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Theorem legbtwn 23143
Description: Deduce betweenness from "less than" relation. Corresponds loosely to Proposition 6.13 of [Schwabhauser] p. 45. (Contributed by Thierry Arnoux, 25-Aug-2019.)
Hypotheses
Ref Expression
legval.p  |-  P  =  ( Base `  G
)
legval.d  |-  .-  =  ( dist `  G )
legval.i  |-  I  =  (Itv `  G )
legval.l  |-  .<_  =  (≤G `  G )
legval.g  |-  ( ph  ->  G  e. TarskiG )
legid.a  |-  ( ph  ->  A  e.  P )
legid.b  |-  ( ph  ->  B  e.  P )
legtrd.c  |-  ( ph  ->  C  e.  P )
legtrd.d  |-  ( ph  ->  D  e.  P )
legbtwn.1  |-  ( ph  ->  ( A  e.  ( C I B )  \/  B  e.  ( C I A ) ) )
legbtwn.2  |-  ( ph  ->  ( C  .-  A
)  .<_  ( C  .-  B ) )
Assertion
Ref Expression
legbtwn  |-  ( ph  ->  A  e.  ( C I B ) )

Proof of Theorem legbtwn
StepHypRef Expression
1 simpr 461 . 2  |-  ( (
ph  /\  A  e.  ( C I B ) )  ->  A  e.  ( C I B ) )
2 legval.p . . . . 5  |-  P  =  ( Base `  G
)
3 legval.d . . . . 5  |-  .-  =  ( dist `  G )
4 legval.i . . . . 5  |-  I  =  (Itv `  G )
5 legval.g . . . . . 6  |-  ( ph  ->  G  e. TarskiG )
65adantr 465 . . . . 5  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  G  e. TarskiG )
7 legid.a . . . . . 6  |-  ( ph  ->  A  e.  P )
87adantr 465 . . . . 5  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  A  e.  P )
9 legid.b . . . . . 6  |-  ( ph  ->  B  e.  P )
109adantr 465 . . . . 5  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  B  e.  P )
11 legtrd.c . . . . . . 7  |-  ( ph  ->  C  e.  P )
1211adantr 465 . . . . . 6  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  C  e.  P )
13 simpr 461 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  B  e.  ( C I A ) )
142, 3, 4, 6, 12, 10, 8, 13tgbtwncom 23056 . . . . . 6  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  B  e.  ( A I C ) )
152, 3, 4, 6, 10, 12tgbtwntriv1 23059 . . . . . 6  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  B  e.  ( B I C ) )
16 legval.l . . . . . . . . 9  |-  .<_  =  (≤G `  G )
172, 3, 4, 16, 6, 12, 10, 8, 13btwnleg 23137 . . . . . . . . 9  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  ( C  .-  B )  .<_  ( C 
.-  A ) )
18 legbtwn.2 . . . . . . . . . 10  |-  ( ph  ->  ( C  .-  A
)  .<_  ( C  .-  B ) )
1918adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  ( C  .-  A )  .<_  ( C 
.-  B ) )
202, 3, 4, 16, 6, 12, 10, 12, 8, 17, 19legtri3 23139 . . . . . . . 8  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  ( C  .-  B )  =  ( C  .-  A ) )
2120eqcomd 2458 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  ( C  .-  A )  =  ( C  .-  B ) )
222, 3, 4, 6, 12, 8, 12, 10, 21tgcgrcomlr 23048 . . . . . 6  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  ( A  .-  C )  =  ( B  .-  C ) )
23 eqidd 2452 . . . . . 6  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  ( B  .-  C )  =  ( B  .-  C ) )
242, 3, 4, 6, 8, 10, 12, 10, 10, 12, 14, 15, 22, 23tgcgrsub 23078 . . . . 5  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  ( A  .-  B )  =  ( B  .-  B ) )
252, 3, 4, 6, 8, 10, 10, 24axtgcgrid 23037 . . . 4  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  A  =  B )
2625, 13eqeltrd 2537 . . 3  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  A  e.  ( C I A ) )
2725oveq2d 6203 . . 3  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  ( C I A )  =  ( C I B ) )
2826, 27eleqtrd 2539 . 2  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  A  e.  ( C I B ) )
29 legbtwn.1 . 2  |-  ( ph  ->  ( A  e.  ( C I B )  \/  B  e.  ( C I A ) ) )
301, 28, 29mpjaodan 784 1  |-  ( ph  ->  A  e.  ( C I B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758   class class class wbr 4387   ` cfv 5513  (class class class)co 6187   Basecbs 14273   distcds 14346  TarskiGcstrkg 23002  Itvcitv 23009  ≤Gcleg 23131
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 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-cnex 9436  ax-resscn 9437  ax-1cn 9438  ax-icn 9439  ax-addcl 9440  ax-addrcl 9441  ax-mulcl 9442  ax-mulrcl 9443  ax-mulcom 9444  ax-addass 9445  ax-mulass 9446  ax-distr 9447  ax-i2m1 9448  ax-1ne0 9449  ax-1rid 9450  ax-rnegex 9451  ax-rrecex 9452  ax-cnre 9453  ax-pre-lttri 9454  ax-pre-lttrn 9455  ax-pre-ltadd 9456  ax-pre-mulgt0 9457
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 2599  df-ne 2644  df-nel 2645  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-int 4224  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-om 6574  df-1st 6674  df-2nd 6675  df-recs 6929  df-rdg 6963  df-1o 7017  df-oadd 7021  df-er 7198  df-pm 7314  df-en 7408  df-dom 7409  df-sdom 7410  df-fin 7411  df-card 8207  df-cda 8435  df-pnf 9518  df-mnf 9519  df-xr 9520  df-ltxr 9521  df-le 9522  df-sub 9695  df-neg 9696  df-nn 10421  df-2 10478  df-3 10479  df-n0 10678  df-z 10745  df-uz 10960  df-fz 11536  df-fzo 11647  df-hash 12202  df-word 12328  df-concat 12330  df-s1 12331  df-s2 12574  df-s3 12575  df-trkgc 23021  df-trkgb 23022  df-trkgcb 23023  df-trkg 23027  df-cgrg 23080  df-leg 23132
This theorem is referenced by:  krippenlem  23207
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