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Theorem tgbtwndiff 23653
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 23617 . . 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 6300 . . . . . . . . . 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 2509 . . . . . . . . 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 23616 . . . . . . . 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 2669 . . . . . . . 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 )
2524neqned 2670 . . . . . 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 2938 . . 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 23647 . 2  |-  ( ph  ->  E. u  e.  P  E. v  e.  P  u  =/=  v )
3229, 31r19.29_2a 3005 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 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815   class class class wbr 4447   ` cfv 5588  (class class class)co 6284    <_ cle 9629   2c2 10585   #chash 12373   Basecbs 14490   distcds 14564  TarskiGcstrkg 23581  Itvcitv 23588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-hash 12374  df-trkgc 23600  df-trkgcb 23602  df-trkg 23606
This theorem is referenced by:  tgifscgr  23656  tgcgrxfr  23665  tgbtwnconn3  23719  legtrid  23733  midexlem  23805
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