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Theorem colmid 23766
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 2468 . . . . 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 23600 . . . . 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 23746 . . . 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 23600 . . . . . . . 8  |-  ( (
ph  /\  A  e.  ( X I B ) )  ->  A  e.  ( B I X ) )
303, 4, 5, 24, 26, 27tgbtwntriv1 23603 . . . . . . . 8  |-  ( (
ph  /\  A  e.  ( X I B ) )  ->  A  e.  ( A I X ) )
313, 4, 5, 8, 10, 13, 10, 15, 17tgcgrcomlr 23592 . . . . . . . . . 10  |-  ( ph  ->  ( A  .-  X
)  =  ( B 
.-  X ) )
3231adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  A  e.  ( X I B ) )  ->  ( A  .-  X )  =  ( B  .-  X ) )
3332eqcomd 2468 . . . . . . . 8  |-  ( (
ph  /\  A  e.  ( X I B ) )  ->  ( B  .-  X )  =  ( A  .-  X ) )
34 eqidd 2461 . . . . . . . 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 23622 . . . . . . 7  |-  ( (
ph  /\  A  e.  ( X I B ) )  ->  ( B  .-  A )  =  ( A  .-  A ) )
363, 4, 5, 24, 25, 26, 26, 35axtgcgrid 23581 . . . . . 6  |-  ( (
ph  /\  A  e.  ( X I B ) )  ->  B  =  A )
3736eqcomd 2468 . . . . 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 23603 . . . . . . 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 2461 . . . . . . 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 23622 . . . . . 6  |-  ( (
ph  /\  B  e.  ( A I X ) )  ->  ( A  .-  B )  =  ( B  .-  B ) )
493, 4, 5, 40, 41, 42, 42, 48axtgcgrid 23581 . . . . 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 2657 . . . . 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 23661 . . . 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 1290 . 2  |-  ( (
ph  /\  A  =/=  B )  ->  ( B  =  ( M `  A )  \/  A  =  B ) )
662, 65pm2.61dane 2778 1  |-  ( ph  ->  ( B  =  ( M `  A )  \/  A  =  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    \/ w3o 967    = wceq 1374    e. wcel 1762    =/= wne 2655   ` cfv 5579  (class class class)co 6275   Basecbs 14479   distcds 14553  TarskiGcstrkg 23546  Itvcitv 23553  LineGclng 23554  pInvGcmir 23739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  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 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-hash 12361  df-trkgc 23565  df-trkgb 23566  df-trkgcb 23567  df-trkg 23571  df-mir 23740
This theorem is referenced by:  symquadlem  23767  midexlem  23770
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