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Theorem colmid 23210
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 461 . . 3  |-  ( (
ph  /\  A  =  B )  ->  A  =  B )
21olcd 393 . 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 725 . . . . 5  |-  ( ( ( ph  /\  A  =/=  B )  /\  X  e.  ( A I B ) )  ->  G  e. TarskiG )
10 colmid.x . . . . . 6  |-  ( ph  ->  X  e.  P )
1110ad2antrr 725 . . . . 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 725 . . . . 5  |-  ( ( ( ph  /\  A  =/=  B )  /\  X  e.  ( A I B ) )  ->  A  e.  P )
15 colmid.b . . . . . 6  |-  ( ph  ->  B  e.  P )
1615ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  A  =/=  B )  /\  X  e.  ( A I B ) )  ->  B  e.  P )
17 colmid.d . . . . . . 7  |-  ( ph  ->  ( X  .-  A
)  =  ( X 
.-  B ) )
1817ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  B )  /\  X  e.  ( A I B ) )  ->  ( X  .-  A )  =  ( X  .-  B
) )
1918eqcomd 2459 . . . . 5  |-  ( ( ( ph  /\  A  =/=  B )  /\  X  e.  ( A I B ) )  ->  ( X  .-  B )  =  ( X  .-  A
) )
20 simpr 461 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  B )  /\  X  e.  ( A I B ) )  ->  X  e.  ( A I B ) )
213, 4, 5, 9, 14, 11, 16, 20tgbtwncom 23061 . . . . 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 23191 . . . 4  |-  ( ( ( ph  /\  A  =/=  B )  /\  X  e.  ( A I B ) )  ->  B  =  ( M `  A ) )
2322orcd 392 . . 3  |-  ( ( ( ph  /\  A  =/=  B )  /\  X  e.  ( A I B ) )  ->  ( B  =  ( M `  A )  \/  A  =  B ) )
248adantr 465 . . . . . . 7  |-  ( (
ph  /\  A  e.  ( X I B ) )  ->  G  e. TarskiG )
2515adantr 465 . . . . . . 7  |-  ( (
ph  /\  A  e.  ( X I B ) )  ->  B  e.  P )
2613adantr 465 . . . . . . 7  |-  ( (
ph  /\  A  e.  ( X I B ) )  ->  A  e.  P )
2710adantr 465 . . . . . . . 8  |-  ( (
ph  /\  A  e.  ( X I B ) )  ->  X  e.  P )
28 simpr 461 . . . . . . . . 9  |-  ( (
ph  /\  A  e.  ( X I B ) )  ->  A  e.  ( X I B ) )
293, 4, 5, 24, 27, 26, 25, 28tgbtwncom 23061 . . . . . . . 8  |-  ( (
ph  /\  A  e.  ( X I B ) )  ->  A  e.  ( B I X ) )
303, 4, 5, 24, 26, 27tgbtwntriv1 23064 . . . . . . . 8  |-  ( (
ph  /\  A  e.  ( X I B ) )  ->  A  e.  ( A I X ) )
313, 4, 5, 8, 10, 13, 10, 15, 17tgcgrcomlr 23053 . . . . . . . . . 10  |-  ( ph  ->  ( A  .-  X
)  =  ( B 
.-  X ) )
3231adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  A  e.  ( X I B ) )  ->  ( A  .-  X )  =  ( B  .-  X ) )
3332eqcomd 2459 . . . . . . . 8  |-  ( (
ph  /\  A  e.  ( X I B ) )  ->  ( B  .-  X )  =  ( A  .-  X ) )
34 eqidd 2452 . . . . . . . 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 23083 . . . . . . 7  |-  ( (
ph  /\  A  e.  ( X I B ) )  ->  ( B  .-  A )  =  ( A  .-  A ) )
363, 4, 5, 24, 25, 26, 26, 35axtgcgrid 23042 . . . . . 6  |-  ( (
ph  /\  A  e.  ( X I B ) )  ->  B  =  A )
3736eqcomd 2459 . . . . 5  |-  ( (
ph  /\  A  e.  ( X I B ) )  ->  A  =  B )
3837adantlr 714 . . . 4  |-  ( ( ( ph  /\  A  =/=  B )  /\  A  e.  ( X I B ) )  ->  A  =  B )
3938olcd 393 . . 3  |-  ( ( ( ph  /\  A  =/=  B )  /\  A  e.  ( X I B ) )  ->  ( B  =  ( M `  A )  \/  A  =  B ) )
408adantr 465 . . . . . 6  |-  ( (
ph  /\  B  e.  ( A I X ) )  ->  G  e. TarskiG )
4113adantr 465 . . . . . 6  |-  ( (
ph  /\  B  e.  ( A I X ) )  ->  A  e.  P )
4215adantr 465 . . . . . 6  |-  ( (
ph  /\  B  e.  ( A I X ) )  ->  B  e.  P )
4310adantr 465 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( A I X ) )  ->  X  e.  P )
44 simpr 461 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( A I X ) )  ->  B  e.  ( A I X ) )
453, 4, 5, 40, 42, 43tgbtwntriv1 23064 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( A I X ) )  ->  B  e.  ( B I X ) )
4631adantr 465 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( A I X ) )  ->  ( A  .-  X )  =  ( B  .-  X ) )
47 eqidd 2452 . . . . . . 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 23083 . . . . . 6  |-  ( (
ph  /\  B  e.  ( A I X ) )  ->  ( A  .-  B )  =  ( B  .-  B ) )
493, 4, 5, 40, 41, 42, 42, 48axtgcgrid 23042 . . . . 5  |-  ( (
ph  /\  B  e.  ( A I X ) )  ->  A  =  B )
5049adantlr 714 . . . 4  |-  ( ( ( ph  /\  A  =/=  B )  /\  B  e.  ( A I X ) )  ->  A  =  B )
5150olcd 393 . . 3  |-  ( ( ( ph  /\  A  =/=  B )  /\  B  e.  ( A I X ) )  ->  ( B  =  ( M `  A )  \/  A  =  B ) )
52 df-ne 2646 . . . . 5  |-  ( A  =/=  B  <->  -.  A  =  B )
53 colmid.c . . . . . . . 8  |-  ( ph  ->  ( X  e.  ( A L B )  \/  A  =  B ) )
5453orcomd 388 . . . . . . 7  |-  ( ph  ->  ( A  =  B  \/  X  e.  ( A L B ) ) )
5554ord 377 . . . . . 6  |-  ( ph  ->  ( -.  A  =  B  ->  X  e.  ( A L B ) ) )
5655imp 429 . . . . 5  |-  ( (
ph  /\  -.  A  =  B )  ->  X  e.  ( A L B ) )
5752, 56sylan2b 475 . . . 4  |-  ( (
ph  /\  A  =/=  B )  ->  X  e.  ( A L B ) )
588adantr 465 . . . . 5  |-  ( (
ph  /\  A  =/=  B )  ->  G  e. TarskiG )
5913adantr 465 . . . . 5  |-  ( (
ph  /\  A  =/=  B )  ->  A  e.  P )
6015adantr 465 . . . . 5  |-  ( (
ph  /\  A  =/=  B )  ->  B  e.  P )
61 simpr 461 . . . . 5  |-  ( (
ph  /\  A  =/=  B )  ->  A  =/=  B )
6210adantr 465 . . . . 5  |-  ( (
ph  /\  A  =/=  B )  ->  X  e.  P )
633, 6, 5, 58, 59, 60, 61, 62tgellng 23108 . . . 4  |-  ( (
ph  /\  A  =/=  B )  ->  ( X  e.  ( A L B )  <->  ( X  e.  ( A I B )  \/  A  e.  ( X I B )  \/  B  e.  ( A I X ) ) ) )
6457, 63mpbid 210 . . 3  |-  ( (
ph  /\  A  =/=  B )  ->  ( X  e.  ( A I B )  \/  A  e.  ( X I B )  \/  B  e.  ( A I X ) ) )
6523, 39, 51, 64mpjao3dan 1286 . 2  |-  ( (
ph  /\  A  =/=  B )  ->  ( B  =  ( M `  A )  \/  A  =  B ) )
662, 65pm2.61dane 2766 1  |-  ( ph  ->  ( B  =  ( M `  A )  \/  A  =  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    \/ w3o 964    = wceq 1370    e. wcel 1758    =/= wne 2644   ` cfv 5518  (class class class)co 6192   Basecbs 14278   distcds 14351  TarskiGcstrkg 23007  Itvcitv 23014  LineGclng 23015  pInvGcmir 23183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-card 8212  df-cda 8440  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-n0 10683  df-z 10750  df-uz 10965  df-fz 11541  df-hash 12207  df-trkgc 23026  df-trkgb 23027  df-trkgcb 23028  df-trkg 23032  df-mir 23184
This theorem is referenced by:  symquadlem  23211  midexlem  23214
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