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Theorem colmid 24266
Description: Colinearity and equidistance implies midpoint. Theorem 7.20 of [Schwabhauser] p. 52. (Contributed by Thierry Arnoux, 30-Jul-2019.)
Hypotheses
Ref Expression
mirval.p  |-  P  =  ( Base `  G
)
mirval.d  |-  .-  =  ( dist `  G )
mirval.i  |-  I  =  (Itv `  G )
mirval.l  |-  L  =  (LineG `  G )
mirval.s  |-  S  =  (pInvG `  G )
mirval.g  |-  ( ph  ->  G  e. TarskiG )
colmid.m  |-  M  =  ( S `  X
)
colmid.a  |-  ( ph  ->  A  e.  P )
colmid.b  |-  ( ph  ->  B  e.  P )
colmid.x  |-  ( ph  ->  X  e.  P )
colmid.c  |-  ( ph  ->  ( X  e.  ( A L B )  \/  A  =  B ) )
colmid.d  |-  ( ph  ->  ( X  .-  A
)  =  ( X 
.-  B ) )
Assertion
Ref Expression
colmid  |-  ( ph  ->  ( B  =  ( M `  A )  \/  A  =  B ) )

Proof of Theorem colmid
StepHypRef Expression
1 simpr 459 . . 3  |-  ( (
ph  /\  A  =  B )  ->  A  =  B )
21olcd 391 . 2  |-  ( (
ph  /\  A  =  B )  ->  ( B  =  ( M `  A )  \/  A  =  B ) )
3 mirval.p . . . . 5  |-  P  =  ( Base `  G
)
4 mirval.d . . . . 5  |-  .-  =  ( dist `  G )
5 mirval.i . . . . 5  |-  I  =  (Itv `  G )
6 mirval.l . . . . 5  |-  L  =  (LineG `  G )
7 mirval.s . . . . 5  |-  S  =  (pInvG `  G )
8 mirval.g . . . . . 6  |-  ( ph  ->  G  e. TarskiG )
98ad2antrr 723 . . . . 5  |-  ( ( ( ph  /\  A  =/=  B )  /\  X  e.  ( A I B ) )  ->  G  e. TarskiG )
10 colmid.x . . . . . 6  |-  ( ph  ->  X  e.  P )
1110ad2antrr 723 . . . . 5  |-  ( ( ( ph  /\  A  =/=  B )  /\  X  e.  ( A I B ) )  ->  X  e.  P )
12 colmid.m . . . . 5  |-  M  =  ( S `  X
)
13 colmid.a . . . . . 6  |-  ( ph  ->  A  e.  P )
1413ad2antrr 723 . . . . 5  |-  ( ( ( ph  /\  A  =/=  B )  /\  X  e.  ( A I B ) )  ->  A  e.  P )
15 colmid.b . . . . . 6  |-  ( ph  ->  B  e.  P )
1615ad2antrr 723 . . . . 5  |-  ( ( ( ph  /\  A  =/=  B )  /\  X  e.  ( A I B ) )  ->  B  e.  P )
17 colmid.d . . . . . . 7  |-  ( ph  ->  ( X  .-  A
)  =  ( X 
.-  B ) )
1817ad2antrr 723 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  B )  /\  X  e.  ( A I B ) )  ->  ( X  .-  A )  =  ( X  .-  B
) )
1918eqcomd 2462 . . . . 5  |-  ( ( ( ph  /\  A  =/=  B )  /\  X  e.  ( A I B ) )  ->  ( X  .-  B )  =  ( X  .-  A
) )
20 simpr 459 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  B )  /\  X  e.  ( A I B ) )  ->  X  e.  ( A I B ) )
213, 4, 5, 9, 14, 11, 16, 20tgbtwncom 24080 . . . . 5  |-  ( ( ( ph  /\  A  =/=  B )  /\  X  e.  ( A I B ) )  ->  X  e.  ( B I A ) )
223, 4, 5, 6, 7, 9, 11, 12, 14, 16, 19, 21ismir 24241 . . . 4  |-  ( ( ( ph  /\  A  =/=  B )  /\  X  e.  ( A I B ) )  ->  B  =  ( M `  A ) )
2322orcd 390 . . 3  |-  ( ( ( ph  /\  A  =/=  B )  /\  X  e.  ( A I B ) )  ->  ( B  =  ( M `  A )  \/  A  =  B ) )
248adantr 463 . . . . . . 7  |-  ( (
ph  /\  A  e.  ( X I B ) )  ->  G  e. TarskiG )
2515adantr 463 . . . . . . 7  |-  ( (
ph  /\  A  e.  ( X I B ) )  ->  B  e.  P )
2613adantr 463 . . . . . . 7  |-  ( (
ph  /\  A  e.  ( X I B ) )  ->  A  e.  P )
2710adantr 463 . . . . . . . 8  |-  ( (
ph  /\  A  e.  ( X I B ) )  ->  X  e.  P )
28 simpr 459 . . . . . . . . 9  |-  ( (
ph  /\  A  e.  ( X I B ) )  ->  A  e.  ( X I B ) )
293, 4, 5, 24, 27, 26, 25, 28tgbtwncom 24080 . . . . . . . 8  |-  ( (
ph  /\  A  e.  ( X I B ) )  ->  A  e.  ( B I X ) )
303, 4, 5, 24, 26, 27tgbtwntriv1 24083 . . . . . . . 8  |-  ( (
ph  /\  A  e.  ( X I B ) )  ->  A  e.  ( A I X ) )
313, 4, 5, 8, 10, 13, 10, 15, 17tgcgrcomlr 24072 . . . . . . . . . 10  |-  ( ph  ->  ( A  .-  X
)  =  ( B 
.-  X ) )
3231adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  A  e.  ( X I B ) )  ->  ( A  .-  X )  =  ( B  .-  X ) )
3332eqcomd 2462 . . . . . . . 8  |-  ( (
ph  /\  A  e.  ( X I B ) )  ->  ( B  .-  X )  =  ( A  .-  X ) )
34 eqidd 2455 . . . . . . . 8  |-  ( (
ph  /\  A  e.  ( X I B ) )  ->  ( A  .-  X )  =  ( A  .-  X ) )
353, 4, 5, 24, 25, 26, 27, 26, 26, 27, 29, 30, 33, 34tgcgrsub 24102 . . . . . . 7  |-  ( (
ph  /\  A  e.  ( X I B ) )  ->  ( B  .-  A )  =  ( A  .-  A ) )
363, 4, 5, 24, 25, 26, 26, 35axtgcgrid 24058 . . . . . 6  |-  ( (
ph  /\  A  e.  ( X I B ) )  ->  B  =  A )
3736eqcomd 2462 . . . . 5  |-  ( (
ph  /\  A  e.  ( X I B ) )  ->  A  =  B )
3837adantlr 712 . . . 4  |-  ( ( ( ph  /\  A  =/=  B )  /\  A  e.  ( X I B ) )  ->  A  =  B )
3938olcd 391 . . 3  |-  ( ( ( ph  /\  A  =/=  B )  /\  A  e.  ( X I B ) )  ->  ( B  =  ( M `  A )  \/  A  =  B ) )
408adantr 463 . . . . . 6  |-  ( (
ph  /\  B  e.  ( A I X ) )  ->  G  e. TarskiG )
4113adantr 463 . . . . . 6  |-  ( (
ph  /\  B  e.  ( A I X ) )  ->  A  e.  P )
4215adantr 463 . . . . . 6  |-  ( (
ph  /\  B  e.  ( A I X ) )  ->  B  e.  P )
4310adantr 463 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( A I X ) )  ->  X  e.  P )
44 simpr 459 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( A I X ) )  ->  B  e.  ( A I X ) )
453, 4, 5, 40, 42, 43tgbtwntriv1 24083 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( A I X ) )  ->  B  e.  ( B I X ) )
4631adantr 463 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( A I X ) )  ->  ( A  .-  X )  =  ( B  .-  X ) )
47 eqidd 2455 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( A I X ) )  ->  ( B  .-  X )  =  ( B  .-  X ) )
483, 4, 5, 40, 41, 42, 43, 42, 42, 43, 44, 45, 46, 47tgcgrsub 24102 . . . . . 6  |-  ( (
ph  /\  B  e.  ( A I X ) )  ->  ( A  .-  B )  =  ( B  .-  B ) )
493, 4, 5, 40, 41, 42, 42, 48axtgcgrid 24058 . . . . 5  |-  ( (
ph  /\  B  e.  ( A I X ) )  ->  A  =  B )
5049adantlr 712 . . . 4  |-  ( ( ( ph  /\  A  =/=  B )  /\  B  e.  ( A I X ) )  ->  A  =  B )
5150olcd 391 . . 3  |-  ( ( ( ph  /\  A  =/=  B )  /\  B  e.  ( A I X ) )  ->  ( B  =  ( M `  A )  \/  A  =  B ) )
52 df-ne 2651 . . . . 5  |-  ( A  =/=  B  <->  -.  A  =  B )
53 colmid.c . . . . . . 7  |-  ( ph  ->  ( X  e.  ( A L B )  \/  A  =  B ) )
5453orcomd 386 . . . . . 6  |-  ( ph  ->  ( A  =  B  \/  X  e.  ( A L B ) ) )
5554orcanai 911 . . . . 5  |-  ( (
ph  /\  -.  A  =  B )  ->  X  e.  ( A L B ) )
5652, 55sylan2b 473 . . . 4  |-  ( (
ph  /\  A  =/=  B )  ->  X  e.  ( A L B ) )
578adantr 463 . . . . 5  |-  ( (
ph  /\  A  =/=  B )  ->  G  e. TarskiG )
5813adantr 463 . . . . 5  |-  ( (
ph  /\  A  =/=  B )  ->  A  e.  P )
5915adantr 463 . . . . 5  |-  ( (
ph  /\  A  =/=  B )  ->  B  e.  P )
60 simpr 459 . . . . 5  |-  ( (
ph  /\  A  =/=  B )  ->  A  =/=  B )
6110adantr 463 . . . . 5  |-  ( (
ph  /\  A  =/=  B )  ->  X  e.  P )
623, 6, 5, 57, 58, 59, 60, 61tgellng 24141 . . . 4  |-  ( (
ph  /\  A  =/=  B )  ->  ( X  e.  ( A L B )  <->  ( X  e.  ( A I B )  \/  A  e.  ( X I B )  \/  B  e.  ( A I X ) ) ) )
6356, 62mpbid 210 . . 3  |-  ( (
ph  /\  A  =/=  B )  ->  ( X  e.  ( A I B )  \/  A  e.  ( X I B )  \/  B  e.  ( A I X ) ) )
6423, 39, 51, 63mpjao3dan 1293 . 2  |-  ( (
ph  /\  A  =/=  B )  ->  ( B  =  ( M `  A )  \/  A  =  B ) )
652, 64pm2.61dane 2772 1  |-  ( ph  ->  ( B  =  ( M `  A )  \/  A  =  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 366    /\ wa 367    \/ w3o 970    = wceq 1398    e. wcel 1823    =/= wne 2649   ` cfv 5570  (class class class)co 6270   Basecbs 14716   distcds 14793  TarskiGcstrkg 24023  Itvcitv 24030  LineGclng 24031  pInvGcmir 24234
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-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-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-hash 12388  df-trkgc 24042  df-trkgb 24043  df-trkgcb 24044  df-trkg 24048  df-mir 24235
This theorem is referenced by:  symquadlem  24267  midexlem  24270
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