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Theorem tgbtwndiff 22971
Description: There is always a  c distinct from  B such that  B lies between  A and  c. Theorem 3.14 of [Schwabhauser] p. 32. The condition "the space is of dimension 1 or more" is written here as  2  <_  (
# `  P ) for simplicity. (Contributed by Thierry Arnoux, 23-Mar-2019.)
Hypotheses
Ref Expression
tgbtwndiff.p  |-  P  =  ( Base `  G
)
tgbtwndiff.d  |-  .-  =  ( dist `  G )
tgbtwndiff.i  |-  I  =  (Itv `  G )
tgbtwndiff.g  |-  ( ph  ->  G  e. TarskiG )
tgbtwndiff.a  |-  ( ph  ->  A  e.  P )
tgbtwndiff.b  |-  ( ph  ->  B  e.  P )
tgbtwndiff.l  |-  ( ph  ->  2  <_  ( # `  P
) )
Assertion
Ref Expression
tgbtwndiff  |-  ( ph  ->  E. c  e.  P  ( B  e.  ( A I c )  /\  B  =/=  c
) )
Distinct variable groups:    .- , c    A, c    B, c    I, c    P, c    ph, c
Allowed substitution hint:    G( c)

Proof of Theorem tgbtwndiff
Dummy variables  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tgbtwndiff.p . . . 4  |-  P  =  ( Base `  G
)
2 tgbtwndiff.d . . . 4  |-  .-  =  ( dist `  G )
3 tgbtwndiff.i . . . 4  |-  I  =  (Itv `  G )
4 tgbtwndiff.g . . . . 5  |-  ( ph  ->  G  e. TarskiG )
54ad3antrrr 729 . . . 4  |-  ( ( ( ( ph  /\  u  e.  P )  /\  v  e.  P
)  /\  u  =/=  v )  ->  G  e. TarskiG )
6 tgbtwndiff.a . . . . 5  |-  ( ph  ->  A  e.  P )
76ad3antrrr 729 . . . 4  |-  ( ( ( ( ph  /\  u  e.  P )  /\  v  e.  P
)  /\  u  =/=  v )  ->  A  e.  P )
8 tgbtwndiff.b . . . . 5  |-  ( ph  ->  B  e.  P )
98ad3antrrr 729 . . . 4  |-  ( ( ( ( ph  /\  u  e.  P )  /\  v  e.  P
)  /\  u  =/=  v )  ->  B  e.  P )
10 simpllr 758 . . . 4  |-  ( ( ( ( ph  /\  u  e.  P )  /\  v  e.  P
)  /\  u  =/=  v )  ->  u  e.  P )
11 simplr 754 . . . 4  |-  ( ( ( ( ph  /\  u  e.  P )  /\  v  e.  P
)  /\  u  =/=  v )  ->  v  e.  P )
121, 2, 3, 5, 7, 9, 10, 11axtgsegcon 22937 . . 3  |-  ( ( ( ( ph  /\  u  e.  P )  /\  v  e.  P
)  /\  u  =/=  v )  ->  E. c  e.  P  ( B  e.  ( A I c )  /\  ( B 
.-  c )  =  ( u  .-  v
) ) )
135ad3antrrr 729 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ph  /\  u  e.  P )  /\  v  e.  P )  /\  u  =/=  v )  /\  c  e.  P )  /\  ( B  .-  c )  =  ( u  .-  v
) )  /\  B  =  c )  ->  G  e. TarskiG )
1410ad3antrrr 729 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ph  /\  u  e.  P )  /\  v  e.  P )  /\  u  =/=  v )  /\  c  e.  P )  /\  ( B  .-  c )  =  ( u  .-  v
) )  /\  B  =  c )  ->  u  e.  P )
1511ad3antrrr 729 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ph  /\  u  e.  P )  /\  v  e.  P )  /\  u  =/=  v )  /\  c  e.  P )  /\  ( B  .-  c )  =  ( u  .-  v
) )  /\  B  =  c )  -> 
v  e.  P )
169ad3antrrr 729 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ph  /\  u  e.  P )  /\  v  e.  P )  /\  u  =/=  v )  /\  c  e.  P )  /\  ( B  .-  c )  =  ( u  .-  v
) )  /\  B  =  c )  ->  B  e.  P )
17 simpr 461 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ph  /\  u  e.  P )  /\  v  e.  P )  /\  u  =/=  v )  /\  c  e.  P )  /\  ( B  .-  c )  =  ( u  .-  v
) )  /\  B  =  c )  ->  B  =  c )
1817oveq2d 6119 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ph  /\  u  e.  P )  /\  v  e.  P )  /\  u  =/=  v )  /\  c  e.  P )  /\  ( B  .-  c )  =  ( u  .-  v
) )  /\  B  =  c )  -> 
( B  .-  B
)  =  ( B 
.-  c ) )
19 simplr 754 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ph  /\  u  e.  P )  /\  v  e.  P )  /\  u  =/=  v )  /\  c  e.  P )  /\  ( B  .-  c )  =  ( u  .-  v
) )  /\  B  =  c )  -> 
( B  .-  c
)  =  ( u 
.-  v ) )
2018, 19eqtr2d 2476 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ph  /\  u  e.  P )  /\  v  e.  P )  /\  u  =/=  v )  /\  c  e.  P )  /\  ( B  .-  c )  =  ( u  .-  v
) )  /\  B  =  c )  -> 
( u  .-  v
)  =  ( B 
.-  B ) )
211, 2, 3, 13, 14, 15, 16, 20axtgcgrid 22936 . . . . . . . 8  |-  ( ( ( ( ( ( ( ph  /\  u  e.  P )  /\  v  e.  P )  /\  u  =/=  v )  /\  c  e.  P )  /\  ( B  .-  c )  =  ( u  .-  v
) )  /\  B  =  c )  ->  u  =  v )
22 simp-4r 766 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ph  /\  u  e.  P )  /\  v  e.  P )  /\  u  =/=  v )  /\  c  e.  P )  /\  ( B  .-  c )  =  ( u  .-  v
) )  /\  B  =  c )  ->  u  =/=  v )
2322neneqd 2636 . . . . . . . 8  |-  ( ( ( ( ( ( ( ph  /\  u  e.  P )  /\  v  e.  P )  /\  u  =/=  v )  /\  c  e.  P )  /\  ( B  .-  c )  =  ( u  .-  v
) )  /\  B  =  c )  ->  -.  u  =  v
)
2421, 23pm2.65da 576 . . . . . . 7  |-  ( ( ( ( ( (
ph  /\  u  e.  P )  /\  v  e.  P )  /\  u  =/=  v )  /\  c  e.  P )  /\  ( B  .-  c )  =  ( u  .-  v
) )  ->  -.  B  =  c )
2524neneqad 2693 . . . . . 6  |-  ( ( ( ( ( (
ph  /\  u  e.  P )  /\  v  e.  P )  /\  u  =/=  v )  /\  c  e.  P )  /\  ( B  .-  c )  =  ( u  .-  v
) )  ->  B  =/=  c )
2625ex 434 . . . . 5  |-  ( ( ( ( ( ph  /\  u  e.  P )  /\  v  e.  P
)  /\  u  =/=  v )  /\  c  e.  P )  ->  (
( B  .-  c
)  =  ( u 
.-  v )  ->  B  =/=  c ) )
2726anim2d 565 . . . 4  |-  ( ( ( ( ( ph  /\  u  e.  P )  /\  v  e.  P
)  /\  u  =/=  v )  /\  c  e.  P )  ->  (
( B  e.  ( A I c )  /\  ( B  .-  c )  =  ( u  .-  v ) )  ->  ( B  e.  ( A I c )  /\  B  =/=  c ) ) )
2827reximdva 2840 . . 3  |-  ( ( ( ( ph  /\  u  e.  P )  /\  v  e.  P
)  /\  u  =/=  v )  ->  ( E. c  e.  P  ( B  e.  ( A I c )  /\  ( B  .-  c )  =  ( u  .-  v ) )  ->  E. c  e.  P  ( B  e.  ( A I c )  /\  B  =/=  c ) ) )
2912, 28mpd 15 . 2  |-  ( ( ( ( ph  /\  u  e.  P )  /\  v  e.  P
)  /\  u  =/=  v )  ->  E. c  e.  P  ( B  e.  ( A I c )  /\  B  =/=  c ) )
30 tgbtwndiff.l . . 3  |-  ( ph  ->  2  <_  ( # `  P
) )
311, 2, 3, 4, 30tglowdim1 22965 . 2  |-  ( ph  ->  E. u  e.  P  E. v  e.  P  u  =/=  v )
3229, 31r19.29_2a 2876 1  |-  ( ph  ->  E. c  e.  P  ( B  e.  ( A I c )  /\  B  =/=  c
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2618   E.wrex 2728   class class class wbr 4304   ` cfv 5430  (class class class)co 6103    <_ cle 9431   2c2 10383   #chash 12115   Basecbs 14186   distcds 14259  TarskiGcstrkg 22901  Itvcitv 22909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-card 8121  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-2 10392  df-n0 10592  df-z 10659  df-uz 10874  df-fz 11450  df-hash 12116  df-trkgc 22921  df-trkgcb 22923  df-trkg 22928
This theorem is referenced by:  tgifscgr  22973  tgcgrxfr  22982  tgbtwnconn3  23021  legtrid  23034  midexlem  23098
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