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Theorem tgbtwnxfr 25842
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 717 . . . 4 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → 𝐺 ∈ TarskiG)
6 simplr 785 . . . 4 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → 𝑒𝑃)
7 tgbtwnxfr.e . . . . 5 (𝜑𝐸𝑃)
87ad2antrr 717 . . . 4 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → 𝐸𝑃)
9 tgbtwnxfr.d . . . . . 6 (𝜑𝐷𝑃)
109ad2antrr 717 . . . . 5 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → 𝐷𝑃)
11 tgbtwnxfr.f . . . . . 6 (𝜑𝐹𝑃)
1211ad2antrr 717 . . . . 5 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → 𝐹𝑃)
13 simprl 787 . . . . 5 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → 𝑒 ∈ (𝐷𝐼𝐹))
14 eqidd 2826 . . . . 5 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → (𝐷 𝐹) = (𝐷 𝐹))
15 eqidd 2826 . . . . 5 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → (𝑒 𝐹) = (𝑒 𝐹))
16 tgcgrxfr.r . . . . . 6 = (cgrG‘𝐺)
17 tgbtwnxfr.a . . . . . . . . 9 (𝜑𝐴𝑃)
1817ad2antrr 717 . . . . . . . 8 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → 𝐴𝑃)
19 tgbtwnxfr.b . . . . . . . . 9 (𝜑𝐵𝑃)
2019ad2antrr 717 . . . . . . . 8 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → 𝐵𝑃)
21 tgbtwnxfr.c . . . . . . . . 9 (𝜑𝐶𝑃)
2221ad2antrr 717 . . . . . . . 8 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → 𝐶𝑃)
23 simprr 789 . . . . . . . . 9 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)
241, 2, 3, 16, 5, 18, 20, 22, 10, 6, 12, 23trgcgrcom 25840 . . . . . . . 8 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → ⟨“𝐷𝑒𝐹”⟩ ⟨“𝐴𝐵𝐶”⟩)
25 tgbtwnxfr.2 . . . . . . . . 9 (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩)
2625ad2antrr 717 . . . . . . . 8 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩)
271, 2, 3, 16, 5, 10, 6, 12, 18, 20, 22, 24, 10, 8, 12, 26cgr3tr 25841 . . . . . . 7 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → ⟨“𝐷𝑒𝐹”⟩ ⟨“𝐷𝐸𝐹”⟩)
281, 2, 3, 16, 5, 10, 6, 12, 10, 8, 12, 27trgcgrcom 25840 . . . . . 6 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → ⟨“𝐷𝐸𝐹”⟩ ⟨“𝐷𝑒𝐹”⟩)
291, 2, 3, 16, 5, 10, 8, 12, 10, 6, 12, 28cgr3simp1 25832 . . . . 5 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → (𝐷 𝐸) = (𝐷 𝑒))
301, 2, 3, 16, 5, 10, 8, 12, 10, 6, 12, 28cgr3simp2 25833 . . . . . 6 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → (𝐸 𝐹) = (𝑒 𝐹))
311, 2, 3, 5, 8, 12, 6, 12, 30tgcgrcomlr 25792 . . . . 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 25820 . . . 4 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → (𝑒 𝐸) = (𝑒 𝑒))
331, 2, 3, 5, 6, 8, 6, 32axtgcgrid 25775 . . 3 (((𝜑𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩)) → 𝑒 = 𝐸)
3433, 13eqeltrrd 2907 . 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 25834 . . . 4 (𝜑 → (𝐶 𝐴) = (𝐹 𝐷))
371, 2, 3, 4, 21, 17, 11, 9, 36tgcgrcomlr 25792 . . 3 (𝜑 → (𝐴 𝐶) = (𝐷 𝐹))
381, 2, 3, 16, 4, 17, 19, 21, 9, 11, 35, 37tgcgrxfr 25830 . 2 (𝜑 → ∃𝑒𝑃 (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝑒𝐹”⟩))
3934, 38r19.29a 3288 1 (𝜑𝐸 ∈ (𝐷𝐼𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1656  wcel 2164   class class class wbr 4873  cfv 6123  (class class class)co 6905  ⟨“cs3 13963  Basecbs 16222  distcds 16314  TarskiGcstrkg 25742  Itvcitv 25748  cgrGccgrg 25822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-oadd 7830  df-er 8009  df-pm 8125  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-card 9078  df-cda 9305  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-nn 11351  df-2 11414  df-3 11415  df-n0 11619  df-xnn0 11691  df-z 11705  df-uz 11969  df-fz 12620  df-fzo 12761  df-hash 13411  df-word 13575  df-concat 13631  df-s1 13656  df-s2 13969  df-s3 13970  df-trkgc 25760  df-trkgb 25761  df-trkgcb 25762  df-trkg 25765  df-cgrg 25823
This theorem is referenced by:  lnxfr  25878  tgfscgr  25880  legov  25897  legov2  25898  legtrd  25901  mirbtwni  25983  cgrabtwn  26134  cgrahl  26135
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