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Theorem tgtrisegint 28296
Description: A line segment between two sides of a triange intersects a segment crossing from the remaining side to the opposite vertex. Theorem 3.17 of [Schwabhauser] p. 33. (Contributed by Thierry Arnoux, 23-Mar-2019.)
Hypotheses
Ref Expression
tkgeom.p 𝑃 = (Base‘𝐺)
tkgeom.d = (dist‘𝐺)
tkgeom.i 𝐼 = (Itv‘𝐺)
tkgeom.g (𝜑𝐺 ∈ TarskiG)
tgbtwnintr.1 (𝜑𝐴𝑃)
tgbtwnintr.2 (𝜑𝐵𝑃)
tgbtwnintr.3 (𝜑𝐶𝑃)
tgbtwnintr.4 (𝜑𝐷𝑃)
tgtrisegint.e (𝜑𝐸𝑃)
tgtrisegint.p (𝜑𝐹𝑃)
tgtrisegint.1 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
tgtrisegint.2 (𝜑𝐸 ∈ (𝐷𝐼𝐶))
tgtrisegint.3 (𝜑𝐹 ∈ (𝐴𝐼𝐷))
Assertion
Ref Expression
tgtrisegint (𝜑 → ∃𝑞𝑃 (𝑞 ∈ (𝐹𝐼𝐶) ∧ 𝑞 ∈ (𝐵𝐼𝐸)))
Distinct variable groups:   ,𝑞   𝐴,𝑞   𝐵,𝑞   𝐶,𝑞   𝐷,𝑞   𝐸,𝑞   𝐹,𝑞   𝐼,𝑞   𝑃,𝑞   𝜑,𝑞
Allowed substitution hint:   𝐺(𝑞)

Proof of Theorem tgtrisegint
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 tkgeom.p . . . 4 𝑃 = (Base‘𝐺)
2 tkgeom.d . . . 4 = (dist‘𝐺)
3 tkgeom.i . . . 4 𝐼 = (Itv‘𝐺)
4 tkgeom.g . . . . 5 (𝜑𝐺 ∈ TarskiG)
54ad2antrr 725 . . . 4 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝐺 ∈ TarskiG)
6 tgtrisegint.e . . . . 5 (𝜑𝐸𝑃)
76ad2antrr 725 . . . 4 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝐸𝑃)
8 tgbtwnintr.3 . . . . 5 (𝜑𝐶𝑃)
98ad2antrr 725 . . . 4 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝐶𝑃)
10 tgbtwnintr.1 . . . . 5 (𝜑𝐴𝑃)
1110ad2antrr 725 . . . 4 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝐴𝑃)
12 simplr 768 . . . 4 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝑟𝑃)
13 tgbtwnintr.2 . . . . 5 (𝜑𝐵𝑃)
1413ad2antrr 725 . . . 4 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝐵𝑃)
15 simprl 770 . . . 4 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝑟 ∈ (𝐸𝐼𝐴))
16 tgtrisegint.1 . . . . . 6 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
1716ad2antrr 725 . . . . 5 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝐵 ∈ (𝐴𝐼𝐶))
181, 2, 3, 5, 11, 14, 9, 17tgbtwncom 28285 . . . 4 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝐵 ∈ (𝐶𝐼𝐴))
191, 2, 3, 5, 7, 9, 11, 12, 14, 15, 18axtgpasch 28264 . . 3 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → ∃𝑞𝑃 (𝑞 ∈ (𝑟𝐼𝐶) ∧ 𝑞 ∈ (𝐵𝐼𝐸)))
205ad2antrr 725 . . . . . . 7 (((((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞𝑃) ∧ 𝑞 ∈ (𝑟𝐼𝐶)) → 𝐺 ∈ TarskiG)
21 tgtrisegint.p . . . . . . . . 9 (𝜑𝐹𝑃)
2221ad2antrr 725 . . . . . . . 8 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝐹𝑃)
2322ad2antrr 725 . . . . . . 7 (((((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞𝑃) ∧ 𝑞 ∈ (𝑟𝐼𝐶)) → 𝐹𝑃)
2412ad2antrr 725 . . . . . . 7 (((((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞𝑃) ∧ 𝑞 ∈ (𝑟𝐼𝐶)) → 𝑟𝑃)
25 simplr 768 . . . . . . 7 (((((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞𝑃) ∧ 𝑞 ∈ (𝑟𝐼𝐶)) → 𝑞𝑃)
269ad2antrr 725 . . . . . . 7 (((((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞𝑃) ∧ 𝑞 ∈ (𝑟𝐼𝐶)) → 𝐶𝑃)
27 simprr 772 . . . . . . . 8 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝑟 ∈ (𝐹𝐼𝐶))
2827ad2antrr 725 . . . . . . 7 (((((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞𝑃) ∧ 𝑞 ∈ (𝑟𝐼𝐶)) → 𝑟 ∈ (𝐹𝐼𝐶))
29 simpr 484 . . . . . . 7 (((((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞𝑃) ∧ 𝑞 ∈ (𝑟𝐼𝐶)) → 𝑞 ∈ (𝑟𝐼𝐶))
301, 2, 3, 20, 23, 24, 25, 26, 28, 29tgbtwnexch2 28293 . . . . . 6 (((((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞𝑃) ∧ 𝑞 ∈ (𝑟𝐼𝐶)) → 𝑞 ∈ (𝐹𝐼𝐶))
3130ex 412 . . . . 5 ((((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞𝑃) → (𝑞 ∈ (𝑟𝐼𝐶) → 𝑞 ∈ (𝐹𝐼𝐶)))
3231anim1d 610 . . . 4 ((((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞𝑃) → ((𝑞 ∈ (𝑟𝐼𝐶) ∧ 𝑞 ∈ (𝐵𝐼𝐸)) → (𝑞 ∈ (𝐹𝐼𝐶) ∧ 𝑞 ∈ (𝐵𝐼𝐸))))
3332reximdva 3164 . . 3 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → (∃𝑞𝑃 (𝑞 ∈ (𝑟𝐼𝐶) ∧ 𝑞 ∈ (𝐵𝐼𝐸)) → ∃𝑞𝑃 (𝑞 ∈ (𝐹𝐼𝐶) ∧ 𝑞 ∈ (𝐵𝐼𝐸))))
3419, 33mpd 15 . 2 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → ∃𝑞𝑃 (𝑞 ∈ (𝐹𝐼𝐶) ∧ 𝑞 ∈ (𝐵𝐼𝐸)))
35 tgbtwnintr.4 . . 3 (𝜑𝐷𝑃)
36 tgtrisegint.2 . . . 4 (𝜑𝐸 ∈ (𝐷𝐼𝐶))
371, 2, 3, 4, 35, 6, 8, 36tgbtwncom 28285 . . 3 (𝜑𝐸 ∈ (𝐶𝐼𝐷))
38 tgtrisegint.3 . . 3 (𝜑𝐹 ∈ (𝐴𝐼𝐷))
391, 2, 3, 4, 8, 10, 35, 6, 21, 37, 38axtgpasch 28264 . 2 (𝜑 → ∃𝑟𝑃 (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶)))
4034, 39r19.29a 3158 1 (𝜑 → ∃𝑞𝑃 (𝑞 ∈ (𝐹𝐼𝐶) ∧ 𝑞 ∈ (𝐵𝐼𝐸)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  wrex 3066  cfv 6542  (class class class)co 7414  Basecbs 17173  distcds 17235  TarskiGcstrkg 28224  Itvcitv 28230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-nul 5300
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-sbc 3776  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-iota 6494  df-fv 6550  df-ov 7417  df-trkgc 28245  df-trkgb 28246  df-trkgcb 28247  df-trkg 28250
This theorem is referenced by:  krippenlem  28487  colperpexlem3  28529
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