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Theorem tgbtwndiff 25201
 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 762 . . . 4 ((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) → 𝐺 ∈ TarskiG)
6 tgbtwndiff.a . . . . 5 (𝜑𝐴𝑃)
76ad3antrrr 762 . . . 4 ((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) → 𝐴𝑃)
8 tgbtwndiff.b . . . . 5 (𝜑𝐵𝑃)
98ad3antrrr 762 . . . 4 ((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) → 𝐵𝑃)
10 simpllr 795 . . . 4 ((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) → 𝑢𝑃)
11 simplr 788 . . . 4 ((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) → 𝑣𝑃)
121, 2, 3, 5, 7, 9, 10, 11axtgsegcon 25163 . . 3 ((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) → ∃𝑐𝑃 (𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 𝑐) = (𝑢 𝑣)))
135ad3antrrr 762 . . . . . . . . 9 (((((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) ∧ 𝑐𝑃) ∧ (𝐵 𝑐) = (𝑢 𝑣)) ∧ 𝐵 = 𝑐) → 𝐺 ∈ TarskiG)
1410ad3antrrr 762 . . . . . . . . 9 (((((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) ∧ 𝑐𝑃) ∧ (𝐵 𝑐) = (𝑢 𝑣)) ∧ 𝐵 = 𝑐) → 𝑢𝑃)
1511ad3antrrr 762 . . . . . . . . 9 (((((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) ∧ 𝑐𝑃) ∧ (𝐵 𝑐) = (𝑢 𝑣)) ∧ 𝐵 = 𝑐) → 𝑣𝑃)
169ad3antrrr 762 . . . . . . . . 9 (((((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) ∧ 𝑐𝑃) ∧ (𝐵 𝑐) = (𝑢 𝑣)) ∧ 𝐵 = 𝑐) → 𝐵𝑃)
17 simpr 476 . . . . . . . . . . 11 (((((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) ∧ 𝑐𝑃) ∧ (𝐵 𝑐) = (𝑢 𝑣)) ∧ 𝐵 = 𝑐) → 𝐵 = 𝑐)
1817oveq2d 6565 . . . . . . . . . 10 (((((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) ∧ 𝑐𝑃) ∧ (𝐵 𝑐) = (𝑢 𝑣)) ∧ 𝐵 = 𝑐) → (𝐵 𝐵) = (𝐵 𝑐))
19 simplr 788 . . . . . . . . . 10 (((((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) ∧ 𝑐𝑃) ∧ (𝐵 𝑐) = (𝑢 𝑣)) ∧ 𝐵 = 𝑐) → (𝐵 𝑐) = (𝑢 𝑣))
2018, 19eqtr2d 2645 . . . . . . . . 9 (((((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) ∧ 𝑐𝑃) ∧ (𝐵 𝑐) = (𝑢 𝑣)) ∧ 𝐵 = 𝑐) → (𝑢 𝑣) = (𝐵 𝐵))
211, 2, 3, 13, 14, 15, 16, 20axtgcgrid 25162 . . . . . . . 8 (((((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) ∧ 𝑐𝑃) ∧ (𝐵 𝑐) = (𝑢 𝑣)) ∧ 𝐵 = 𝑐) → 𝑢 = 𝑣)
22 simp-4r 803 . . . . . . . . 9 (((((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) ∧ 𝑐𝑃) ∧ (𝐵 𝑐) = (𝑢 𝑣)) ∧ 𝐵 = 𝑐) → 𝑢𝑣)
2322neneqd 2787 . . . . . . . 8 (((((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) ∧ 𝑐𝑃) ∧ (𝐵 𝑐) = (𝑢 𝑣)) ∧ 𝐵 = 𝑐) → ¬ 𝑢 = 𝑣)
2421, 23pm2.65da 598 . . . . . . 7 ((((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) ∧ 𝑐𝑃) ∧ (𝐵 𝑐) = (𝑢 𝑣)) → ¬ 𝐵 = 𝑐)
2524neqned 2789 . . . . . 6 ((((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) ∧ 𝑐𝑃) ∧ (𝐵 𝑐) = (𝑢 𝑣)) → 𝐵𝑐)
2625ex 449 . . . . 5 (((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) ∧ 𝑐𝑃) → ((𝐵 𝑐) = (𝑢 𝑣) → 𝐵𝑐))
2726anim2d 587 . . . 4 (((((𝜑𝑢𝑃) ∧ 𝑣𝑃) ∧ 𝑢𝑣) ∧ 𝑐𝑃) → ((𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 𝑐) = (𝑢 𝑣)) → (𝐵 ∈ (𝐴𝐼𝑐) ∧ 𝐵𝑐)))
2827reximdva 3000 . . 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 25195 . 2 (𝜑 → ∃𝑢𝑃𝑣𝑃 𝑢𝑣)
3229, 31r19.29vva 3062 1 (𝜑 → ∃𝑐𝑃 (𝐵 ∈ (𝐴𝐼𝑐) ∧ 𝐵𝑐))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549   ≤ cle 9954  2c2 10947  #chash 12979  Basecbs 15695  distcds 15777  TarskiGcstrkg 25129  Itvcitv 25135 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-trkgc 25147  df-trkgcb 25149  df-trkg 25152 This theorem is referenced by:  tgifscgr  25203  tgcgrxfr  25213  tgbtwnconn3  25272  legtrid  25286  hlcgrex  25311  hlcgreulem  25312  midexlem  25387  hpgerlem  25457
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