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Theorem ismidb 24820
Description: Property of the midpoint. (Contributed by Thierry Arnoux, 1-Dec-2019.)
Hypotheses
Ref Expression
ismid.p  |-  P  =  ( Base `  G
)
ismid.d  |-  .-  =  ( dist `  G )
ismid.i  |-  I  =  (Itv `  G )
ismid.g  |-  ( ph  ->  G  e. TarskiG )
ismid.1  |-  ( ph  ->  GDimTarskiG 2 )
midcl.1  |-  ( ph  ->  A  e.  P )
midcl.2  |-  ( ph  ->  B  e.  P )
ismidb.s  |-  S  =  (pInvG `  G )
ismidb.m  |-  ( ph  ->  M  e.  P )
Assertion
Ref Expression
ismidb  |-  ( ph  ->  ( B  =  ( ( S `  M
) `  A )  <->  ( A (midG `  G
) B )  =  M ) )

Proof of Theorem ismidb
Dummy variables  m  a  b  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ismidb.m . . 3  |-  ( ph  ->  M  e.  P )
2 ismid.p . . . 4  |-  P  =  ( Base `  G
)
3 ismid.d . . . 4  |-  .-  =  ( dist `  G )
4 ismid.i . . . 4  |-  I  =  (Itv `  G )
5 eqid 2451 . . . 4  |-  (LineG `  G )  =  (LineG `  G )
6 ismid.g . . . 4  |-  ( ph  ->  G  e. TarskiG )
7 ismidb.s . . . 4  |-  S  =  (pInvG `  G )
8 midcl.1 . . . 4  |-  ( ph  ->  A  e.  P )
9 midcl.2 . . . 4  |-  ( ph  ->  B  e.  P )
10 ismid.1 . . . 4  |-  ( ph  ->  GDimTarskiG 2 )
112, 3, 4, 5, 6, 7, 8, 9, 10mideu 24780 . . 3  |-  ( ph  ->  E! m  e.  P  B  =  ( ( S `  m ) `  A ) )
12 fveq2 5865 . . . . . 6  |-  ( m  =  M  ->  ( S `  m )  =  ( S `  M ) )
1312fveq1d 5867 . . . . 5  |-  ( m  =  M  ->  (
( S `  m
) `  A )  =  ( ( S `
 M ) `  A ) )
1413eqeq2d 2461 . . . 4  |-  ( m  =  M  ->  ( B  =  ( ( S `  m ) `  A )  <->  B  =  ( ( S `  M ) `  A
) ) )
1514riota2 6274 . . 3  |-  ( ( M  e.  P  /\  E! m  e.  P  B  =  ( ( S `  m ) `  A ) )  -> 
( B  =  ( ( S `  M
) `  A )  <->  (
iota_ m  e.  P  B  =  ( ( S `  m ) `  A ) )  =  M ) )
161, 11, 15syl2anc 667 . 2  |-  ( ph  ->  ( B  =  ( ( S `  M
) `  A )  <->  (
iota_ m  e.  P  B  =  ( ( S `  m ) `  A ) )  =  M ) )
17 df-mid 24816 . . . . . 6  |- midG  =  ( g  e.  _V  |->  ( a  e.  ( Base `  g ) ,  b  e.  ( Base `  g
)  |->  ( iota_ m  e.  ( Base `  g
) b  =  ( ( (pInvG `  g
) `  m ) `  a ) ) ) )
1817a1i 11 . . . . 5  |-  ( ph  -> midG  =  ( g  e. 
_V  |->  ( a  e.  ( Base `  g
) ,  b  e.  ( Base `  g
)  |->  ( iota_ m  e.  ( Base `  g
) b  =  ( ( (pInvG `  g
) `  m ) `  a ) ) ) ) )
19 fveq2 5865 . . . . . . . 8  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
2019, 2syl6eqr 2503 . . . . . . 7  |-  ( g  =  G  ->  ( Base `  g )  =  P )
21 fveq2 5865 . . . . . . . . . . . 12  |-  ( g  =  G  ->  (pInvG `  g )  =  (pInvG `  G ) )
2221, 7syl6eqr 2503 . . . . . . . . . . 11  |-  ( g  =  G  ->  (pInvG `  g )  =  S )
2322fveq1d 5867 . . . . . . . . . 10  |-  ( g  =  G  ->  (
(pInvG `  g ) `  m )  =  ( S `  m ) )
2423fveq1d 5867 . . . . . . . . 9  |-  ( g  =  G  ->  (
( (pInvG `  g
) `  m ) `  a )  =  ( ( S `  m
) `  a )
)
2524eqeq2d 2461 . . . . . . . 8  |-  ( g  =  G  ->  (
b  =  ( ( (pInvG `  g ) `  m ) `  a
)  <->  b  =  ( ( S `  m
) `  a )
) )
2620, 25riotaeqbidv 6255 . . . . . . 7  |-  ( g  =  G  ->  ( iota_ m  e.  ( Base `  g ) b  =  ( ( (pInvG `  g ) `  m
) `  a )
)  =  ( iota_ m  e.  P  b  =  ( ( S `  m ) `  a
) ) )
2720, 20, 26mpt2eq123dv 6353 . . . . . 6  |-  ( g  =  G  ->  (
a  e.  ( Base `  g ) ,  b  e.  ( Base `  g
)  |->  ( iota_ m  e.  ( Base `  g
) b  =  ( ( (pInvG `  g
) `  m ) `  a ) ) )  =  ( a  e.  P ,  b  e.  P  |->  ( iota_ m  e.  P  b  =  ( ( S `  m
) `  a )
) ) )
2827adantl 468 . . . . 5  |-  ( (
ph  /\  g  =  G )  ->  (
a  e.  ( Base `  g ) ,  b  e.  ( Base `  g
)  |->  ( iota_ m  e.  ( Base `  g
) b  =  ( ( (pInvG `  g
) `  m ) `  a ) ) )  =  ( a  e.  P ,  b  e.  P  |->  ( iota_ m  e.  P  b  =  ( ( S `  m
) `  a )
) ) )
29 elex 3054 . . . . . 6  |-  ( G  e. TarskiG  ->  G  e.  _V )
306, 29syl 17 . . . . 5  |-  ( ph  ->  G  e.  _V )
31 fvex 5875 . . . . . . . 8  |-  ( Base `  G )  e.  _V
322, 31eqeltri 2525 . . . . . . 7  |-  P  e. 
_V
3332, 32mpt2ex 6870 . . . . . 6  |-  ( a  e.  P ,  b  e.  P  |->  ( iota_ m  e.  P  b  =  ( ( S `  m ) `  a
) ) )  e. 
_V
3433a1i 11 . . . . 5  |-  ( ph  ->  ( a  e.  P ,  b  e.  P  |->  ( iota_ m  e.  P  b  =  ( ( S `  m ) `  a ) ) )  e.  _V )
3518, 28, 30, 34fvmptd 5954 . . . 4  |-  ( ph  ->  (midG `  G )  =  ( a  e.  P ,  b  e.  P  |->  ( iota_ m  e.  P  b  =  ( ( S `  m
) `  a )
) ) )
36 simprr 766 . . . . . 6  |-  ( (
ph  /\  ( a  =  A  /\  b  =  B ) )  -> 
b  =  B )
37 simprl 764 . . . . . . 7  |-  ( (
ph  /\  ( a  =  A  /\  b  =  B ) )  -> 
a  =  A )
3837fveq2d 5869 . . . . . 6  |-  ( (
ph  /\  ( a  =  A  /\  b  =  B ) )  -> 
( ( S `  m ) `  a
)  =  ( ( S `  m ) `
 A ) )
3936, 38eqeq12d 2466 . . . . 5  |-  ( (
ph  /\  ( a  =  A  /\  b  =  B ) )  -> 
( b  =  ( ( S `  m
) `  a )  <->  B  =  ( ( S `
 m ) `  A ) ) )
4039riotabidv 6254 . . . 4  |-  ( (
ph  /\  ( a  =  A  /\  b  =  B ) )  -> 
( iota_ m  e.  P  b  =  ( ( S `  m ) `  a ) )  =  ( iota_ m  e.  P  B  =  ( ( S `  m ) `  A ) ) )
41 riotacl 6266 . . . . 5  |-  ( E! m  e.  P  B  =  ( ( S `
 m ) `  A )  ->  ( iota_ m  e.  P  B  =  ( ( S `
 m ) `  A ) )  e.  P )
4211, 41syl 17 . . . 4  |-  ( ph  ->  ( iota_ m  e.  P  B  =  ( ( S `  m ) `  A ) )  e.  P )
4335, 40, 8, 9, 42ovmpt2d 6424 . . 3  |-  ( ph  ->  ( A (midG `  G ) B )  =  ( iota_ m  e.  P  B  =  ( ( S `  m
) `  A )
) )
4443eqeq1d 2453 . 2  |-  ( ph  ->  ( ( A (midG `  G ) B )  =  M  <->  ( iota_ m  e.  P  B  =  ( ( S `  m ) `  A
) )  =  M ) )
4516, 44bitr4d 260 1  |-  ( ph  ->  ( B  =  ( ( S `  M
) `  A )  <->  ( A (midG `  G
) B )  =  M ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887   E!wreu 2739   _Vcvv 3045   class class class wbr 4402    |-> cmpt 4461   ` cfv 5582   iota_crio 6251  (class class class)co 6290    |-> cmpt2 6292   2c2 10659   Basecbs 15121   distcds 15199  TarskiGcstrkg 24478  DimTarskiGcstrkgld 24482  Itvcitv 24484  LineGclng 24485  pInvGcmir 24697  midGcmid 24814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-fzo 11916  df-hash 12516  df-word 12664  df-concat 12666  df-s1 12667  df-s2 12944  df-s3 12945  df-trkgc 24496  df-trkgb 24497  df-trkgcb 24498  df-trkgld 24500  df-trkg 24501  df-cgrg 24556  df-leg 24628  df-mir 24698  df-rag 24739  df-perpg 24741  df-mid 24816
This theorem is referenced by:  midbtwn  24821  midcgr  24822  midcom  24824  mirmid  24825  lmieu  24826  lmimid  24836  lmiisolem  24838  hypcgrlem1  24841  hypcgrlem2  24842  hypcgr  24843  trgcopyeulem  24847
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