MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  legbtwn Structured version   Unicode version

Theorem legbtwn 24185
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 459 . 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 463 . . . . 5  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  G  e. TarskiG )
7 legid.a . . . . . 6  |-  ( ph  ->  A  e.  P )
87adantr 463 . . . . 5  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  A  e.  P )
9 legid.b . . . . . 6  |-  ( ph  ->  B  e.  P )
109adantr 463 . . . . 5  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  B  e.  P )
11 legtrd.c . . . . . . 7  |-  ( ph  ->  C  e.  P )
1211adantr 463 . . . . . 6  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  C  e.  P )
13 simpr 459 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  B  e.  ( C I A ) )
142, 3, 4, 6, 12, 10, 8, 13tgbtwncom 24083 . . . . . 6  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  B  e.  ( A I C ) )
152, 3, 4, 6, 10, 12tgbtwntriv1 24086 . . . . . 6  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  B  e.  ( B I C ) )
16 legval.l . . . . . . . 8  |-  .<_  =  (≤G `  G )
17 legbtwn.2 . . . . . . . . 9  |-  ( ph  ->  ( C  .-  A
)  .<_  ( C  .-  B ) )
1817adantr 463 . . . . . . . 8  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  ( C  .-  A )  .<_  ( C 
.-  B ) )
192, 3, 4, 16, 6, 12, 10, 8, 13btwnleg 24179 . . . . . . . 8  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  ( C  .-  B )  .<_  ( C 
.-  A ) )
202, 3, 4, 16, 6, 12, 8, 12, 10, 18, 19legtri3 24181 . . . . . . 7  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  ( C  .-  A )  =  ( C  .-  B ) )
212, 3, 4, 6, 12, 8, 12, 10, 20tgcgrcomlr 24075 . . . . . 6  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  ( A  .-  C )  =  ( B  .-  C ) )
22 eqidd 2455 . . . . . 6  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  ( B  .-  C )  =  ( B  .-  C ) )
232, 3, 4, 6, 8, 10, 12, 10, 10, 12, 14, 15, 21, 22tgcgrsub 24105 . . . . 5  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  ( A  .-  B )  =  ( B  .-  B ) )
242, 3, 4, 6, 8, 10, 10, 23axtgcgrid 24061 . . . 4  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  A  =  B )
2524, 13eqeltrd 2542 . . 3  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  A  e.  ( C I A ) )
2624oveq2d 6286 . . 3  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  ( C I A )  =  ( C I B ) )
2725, 26eleqtrd 2544 . 2  |-  ( (
ph  /\  B  e.  ( C I A ) )  ->  A  e.  ( C I B ) )
28 legbtwn.1 . 2  |-  ( ph  ->  ( A  e.  ( C I B )  \/  B  e.  ( C I A ) ) )
291, 27, 28mpjaodan 784 1  |-  ( ph  ->  A  e.  ( C I B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367    = wceq 1398    e. wcel 1823   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14719   distcds 14796  TarskiGcstrkg 24026  Itvcitv 24033  ≤Gcleg 24173
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-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-trkg 24051  df-cgrg 24107  df-leg 24174
This theorem is referenced by:  krippenlem  24271
  Copyright terms: Public domain W3C validator