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Theorem tgbtwnxfr 25338
Description: A condition for extending betweenness to a new set of points based on congruence with another set of points. Theorem 4.6 of [Schwabhauser] p. 36. (Contributed by Thierry Arnoux, 27-Apr-2019.)
Hypotheses
Ref Expression
tgcgrxfr.p 𝑃 = (Base‘𝐺)
tgcgrxfr.m = (dist‘𝐺)
tgcgrxfr.i 𝐼 = (Itv‘𝐺)
tgcgrxfr.r = (cgrG‘𝐺)
tgcgrxfr.g (𝜑𝐺 ∈ TarskiG)
tgbtwnxfr.a (𝜑𝐴𝑃)
tgbtwnxfr.b (𝜑𝐵𝑃)
tgbtwnxfr.c (𝜑𝐶𝑃)
tgbtwnxfr.d (𝜑𝐷𝑃)
tgbtwnxfr.e (𝜑𝐸𝑃)
tgbtwnxfr.f (𝜑𝐹𝑃)
tgbtwnxfr.2 (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩)
tgbtwnxfr.1 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
Assertion
Ref Expression
tgbtwnxfr (𝜑𝐸 ∈ (𝐷𝐼𝐹))

Proof of Theorem tgbtwnxfr
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 tgcgrxfr.p . . . 4 𝑃 = (Base‘𝐺)
2 tgcgrxfr.m . . . 4 = (dist‘𝐺)
3 tgcgrxfr.i . . . 4 𝐼 = (Itv‘𝐺)
4 tgcgrxfr.g . . . . 5 (𝜑𝐺 ∈ TarskiG)
54ad2antrr 761 . . . 4 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → 𝐺 ∈ TarskiG)
6 simplr 791 . . . 4 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → 𝑒𝑃)
7 tgbtwnxfr.e . . . . 5 (𝜑𝐸𝑃)
87ad2antrr 761 . . . 4 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → 𝐸𝑃)
9 tgbtwnxfr.d . . . . . 6 (𝜑𝐷𝑃)
109ad2antrr 761 . . . . 5 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → 𝐷𝑃)
11 tgbtwnxfr.f . . . . . 6 (𝜑𝐹𝑃)
1211ad2antrr 761 . . . . 5 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → 𝐹𝑃)
13 simprl 793 . . . . 5 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → 𝑒 ∈ (𝐷𝐼𝐹))
14 eqidd 2622 . . . . 5 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → (𝐷 𝐹) = (𝐷 𝐹))
15 eqidd 2622 . . . . 5 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → (𝑒 𝐹) = (𝑒 𝐹))
16 tgcgrxfr.r . . . . . 6 = (cgrG‘𝐺)
17 tgbtwnxfr.a . . . . . . . . 9 (𝜑𝐴𝑃)
1817ad2antrr 761 . . . . . . . 8 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → 𝐴𝑃)
19 tgbtwnxfr.b . . . . . . . . 9 (𝜑𝐵𝑃)
2019ad2antrr 761 . . . . . . . 8 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → 𝐵𝑃)
21 tgbtwnxfr.c . . . . . . . . 9 (𝜑𝐶𝑃)
2221ad2antrr 761 . . . . . . . 8 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → 𝐶𝑃)
23 simprr 795 . . . . . . . . 9 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)
241, 2, 3, 16, 5, 18, 20, 22, 10, 6, 12, 23trgcgrcom 25336 . . . . . . . 8 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → ⟨“𝐷𝑒𝐹”⟩ ⟨“𝐴𝐵𝐶”⟩)
25 tgbtwnxfr.2 . . . . . . . . 9 (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩)
2625ad2antrr 761 . . . . . . . 8 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩)
271, 2, 3, 16, 5, 10, 6, 12, 18, 20, 22, 24, 10, 8, 12, 26cgr3tr 25337 . . . . . . 7 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → ⟨“𝐷𝑒𝐹”⟩ ⟨“𝐷𝐸𝐹”⟩)
281, 2, 3, 16, 5, 10, 6, 12, 10, 8, 12, 27trgcgrcom 25336 . . . . . 6 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → ⟨“𝐷𝐸𝐹”⟩ ⟨“𝐷𝑒𝐹”⟩)
291, 2, 3, 16, 5, 10, 8, 12, 10, 6, 12, 28cgr3simp1 25328 . . . . 5 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → (𝐷 𝐸) = (𝐷 𝑒))
301, 2, 3, 16, 5, 10, 8, 12, 10, 6, 12, 28cgr3simp2 25329 . . . . . 6 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → (𝐸 𝐹) = (𝑒 𝐹))
311, 2, 3, 5, 8, 12, 6, 12, 30tgcgrcomlr 25288 . . . . 5 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → (𝐹 𝐸) = (𝐹 𝑒))
321, 2, 3, 5, 10, 6, 12, 8, 10, 6, 12, 6, 13, 13, 14, 15, 29, 31tgifscgr 25316 . . . 4 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → (𝑒 𝐸) = (𝑒 𝑒))
331, 2, 3, 5, 6, 8, 6, 32axtgcgrid 25275 . . 3 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → 𝑒 = 𝐸)
3433, 13eqeltrrd 2699 . 2 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → 𝐸 ∈ (𝐷𝐼𝐹))
35 tgbtwnxfr.1 . . 3 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
361, 2, 3, 16, 4, 17, 19, 21, 9, 7, 11, 25cgr3simp3 25330 . . . 4 (𝜑 → (𝐶 𝐴) = (𝐹 𝐷))
371, 2, 3, 4, 21, 17, 11, 9, 36tgcgrcomlr 25288 . . 3 (𝜑 → (𝐴 𝐶) = (𝐷 𝐹))
381, 2, 3, 16, 4, 17, 19, 21, 9, 11, 35, 37tgcgrxfr 25326 . 2 (𝜑 → ∃𝑒𝑃 (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩))
3934, 38r19.29a 3072 1 (𝜑𝐸 ∈ (𝐷𝐼𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987   class class class wbr 4618  cfv 5852  (class class class)co 6610  ⟨“cs3 13531  Basecbs 15788  distcds 15878  TarskiGcstrkg 25242  Itvcitv 25248  cgrGccgrg 25318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-pm 7812  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-card 8716  df-cda 8941  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-nn 10972  df-2 11030  df-3 11031  df-n0 11244  df-xnn0 11315  df-z 11329  df-uz 11639  df-fz 12276  df-fzo 12414  df-hash 13065  df-word 13245  df-concat 13247  df-s1 13248  df-s2 13537  df-s3 13538  df-trkgc 25260  df-trkgb 25261  df-trkgcb 25262  df-trkg 25265  df-cgrg 25319
This theorem is referenced by:  lnxfr  25374  tgfscgr  25376  legov  25393  legov2  25394  legtrd  25397  mirbtwni  25479  cgrabtwn  25630  cgrahl  25631
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