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Theorem ismidb 24348
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 2454 . . . 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 24316 . . 3  |-  ( ph  ->  E! m  e.  P  B  =  ( ( S `  m ) `  A ) )
12 fveq2 5848 . . . . . 6  |-  ( m  =  M  ->  ( S `  m )  =  ( S `  M ) )
1312fveq1d 5850 . . . . 5  |-  ( m  =  M  ->  (
( S `  m
) `  A )  =  ( ( S `
 M ) `  A ) )
1413eqeq2d 2468 . . . 4  |-  ( m  =  M  ->  ( B  =  ( ( S `  m ) `  A )  <->  B  =  ( ( S `  M ) `  A
) ) )
1514riota2 6254 . . 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 659 . 2  |-  ( ph  ->  ( B  =  ( ( S `  M
) `  A )  <->  (
iota_ m  e.  P  B  =  ( ( S `  m ) `  A ) )  =  M ) )
17 df-mid 24344 . . . . . 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 5848 . . . . . . . 8  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
2019, 2syl6eqr 2513 . . . . . . 7  |-  ( g  =  G  ->  ( Base `  g )  =  P )
21 fveq2 5848 . . . . . . . . . . . 12  |-  ( g  =  G  ->  (pInvG `  g )  =  (pInvG `  G ) )
2221, 7syl6eqr 2513 . . . . . . . . . . 11  |-  ( g  =  G  ->  (pInvG `  g )  =  S )
2322fveq1d 5850 . . . . . . . . . 10  |-  ( g  =  G  ->  (
(pInvG `  g ) `  m )  =  ( S `  m ) )
2423fveq1d 5850 . . . . . . . . 9  |-  ( g  =  G  ->  (
( (pInvG `  g
) `  m ) `  a )  =  ( ( S `  m
) `  a )
)
2524eqeq2d 2468 . . . . . . . 8  |-  ( g  =  G  ->  (
b  =  ( ( (pInvG `  g ) `  m ) `  a
)  <->  b  =  ( ( S `  m
) `  a )
) )
2620, 25riotaeqbidv 6235 . . . . . . 7  |-  ( g  =  G  ->  ( iota_ m  e.  ( Base `  g ) b  =  ( ( (pInvG `  g ) `  m
) `  a )
)  =  ( iota_ m  e.  P  b  =  ( ( S `  m ) `  a
) ) )
2720, 20, 26mpt2eq123dv 6332 . . . . . 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 464 . . . . 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 3115 . . . . . 6  |-  ( G  e. TarskiG  ->  G  e.  _V )
306, 29syl 16 . . . . 5  |-  ( ph  ->  G  e.  _V )
31 fvex 5858 . . . . . . . 8  |-  ( Base `  G )  e.  _V
322, 31eqeltri 2538 . . . . . . 7  |-  P  e. 
_V
3332, 32mpt2ex 6850 . . . . . 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 5936 . . . 4  |-  ( ph  ->  (midG `  G )  =  ( a  e.  P ,  b  e.  P  |->  ( iota_ m  e.  P  b  =  ( ( S `  m
) `  a )
) ) )
36 simprr 755 . . . . . 6  |-  ( (
ph  /\  ( a  =  A  /\  b  =  B ) )  -> 
b  =  B )
37 simprl 754 . . . . . . 7  |-  ( (
ph  /\  ( a  =  A  /\  b  =  B ) )  -> 
a  =  A )
3837fveq2d 5852 . . . . . 6  |-  ( (
ph  /\  ( a  =  A  /\  b  =  B ) )  -> 
( ( S `  m ) `  a
)  =  ( ( S `  m ) `
 A ) )
3936, 38eqeq12d 2476 . . . . 5  |-  ( (
ph  /\  ( a  =  A  /\  b  =  B ) )  -> 
( b  =  ( ( S `  m
) `  a )  <->  B  =  ( ( S `
 m ) `  A ) ) )
4039riotabidv 6234 . . . 4  |-  ( (
ph  /\  ( a  =  A  /\  b  =  B ) )  -> 
( iota_ m  e.  P  b  =  ( ( S `  m ) `  a ) )  =  ( iota_ m  e.  P  B  =  ( ( S `  m ) `  A ) ) )
41 riotacl 6246 . . . . 5  |-  ( E! m  e.  P  B  =  ( ( S `
 m ) `  A )  ->  ( iota_ m  e.  P  B  =  ( ( S `
 m ) `  A ) )  e.  P )
4211, 41syl 16 . . . 4  |-  ( ph  ->  ( iota_ m  e.  P  B  =  ( ( S `  m ) `  A ) )  e.  P )
4335, 40, 8, 9, 42ovmpt2d 6403 . . 3  |-  ( ph  ->  ( A (midG `  G ) B )  =  ( iota_ m  e.  P  B  =  ( ( S `  m
) `  A )
) )
4443eqeq1d 2456 . 2  |-  ( ph  ->  ( ( A (midG `  G ) B )  =  M  <->  ( iota_ m  e.  P  B  =  ( ( S `  m ) `  A
) )  =  M ) )
4516, 44bitr4d 256 1  |-  ( ph  ->  ( B  =  ( ( S `  M
) `  A )  <->  ( A (midG `  G
) B )  =  M ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   E!wreu 2806   _Vcvv 3106   class class class wbr 4439    |-> cmpt 4497   ` cfv 5570   iota_crio 6231  (class class class)co 6270    |-> cmpt2 6272   2c2 10581   Basecbs 14719   distcds 14796  TarskiGcstrkg 24026  DimTarskiGcstrkgld 24030  Itvcitv 24033  LineGclng 24034  pInvGcmir 24237  midGcmid 24342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-hash 12391  df-word 12529  df-concat 12531  df-s1 12532  df-s2 12807  df-s3 12808  df-trkgc 24045  df-trkgb 24046  df-trkgcb 24047  df-trkgld 24049  df-trkg 24051  df-cgrg 24107  df-leg 24174  df-mir 24238  df-rag 24275  df-perpg 24277  df-mid 24344
This theorem is referenced by:  midbtwn  24349  midcgr  24350  midcom  24352  mirmid  24353  lmieu  24354  lmimid  24363  lmiisolem  24365  hypcgrlem1  24368  hypcgrlem2  24369  hypcgr  24370
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