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Theorem lmif 24352
Description: Line mirror as a function. (Contributed by Thierry Arnoux, 11-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 )
lmif.m  |-  M  =  ( (lInvG `  G
) `  D )
lmif.l  |-  L  =  (LineG `  G )
lmif.d  |-  ( ph  ->  D  e.  ran  L
)
Assertion
Ref Expression
lmif  |-  ( ph  ->  M : P --> P )

Proof of Theorem lmif
Dummy variables  a 
b  g  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ismid.p . . . . 5  |-  P  =  ( Base `  G
)
2 ismid.d . . . . 5  |-  .-  =  ( dist `  G )
3 ismid.i . . . . 5  |-  I  =  (Itv `  G )
4 ismid.g . . . . . 6  |-  ( ph  ->  G  e. TarskiG )
54adantr 463 . . . . 5  |-  ( (
ph  /\  a  e.  P )  ->  G  e. TarskiG )
6 ismid.1 . . . . . 6  |-  ( ph  ->  GDimTarskiG 2 )
76adantr 463 . . . . 5  |-  ( (
ph  /\  a  e.  P )  ->  GDimTarskiG 2
)
8 lmif.l . . . . 5  |-  L  =  (LineG `  G )
9 lmif.d . . . . . 6  |-  ( ph  ->  D  e.  ran  L
)
109adantr 463 . . . . 5  |-  ( (
ph  /\  a  e.  P )  ->  D  e.  ran  L )
11 simpr 459 . . . . 5  |-  ( (
ph  /\  a  e.  P )  ->  a  e.  P )
121, 2, 3, 5, 7, 8, 10, 11lmieu 24351 . . . 4  |-  ( (
ph  /\  a  e.  P )  ->  E! b  e.  P  (
( a (midG `  G ) b )  e.  D  /\  ( D (⟂G `  G )
( a L b )  \/  a  =  b ) ) )
13 riotacl 6246 . . . 4  |-  ( E! b  e.  P  ( ( a (midG `  G ) b )  e.  D  /\  ( D (⟂G `  G )
( a L b )  \/  a  =  b ) )  -> 
( iota_ b  e.  P  ( ( a (midG `  G ) b )  e.  D  /\  ( D (⟂G `  G )
( a L b )  \/  a  =  b ) ) )  e.  P )
1412, 13syl 16 . . 3  |-  ( (
ph  /\  a  e.  P )  ->  ( iota_ b  e.  P  ( ( a (midG `  G ) b )  e.  D  /\  ( D (⟂G `  G )
( a L b )  \/  a  =  b ) ) )  e.  P )
15 eqid 2454 . . 3  |-  ( a  e.  P  |->  ( iota_ b  e.  P  ( ( a (midG `  G
) b )  e.  D  /\  ( D (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) )  =  ( a  e.  P  |->  ( iota_ b  e.  P  ( ( a (midG `  G
) b )  e.  D  /\  ( D (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) )
1614, 15fmptd 6031 . 2  |-  ( ph  ->  ( a  e.  P  |->  ( iota_ b  e.  P  ( ( a (midG `  G ) b )  e.  D  /\  ( D (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) ) : P --> P )
17 lmif.m . . . 4  |-  M  =  ( (lInvG `  G
) `  D )
18 df-lmi 24342 . . . . . . 7  |- lInvG  =  ( g  e.  _V  |->  ( d  e.  ran  (LineG `  g )  |->  ( a  e.  ( Base `  g
)  |->  ( iota_ b  e.  ( Base `  g
) ( ( a (midG `  g )
b )  e.  d  /\  ( d (⟂G `  g ) ( a (LineG `  g )
b )  \/  a  =  b ) ) ) ) ) )
1918a1i 11 . . . . . 6  |-  ( ph  -> lInvG  =  ( g  e. 
_V  |->  ( d  e. 
ran  (LineG `  g )  |->  ( a  e.  (
Base `  g )  |->  ( iota_ b  e.  (
Base `  g )
( ( a (midG `  g ) b )  e.  d  /\  (
d (⟂G `  g )
( a (LineG `  g ) b )  \/  a  =  b ) ) ) ) ) ) )
20 fveq2 5848 . . . . . . . . . 10  |-  ( g  =  G  ->  (LineG `  g )  =  (LineG `  G ) )
2120, 8syl6eqr 2513 . . . . . . . . 9  |-  ( g  =  G  ->  (LineG `  g )  =  L )
2221rneqd 5219 . . . . . . . 8  |-  ( g  =  G  ->  ran  (LineG `  g )  =  ran  L )
23 fveq2 5848 . . . . . . . . . 10  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
2423, 1syl6eqr 2513 . . . . . . . . 9  |-  ( g  =  G  ->  ( Base `  g )  =  P )
25 fveq2 5848 . . . . . . . . . . . . 13  |-  ( g  =  G  ->  (midG `  g )  =  (midG `  G ) )
2625oveqd 6287 . . . . . . . . . . . 12  |-  ( g  =  G  ->  (
a (midG `  g
) b )  =  ( a (midG `  G ) b ) )
2726eleq1d 2523 . . . . . . . . . . 11  |-  ( g  =  G  ->  (
( a (midG `  g ) b )  e.  d  <->  ( a
(midG `  G )
b )  e.  d ) )
28 eqidd 2455 . . . . . . . . . . . . 13  |-  ( g  =  G  ->  d  =  d )
29 fveq2 5848 . . . . . . . . . . . . 13  |-  ( g  =  G  ->  (⟂G `  g )  =  (⟂G `  G ) )
3021oveqd 6287 . . . . . . . . . . . . 13  |-  ( g  =  G  ->  (
a (LineG `  g
) b )  =  ( a L b ) )
3128, 29, 30breq123d 4453 . . . . . . . . . . . 12  |-  ( g  =  G  ->  (
d (⟂G `  g )
( a (LineG `  g ) b )  <-> 
d (⟂G `  G )
( a L b ) ) )
3231orbi1d 700 . . . . . . . . . . 11  |-  ( g  =  G  ->  (
( d (⟂G `  g
) ( a (LineG `  g ) b )  \/  a  =  b )  <->  ( d (⟂G `  G ) ( a L b )  \/  a  =  b ) ) )
3327, 32anbi12d 708 . . . . . . . . . 10  |-  ( g  =  G  ->  (
( ( a (midG `  g ) b )  e.  d  /\  (
d (⟂G `  g )
( a (LineG `  g ) b )  \/  a  =  b ) )  <->  ( (
a (midG `  G
) b )  e.  d  /\  ( d (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) )
3424, 33riotaeqbidv 6235 . . . . . . . . 9  |-  ( g  =  G  ->  ( iota_ b  e.  ( Base `  g ) ( ( a (midG `  g
) b )  e.  d  /\  ( d (⟂G `  g )
( a (LineG `  g ) b )  \/  a  =  b ) ) )  =  ( iota_ b  e.  P  ( ( a (midG `  G ) b )  e.  d  /\  (
d (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) )
3524, 34mpteq12dv 4517 . . . . . . . 8  |-  ( g  =  G  ->  (
a  e.  ( Base `  g )  |->  ( iota_ b  e.  ( Base `  g
) ( ( a (midG `  g )
b )  e.  d  /\  ( d (⟂G `  g ) ( a (LineG `  g )
b )  \/  a  =  b ) ) ) )  =  ( a  e.  P  |->  (
iota_ b  e.  P  ( ( a (midG `  G ) b )  e.  d  /\  (
d (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) ) )
3622, 35mpteq12dv 4517 . . . . . . 7  |-  ( g  =  G  ->  (
d  e.  ran  (LineG `  g )  |->  ( a  e.  ( Base `  g
)  |->  ( iota_ b  e.  ( Base `  g
) ( ( a (midG `  g )
b )  e.  d  /\  ( d (⟂G `  g ) ( a (LineG `  g )
b )  \/  a  =  b ) ) ) ) )  =  ( d  e.  ran  L 
|->  ( a  e.  P  |->  ( iota_ b  e.  P  ( ( a (midG `  G ) b )  e.  d  /\  (
d (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) ) ) )
3736adantl 464 . . . . . 6  |-  ( (
ph  /\  g  =  G )  ->  (
d  e.  ran  (LineG `  g )  |->  ( a  e.  ( Base `  g
)  |->  ( iota_ b  e.  ( Base `  g
) ( ( a (midG `  g )
b )  e.  d  /\  ( d (⟂G `  g ) ( a (LineG `  g )
b )  \/  a  =  b ) ) ) ) )  =  ( d  e.  ran  L 
|->  ( a  e.  P  |->  ( iota_ b  e.  P  ( ( a (midG `  G ) b )  e.  d  /\  (
d (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) ) ) )
38 elex 3115 . . . . . . 7  |-  ( G  e. TarskiG  ->  G  e.  _V )
394, 38syl 16 . . . . . 6  |-  ( ph  ->  G  e.  _V )
40 fvex 5858 . . . . . . . . 9  |-  (LineG `  G )  e.  _V
418, 40eqeltri 2538 . . . . . . . 8  |-  L  e. 
_V
42 rnexg 6705 . . . . . . . 8  |-  ( L  e.  _V  ->  ran  L  e.  _V )
43 mptexg 6117 . . . . . . . 8  |-  ( ran 
L  e.  _V  ->  ( d  e.  ran  L  |->  ( a  e.  P  |->  ( iota_ b  e.  P  ( ( a (midG `  G ) b )  e.  d  /\  (
d (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) ) )  e.  _V )
4441, 42, 43mp2b 10 . . . . . . 7  |-  ( d  e.  ran  L  |->  ( a  e.  P  |->  (
iota_ b  e.  P  ( ( a (midG `  G ) b )  e.  d  /\  (
d (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) ) )  e.  _V
4544a1i 11 . . . . . 6  |-  ( ph  ->  ( d  e.  ran  L 
|->  ( a  e.  P  |->  ( iota_ b  e.  P  ( ( a (midG `  G ) b )  e.  d  /\  (
d (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) ) )  e.  _V )
4619, 37, 39, 45fvmptd 5936 . . . . 5  |-  ( ph  ->  (lInvG `  G )  =  ( d  e. 
ran  L  |->  ( a  e.  P  |->  ( iota_ b  e.  P  ( ( a (midG `  G
) b )  e.  d  /\  ( d (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) ) ) )
47 eleq2 2527 . . . . . . . . 9  |-  ( d  =  D  ->  (
( a (midG `  G ) b )  e.  d  <->  ( a
(midG `  G )
b )  e.  D
) )
48 breq1 4442 . . . . . . . . . 10  |-  ( d  =  D  ->  (
d (⟂G `  G )
( a L b )  <->  D (⟂G `  G
) ( a L b ) ) )
4948orbi1d 700 . . . . . . . . 9  |-  ( d  =  D  ->  (
( d (⟂G `  G
) ( a L b )  \/  a  =  b )  <->  ( D
(⟂G `  G ) ( a L b )  \/  a  =  b ) ) )
5047, 49anbi12d 708 . . . . . . . 8  |-  ( d  =  D  ->  (
( ( a (midG `  G ) b )  e.  d  /\  (
d (⟂G `  G )
( a L b )  \/  a  =  b ) )  <->  ( (
a (midG `  G
) b )  e.  D  /\  ( D (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) )
5150riotabidv 6234 . . . . . . 7  |-  ( d  =  D  ->  ( iota_ b  e.  P  ( ( a (midG `  G ) b )  e.  d  /\  (
d (⟂G `  G )
( a L b )  \/  a  =  b ) ) )  =  ( iota_ b  e.  P  ( ( a (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( a L b )  \/  a  =  b ) ) ) )
5251mpteq2dv 4526 . . . . . 6  |-  ( d  =  D  ->  (
a  e.  P  |->  (
iota_ b  e.  P  ( ( a (midG `  G ) b )  e.  d  /\  (
d (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) )  =  ( a  e.  P  |->  ( iota_ b  e.  P  ( ( a (midG `  G
) b )  e.  D  /\  ( D (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) ) )
5352adantl 464 . . . . 5  |-  ( (
ph  /\  d  =  D )  ->  (
a  e.  P  |->  (
iota_ b  e.  P  ( ( a (midG `  G ) b )  e.  d  /\  (
d (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) )  =  ( a  e.  P  |->  ( iota_ b  e.  P  ( ( a (midG `  G
) b )  e.  D  /\  ( D (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) ) )
54 fvex 5858 . . . . . . . 8  |-  ( Base `  G )  e.  _V
551, 54eqeltri 2538 . . . . . . 7  |-  P  e. 
_V
5655mptex 6118 . . . . . 6  |-  ( a  e.  P  |->  ( iota_ b  e.  P  ( ( a (midG `  G
) b )  e.  D  /\  ( D (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) )  e.  _V
5756a1i 11 . . . . 5  |-  ( ph  ->  ( a  e.  P  |->  ( iota_ b  e.  P  ( ( a (midG `  G ) b )  e.  D  /\  ( D (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) )  e.  _V )
5846, 53, 9, 57fvmptd 5936 . . . 4  |-  ( ph  ->  ( (lInvG `  G
) `  D )  =  ( a  e.  P  |->  ( iota_ b  e.  P  ( ( a (midG `  G )
b )  e.  D  /\  ( D (⟂G `  G
) ( a L b )  \/  a  =  b ) ) ) ) )
5917, 58syl5eq 2507 . . 3  |-  ( ph  ->  M  =  ( a  e.  P  |->  ( iota_ b  e.  P  ( ( a (midG `  G
) b )  e.  D  /\  ( D (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) ) )
6059feq1d 5699 . 2  |-  ( ph  ->  ( M : P --> P 
<->  ( a  e.  P  |->  ( iota_ b  e.  P  ( ( a (midG `  G ) b )  e.  D  /\  ( D (⟂G `  G )
( a L b )  \/  a  =  b ) ) ) ) : P --> P ) )
6116, 60mpbird 232 1  |-  ( ph  ->  M : P --> P )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367    = wceq 1398    e. wcel 1823   E!wreu 2806   _Vcvv 3106   class class class wbr 4439    |-> cmpt 4497   ran crn 4989   -->wf 5566   ` cfv 5570   iota_crio 6231  (class class class)co 6270   2c2 10581   Basecbs 14716   distcds 14793  TarskiGcstrkg 24023  DimTarskiGcstrkgld 24027  Itvcitv 24030  LineGclng 24031  ⟂Gcperpg 24273  midGcmid 24339  lInvGclmi 24340
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 12388  df-word 12526  df-concat 12528  df-s1 12529  df-s2 12804  df-s3 12805  df-trkgc 24042  df-trkgb 24043  df-trkgcb 24044  df-trkgld 24046  df-trkg 24048  df-cgrg 24104  df-leg 24171  df-mir 24235  df-rag 24272  df-perpg 24274  df-mid 24341  df-lmi 24342
This theorem is referenced by:  lmicl  24353  lmif1o  24361
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