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Theorem tgbtwndiff 26206
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 (𝜑𝐺 ∈ TarskiG)
tgbtwndiff.a (𝜑𝐴𝑃)
tgbtwndiff.b (𝜑𝐵𝑃)
tgbtwndiff.l (𝜑 → 2 ≤ (♯‘𝑃))
Assertion
Ref Expression
tgbtwndiff (𝜑 → ∃𝑐𝑃 (𝐵 ∈ (𝐴𝐼𝑐) ∧ 𝐵𝑐))
Distinct variable groups:   ,𝑐   𝐴,𝑐   𝐵,𝑐   𝐼,𝑐   𝑃,𝑐   𝜑,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 (𝜑𝐺 ∈ TarskiG)
54ad3antrrr 726 . . . 4 ((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) → 𝐺 ∈ TarskiG)
6 tgbtwndiff.a . . . . 5 (𝜑𝐴𝑃)
76ad3antrrr 726 . . . 4 ((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) → 𝐴𝑃)
8 tgbtwndiff.b . . . . 5 (𝜑𝐵𝑃)
98ad3antrrr 726 . . . 4 ((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) → 𝐵𝑃)
10 simpllr 772 . . . 4 ((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) → 𝑢𝑃)
11 simplr 765 . . . 4 ((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) → 𝑣𝑃)
121, 2, 3, 5, 7, 9, 10, 11axtgsegcon 26164 . . 3 ((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) → ∃𝑐𝑃 (𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 𝑐) = (𝑢 𝑣)))
135ad3antrrr 726 . . . . . . . . 9 (((((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) ∧ 𝑐𝑃) ∧ (𝐵 𝑐) = (𝑢 𝑣)) ∧ 𝐵 = 𝑐) → 𝐺 ∈ TarskiG)
1410ad3antrrr 726 . . . . . . . . 9 (((((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) ∧ 𝑐𝑃) ∧ (𝐵 𝑐) = (𝑢 𝑣)) ∧ 𝐵 = 𝑐) → 𝑢𝑃)
1511ad3antrrr 726 . . . . . . . . 9 (((((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) ∧ 𝑐𝑃) ∧ (𝐵 𝑐) = (𝑢 𝑣)) ∧ 𝐵 = 𝑐) → 𝑣𝑃)
169ad3antrrr 726 . . . . . . . . 9 (((((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) ∧ 𝑐𝑃) ∧ (𝐵 𝑐) = (𝑢 𝑣)) ∧ 𝐵 = 𝑐) → 𝐵𝑃)
17 simpr 485 . . . . . . . . . . 11 (((((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) ∧ 𝑐𝑃) ∧ (𝐵 𝑐) = (𝑢 𝑣)) ∧ 𝐵 = 𝑐) → 𝐵 = 𝑐)
1817oveq2d 7164 . . . . . . . . . 10 (((((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) ∧ 𝑐𝑃) ∧ (𝐵 𝑐) = (𝑢 𝑣)) ∧ 𝐵 = 𝑐) → (𝐵 𝐵) = (𝐵 𝑐))
19 simplr 765 . . . . . . . . . 10 (((((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) ∧ 𝑐𝑃) ∧ (𝐵 𝑐) = (𝑢 𝑣)) ∧ 𝐵 = 𝑐) → (𝐵 𝑐) = (𝑢 𝑣))
2018, 19eqtr2d 2862 . . . . . . . . 9 (((((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) ∧ 𝑐𝑃) ∧ (𝐵 𝑐) = (𝑢 𝑣)) ∧ 𝐵 = 𝑐) → (𝑢 𝑣) = (𝐵 𝐵))
211, 2, 3, 13, 14, 15, 16, 20axtgcgrid 26163 . . . . . . . 8 (((((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) ∧ 𝑐𝑃) ∧ (𝐵 𝑐) = (𝑢 𝑣)) ∧ 𝐵 = 𝑐) → 𝑢 = 𝑣)
22 simp-4r 780 . . . . . . . . 9 (((((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) ∧ 𝑐𝑃) ∧ (𝐵 𝑐) = (𝑢 𝑣)) ∧ 𝐵 = 𝑐) → 𝑢𝑣)
2322neneqd 3026 . . . . . . . 8 (((((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) ∧ 𝑐𝑃) ∧ (𝐵 𝑐) = (𝑢 𝑣)) ∧ 𝐵 = 𝑐) → ¬ 𝑢 = 𝑣)
2421, 23pm2.65da 813 . . . . . . 7 ((((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) ∧ 𝑐𝑃) ∧ (𝐵 𝑐) = (𝑢 𝑣)) → ¬ 𝐵 = 𝑐)
2524neqned 3028 . . . . . 6 ((((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) ∧ 𝑐𝑃) ∧ (𝐵 𝑐) = (𝑢 𝑣)) → 𝐵𝑐)
2625ex 413 . . . . 5 (((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) ∧ 𝑐𝑃) → ((𝐵 𝑐) = (𝑢 𝑣) → 𝐵𝑐))
2726anim2d 611 . . . 4 (((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) ∧ 𝑐𝑃) → ((𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 𝑐) = (𝑢 𝑣)) → (𝐵 ∈ (𝐴𝐼𝑐) ∧ 𝐵𝑐)))
2827reximdva 3279 . . 3 ((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) → (∃𝑐𝑃 (𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 𝑐) = (𝑢 𝑣)) → ∃𝑐𝑃 (𝐵 ∈ (𝐴𝐼𝑐) ∧ 𝐵𝑐)))
2912, 28mpd 15 . 2 ((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) → ∃𝑐𝑃 (𝐵 ∈ (𝐴𝐼𝑐) ∧ 𝐵𝑐))
30 tgbtwndiff.l . . 3 (𝜑 → 2 ≤ (♯‘𝑃))
311, 2, 3, 4, 30tglowdim1 26200 . 2 (𝜑 → ∃𝑢𝑃𝑣𝑃 𝑢𝑣)
3229, 31r19.29vva 3341 1 (𝜑 → ∃𝑐𝑃 (𝐵 ∈ (𝐴𝐼𝑐) ∧ 𝐵𝑐))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  wne 3021  wrex 3144   class class class wbr 5063  cfv 6352  (class class class)co 7148  cle 10665  2c2 11681  chash 13680  Basecbs 16473  distcds 16564  TarskiGcstrkg 26130  Itvcitv 26136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-er 8279  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-card 9357  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-2 11689  df-n0 11887  df-xnn0 11957  df-z 11971  df-uz 12233  df-fz 12883  df-hash 13681  df-trkgc 26148  df-trkgcb 26150  df-trkg 26153
This theorem is referenced by:  tgifscgr  26208  tgcgrxfr  26218  tgbtwnconn3  26277  legtrid  26291  hlcgrex  26316  hlcgreulem  26317  midexlem  26392  hpgerlem  26465
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