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Theorem lmif1o 24364
Description: The line mirroring function  M is a bijection. Theorem 10.9 of [Schwabhauser] p. 89. (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
lmif1o  |-  ( ph  ->  M : P -1-1-onto-> P )

Proof of Theorem lmif1o
Dummy variable  b is distinct from all other variables.
StepHypRef Expression
1 ismid.p . . . 4  |-  P  =  ( Base `  G
)
2 ismid.d . . . 4  |-  .-  =  ( dist `  G )
3 ismid.i . . . 4  |-  I  =  (Itv `  G )
4 ismid.g . . . 4  |-  ( ph  ->  G  e. TarskiG )
5 ismid.1 . . . 4  |-  ( ph  ->  GDimTarskiG 2 )
6 lmif.m . . . 4  |-  M  =  ( (lInvG `  G
) `  D )
7 lmif.l . . . 4  |-  L  =  (LineG `  G )
8 lmif.d . . . 4  |-  ( ph  ->  D  e.  ran  L
)
91, 2, 3, 4, 5, 6, 7, 8lmif 24355 . . 3  |-  ( ph  ->  M : P --> P )
10 ffn 5713 . . 3  |-  ( M : P --> P  ->  M  Fn  P )
119, 10syl 16 . 2  |-  ( ph  ->  M  Fn  P )
124adantr 463 . . . . 5  |-  ( (
ph  /\  b  e.  P )  ->  G  e. TarskiG )
135adantr 463 . . . . 5  |-  ( (
ph  /\  b  e.  P )  ->  GDimTarskiG 2
)
148adantr 463 . . . . 5  |-  ( (
ph  /\  b  e.  P )  ->  D  e.  ran  L )
15 simpr 459 . . . . 5  |-  ( (
ph  /\  b  e.  P )  ->  b  e.  P )
161, 2, 3, 12, 13, 6, 7, 14, 15lmilmi 24359 . . . 4  |-  ( (
ph  /\  b  e.  P )  ->  ( M `  ( M `  b ) )  =  b )
1716ralrimiva 2868 . . 3  |-  ( ph  ->  A. b  e.  P  ( M `  ( M `
 b ) )  =  b )
18 nvocnv 6162 . . 3  |-  ( ( M : P --> P  /\  A. b  e.  P  ( M `  ( M `
 b ) )  =  b )  ->  `' M  =  M
)
199, 17, 18syl2anc 659 . 2  |-  ( ph  ->  `' M  =  M
)
20 nvof1o 6161 . 2  |-  ( ( M  Fn  P  /\  `' M  =  M
)  ->  M : P
-1-1-onto-> P )
2111, 19, 20syl2anc 659 1  |-  ( ph  ->  M : P -1-1-onto-> P )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   class class class wbr 4439   `'ccnv 4987   ran crn 4989    Fn wfn 5565   -->wf 5566   -1-1-onto->wf1o 5569   ` cfv 5570   2c2 10581   Basecbs 14719   distcds 14796  TarskiGcstrkg 24026  DimTarskiGcstrkgld 24030  Itvcitv 24033  LineGclng 24034  lInvGclmi 24343
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  df-lmi 24345
This theorem is referenced by:  lmimot  24367
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