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Theorem tgtrisegint 25111
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 757 . . . 4 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝐺 ∈ TarskiG)
6 tgtrisegint.e . . . . 5 (𝜑𝐸𝑃)
76ad2antrr 757 . . . 4 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝐸𝑃)
8 tgbtwnintr.3 . . . . 5 (𝜑𝐶𝑃)
98ad2antrr 757 . . . 4 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝐶𝑃)
10 tgbtwnintr.1 . . . . 5 (𝜑𝐴𝑃)
1110ad2antrr 757 . . . 4 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝐴𝑃)
12 simplr 787 . . . 4 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝑟𝑃)
13 tgbtwnintr.2 . . . . 5 (𝜑𝐵𝑃)
1413ad2antrr 757 . . . 4 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝐵𝑃)
15 simprl 789 . . . 4 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝑟 ∈ (𝐸𝐼𝐴))
16 tgtrisegint.1 . . . . . 6 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
1716ad2antrr 757 . . . . 5 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝐵 ∈ (𝐴𝐼𝐶))
181, 2, 3, 5, 11, 14, 9, 17tgbtwncom 25100 . . . 4 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝐵 ∈ (𝐶𝐼𝐴))
191, 2, 3, 5, 7, 9, 11, 12, 14, 15, 18axtgpasch 25083 . . 3 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → ∃𝑞𝑃 (𝑞 ∈ (𝑟𝐼𝐶) ∧ 𝑞 ∈ (𝐵𝐼𝐸)))
205ad2antrr 757 . . . . . . 7 (((((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞𝑃) ∧ 𝑞 ∈ (𝑟𝐼𝐶)) → 𝐺 ∈ TarskiG)
21 tgtrisegint.p . . . . . . . . 9 (𝜑𝐹𝑃)
2221ad2antrr 757 . . . . . . . 8 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝐹𝑃)
2322ad2antrr 757 . . . . . . 7 (((((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞𝑃) ∧ 𝑞 ∈ (𝑟𝐼𝐶)) → 𝐹𝑃)
2412ad2antrr 757 . . . . . . 7 (((((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞𝑃) ∧ 𝑞 ∈ (𝑟𝐼𝐶)) → 𝑟𝑃)
25 simplr 787 . . . . . . 7 (((((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞𝑃) ∧ 𝑞 ∈ (𝑟𝐼𝐶)) → 𝑞𝑃)
269ad2antrr 757 . . . . . . 7 (((((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞𝑃) ∧ 𝑞 ∈ (𝑟𝐼𝐶)) → 𝐶𝑃)
27 simprr 791 . . . . . . . 8 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝑟 ∈ (𝐹𝐼𝐶))
2827ad2antrr 757 . . . . . . 7 (((((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞𝑃) ∧ 𝑞 ∈ (𝑟𝐼𝐶)) → 𝑟 ∈ (𝐹𝐼𝐶))
29 simpr 475 . . . . . . 7 (((((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞𝑃) ∧ 𝑞 ∈ (𝑟𝐼𝐶)) → 𝑞 ∈ (𝑟𝐼𝐶))
301, 2, 3, 20, 23, 24, 25, 26, 28, 29tgbtwnexch2 25108 . . . . . 6 (((((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞𝑃) ∧ 𝑞 ∈ (𝑟𝐼𝐶)) → 𝑞 ∈ (𝐹𝐼𝐶))
3130ex 448 . . . . 5 ((((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞𝑃) → (𝑞 ∈ (𝑟𝐼𝐶) → 𝑞 ∈ (𝐹𝐼𝐶)))
3231anim1d 585 . . . 4 ((((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞𝑃) → ((𝑞 ∈ (𝑟𝐼𝐶) ∧ 𝑞 ∈ (𝐵𝐼𝐸)) → (𝑞 ∈ (𝐹𝐼𝐶) ∧ 𝑞 ∈ (𝐵𝐼𝐸))))
3332reximdva 2999 . . 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 25100 . . 3 (𝜑𝐸 ∈ (𝐶𝐼𝐷))
38 tgtrisegint.3 . . 3 (𝜑𝐹 ∈ (𝐴𝐼𝐷))
391, 2, 3, 4, 8, 10, 35, 6, 21, 37, 38axtgpasch 25083 . 2 (𝜑 → ∃𝑟𝑃 (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶)))
4034, 39r19.29a 3059 1 (𝜑 → ∃𝑞𝑃 (𝑞 ∈ (𝐹𝐼𝐶) ∧ 𝑞 ∈ (𝐵𝐼𝐸)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  wrex 2896  cfv 5790  (class class class)co 6527  Basecbs 15641  distcds 15723  TarskiGcstrkg 25046  Itvcitv 25052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-nul 4712
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-iota 5754  df-fv 5798  df-ov 6530  df-trkgc 25064  df-trkgb 25065  df-trkgcb 25066  df-trkg 25069
This theorem is referenced by:  krippenlem  25303  colperpexlem3  25342
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