ( dist `  g )  =  ( dist `  G
) )
28 israg.d . . . . . . . . 9  |-  .-  =  ( dist `  G )
2927, 28syl6eqr 2493 . . . . . . . 8  |-  ( (
ph  /\  g  =  G )  ->  ( dist `  g )  = 
.-  )
3029oveqd 6108 . . . . . . 7  |-  ( (
ph  /\  g  =  G )  ->  (
( w `  0
) ( dist `  g
) ( w ` 
2 ) )  =  ( ( w ` 
0 )  .-  (
w `  2 )
) )
31 eqidd 2444 . . . . . . . 8  |-  ( (
ph  /\  g  =  G )  ->  (
w `  0 )  =  ( w ` 
0 ) )
3220fveq2d 5695 . . . . . . . . . . 11  |-  ( (
ph  /\  g  =  G )  ->  (pInvG `  g )  =  (pInvG `  G ) )
33 israg.s . . . . . . . . . . 11  |-  S  =  (pInvG `  G )
3432, 33syl6eqr 2493 . . . . . . . . . 10  |-  ( (
ph  /\  g  =  G )  ->  (pInvG `  g )  =  S )
35 eqidd 2444 . . . . . . . . . 10  |-  ( (
ph  /\  g  =  G )  ->  (
w `  1 )  =  ( w ` 
1 ) )
3634, 35fveq12d 5697 . . . . . . . . 9  |-  ( (
ph  /\  g  =  G )  ->  (
(pInvG `  g ) `  ( w `  1
) )  =  ( S `  ( w `
 1 ) ) )
37 eqidd 2444 . . . . . . . . 9  |-  ( (
ph  /\  g  =  G )  ->  (
w `  2 )  =  ( w ` 
2 ) )
3836, 37fveq12d 5697 . . . . . . . 8  |-  ( (
ph  /\  g  =  G )  ->  (
( (pInvG `  g
) `  ( w `  1 ) ) `
 ( w ` 
2 ) )  =  ( ( S `  ( w `  1
) ) `  (
w `  2 )
) )
3929, 31, 38oveq123d 6112 . . . . . . 7  |-  ( (
ph  /\  g  =  G )  ->  (
( w `  0
) ( dist `  g
) ( ( (pInvG `  g ) `  (
w `  1 )
) `  ( w `  2 ) ) )  =  ( ( w `  0 ) 
.-  ( ( S `
 ( w ` 
1 ) ) `  ( w `  2
) ) ) )
4030, 39eqeq12d 2457 . . . . . 6  |-  ( (
ph  /\  g  =  G )  ->  (
( ( w ` 
0 ) ( dist `  g ) ( w `
 2 ) )  =  ( ( w `
 0 ) (
dist `  g )
( ( (pInvG `  g ) `  (
w `  1 )
) `  ( w `  2 ) ) )  <->  ( ( w `
 0 )  .-  ( w `  2
) )  =  ( ( w `  0
)  .-  ( ( S `  ( w `  1 ) ) `
 ( w ` 
2 ) ) ) ) )
4126, 40anbi12d 710 . . . . 5  |-  ( (
ph  /\  g  =  G )  ->  (
( ( # `  w
)  =  3  /\  ( ( w ` 
0 ) ( dist `  g ) ( w `
 2 ) )  =  ( ( w `
 0 ) (
dist `  g )
( ( (pInvG `  g ) `  (
w `  1 )
) `  ( w `  2 ) ) ) )  <->  ( ( # `
 w )  =  3  /\  ( ( w `  0 ) 
.-  ( w ` 
2 ) )  =  ( ( w ` 
0 )  .-  (
( S `  (
w `  1 )
) `  ( w `  2 ) ) ) ) ) )
4225, 41rabeqbidv 2967 . . . 4  |-  ( (
ph  /\  g  =  G )  ->  { w  e. Word  ( Base `  g
)  |  ( (
# `  w )  =  3  /\  (
( w `  0
) ( dist `  g
) ( w ` 
2 ) )  =  ( ( w ` 
0 ) ( dist `  g ) ( ( (pInvG `  g ) `  ( w `  1
) ) `  (
w `  2 )
) ) ) }  =  { w  e. Word  P  |  ( ( # `
 w )  =  3  /\  ( ( w `  0 ) 
.-  ( w ` 
2 ) )  =  ( ( w ` 
0 )  .-  (
( S `  (
w `  1 )
) `  ( w `  2 ) ) ) ) } )
43 israg.g . . . . 5  |-  ( ph  ->  G  e. TarskiG )
44 elex 2981 . . . . 5  |-  ( G  e. TarskiG  ->  G  e.  _V )
4543, 44syl 16 . . . 4  |-  ( ph  ->  G  e.  _V )
46 fvex 5701 . . . . . . . 8  |-  ( Base `  G )  e.  _V
4722, 46eqeltri 2513 . . . . . . 7  |-  P  e. 
_V
48 wrdexg 12244 . . . . . . 7  |-  ( P  e.  _V  -> Word  P  e. 
_V )
4947, 48ax-mp 5 . . . . . 6  |- Word  P  e. 
_V
5049rabex 4443 . . . . 5  |-  { w  e. Word  P  |  ( (
# `  w )  =  3  /\  (
( w `  0
)  .-  ( w `  2 ) )  =  ( ( w `
 0 )  .-  ( ( S `  ( w `  1
) ) `  (
w `  2 )
) ) ) }  e.  _V
5150a1i 11 . . . 4  |-  ( ph  ->  { w  e. Word  P  |  ( ( # `  w )  =  3  /\  ( ( w `
 0 )  .-  ( w `  2
) )  =  ( ( w `  0
)  .-  ( ( S `  ( w `  1 ) ) `
 ( w ` 
2 ) ) ) ) }  e.  _V )
5219, 42, 45, 51fvmptd 5779 . . 3  |-  ( ph  ->  (∟G `  G )  =  { w  e. Word  P  |  ( ( # `  w )  =  3  /\  ( ( w `
 0 )  .-  ( w `  2
) )  =  ( ( w `  0
)  .-  ( ( S `  ( w `  1 ) ) `
 ( w ` 
2 ) ) ) ) } )
5352eleq2d 2510 . 2  |-  ( ph  ->  ( <" A B C ">  e.  (∟G `  G )  <->  <" A B C ">  e.  { w  e. Word  P  | 
( ( # `  w
)  =  3  /\  ( ( w ` 
0 )  .-  (
w `  2 )
)  =  ( ( w `  0 ) 
.-  ( ( S `
 ( w ` 
1 ) ) `  ( w `  2
) ) ) ) } ) )
54 s3fv0 12515 . . . . . . 7  |-  ( A  e.  P  ->  ( <" A B C "> `  0
)  =  A )
551, 54syl 16 . . . . . 6  |-  ( ph  ->  ( <" A B C "> `  0
)  =  A )
5655eqcomd 2448 . . . . 5  |-  ( ph  ->  A  =  ( <" A B C "> `  0
) )
57 s3fv2 12517 . . . . . . 7  |-  ( C  e.  P  ->  ( <" A B C "> `  2
)  =  C )
583, 57syl 16 . . . . . 6  |-  ( ph  ->  ( <" A B C "> `  2
)  =  C )
5958eqcomd 2448 . . . . 5  |-  ( ph  ->  C  =  ( <" A B C "> `  2
) )
6056, 59oveq12d 6109 . . . 4  |-  ( ph  ->  ( A  .-  C
)  =  ( (
<" A B C "> `  0
)  .-  ( <" A B C "> `  2 ) ) )
61 s3fv1 12516 . . . . . . . . 9  |-  ( B  e.  P  ->  ( <" A B C "> `  1
)  =  B )
622, 61syl 16 . . . . . . . 8  |-  ( ph  ->  ( <" A B C "> `  1
)  =  B )
6362eqcomd 2448 . . . . . . 7  |-  ( ph  ->  B  =  ( <" A B C "> `  1
) )
6463fveq2d 5695 . . . . . 6  |-  ( ph  ->  ( S `  B
)  =  ( S `
 ( <" A B C "> `  1
) ) )
6564, 59fveq12d 5697 . . . . 5  |-  ( ph  ->  ( ( S `  B ) `  C
)  =  ( ( S `  ( <" A B C "> `  1
) ) `  ( <" A B C "> `  2
) ) )
6656, 65oveq12d 6109 . . . 4  |-  ( ph  ->  ( A  .-  (
( S `  B
) `  C )
)  =  ( (
<" A B C "> `  0
)  .-  ( ( S `  ( <" A B C "> `  1 ) ) `
 ( <" A B C "> `  2
) ) ) )
6760, 66eqeq12d 2457 . . 3  |-  ( ph  ->  ( ( A  .-  C )  =  ( A  .-  ( ( S `  B ) `
 C ) )  <-> 
( ( <" A B C "> `  0
)  .-  ( <" A B C "> `  2 ) )  =  ( ( <" A B C "> `  0
)  .-  ( ( S `  ( <" A B C "> `  1 ) ) `
 ( <" A B C "> `  2
) ) ) ) )
68 s3len 12518 . . . . 5  |-  ( # `  <" A B C "> )  =  3
6968a1i 11 . . . 4  |-  ( ph  ->  ( # `  <" A B C "> )  =  3
)
7069biantrurd 508 . . 3  |-  ( ph  ->  ( ( ( <" A B C "> `  0
)  .-  ( <" A B C "> `  2 ) )  =  ( ( <" A B C "> `  0
)  .-  ( ( S `  ( <" A B C "> `  1 ) ) `
 ( <" A B C "> `  2
) ) )  <->  ( ( # `
 <" A B C "> )  =  3  /\  (
( <" A B C "> `  0
)  .-  ( <" A B C "> `  2 ) )  =  ( ( <" A B C "> `  0
)  .-  ( ( S `  ( <" A B C "> `  1 ) ) `
 ( <" A B C "> `  2
) ) ) ) ) )
7167, 70bitrd 253 . 2  |-  ( ph  ->  ( ( A  .-  C )  =  ( A  .-  ( ( S `  B ) `
 C ) )  <-> 
( ( # `  <" A B C "> )  =  3  /\  ( ( <" A B C "> `  0
)  .-  ( <" A B C "> `  2 ) )  =  ( ( <" A B C "> `  0
)  .-  ( ( S `  ( <" A B C "> `  1 ) ) `
 ( <" A B C "> `  2
) ) ) ) ) )
7217, 53, 713bitr4d 285 1  |-  ( ph  ->  ( <" A B C ">  e.  (∟G `  G )  <->  ( A  .-  C )  =  ( A  .-  ( ( S `  B ) `
 C ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   {crab 2719   _Vcvv 2972    e. cmpt 4350   ` cfv 5418  (class class class)co 6091   0cc0 9282   1c1 9283   2c2 10371   3c3 10372   #chash 12103  Word cword 12221   <"cs3 12469   Basecbs 14174   distcds 14247  TarskiGcstrkg 22889  Itvcitv 22897  LineGclng 22898  pInvGcmir 23055  ∟Gcrag 23087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-fzo 11549  df-hash 12104  df-word 12229  df-concat 12231  df-s1 12232  df-s2 12475  df-s3 12476  df-rag 23088
This theorem is referenced by:  ragcom  23092  ragcol  23093  ragmir  23094  mirrag  23095  ragtrivb  23096  ragflat2  23097  ragflat  23098  ragcgr  23101  footex  23109  colperplem1  23112
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