Theorem tgtrisegint 26287

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	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tgbtwnintr.1	⊢ (𝜑 → 𝐴 ∈ 𝑃)
tgbtwnintr.2	⊢ (𝜑 → 𝐵 ∈ 𝑃)
tgbtwnintr.3	⊢ (𝜑 → 𝐶 ∈ 𝑃)
tgbtwnintr.4	⊢ (𝜑 → 𝐷 ∈ 𝑃)
tgtrisegint.e	⊢ (𝜑 → 𝐸 ∈ 𝑃)
tgtrisegint.p	⊢ (𝜑 → 𝐹 ∈ 𝑃)
tgtrisegint.1	⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
tgtrisegint.2	⊢ (𝜑 → 𝐸 ∈ (𝐷𝐼𝐶))
tgtrisegint.3	⊢ (𝜑 → 𝐹 ∈ (𝐴𝐼𝐷))

Assertion

Ref	Expression
tgtrisegint	⊢ (𝜑 → ∃𝑞 ∈ 𝑃 (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.

Step	Hyp	Ref	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 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝐺 ∈ TarskiG)
6		tgtrisegint.e	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
7	6	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝐸 ∈ 𝑃)
8		tgbtwnintr.3	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
9	8	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝐶 ∈ 𝑃)
10		tgbtwnintr.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
11	10	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝐴 ∈ 𝑃)
12		simplr 767	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝑟 ∈ 𝑃)
13		tgbtwnintr.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
14	13	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝐵 ∈ 𝑃)
15		simprl 769	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝑟 ∈ (𝐸𝐼𝐴))
16		tgtrisegint.1	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
17	16	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝐵 ∈ (𝐴𝐼𝐶))
18	1, 2, 3, 5, 11, 14, 9, 17	tgbtwncom 26276	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝐵 ∈ (𝐶𝐼𝐴))
19	1, 2, 3, 5, 7, 9, 11, 12, 14, 15, 18	axtgpasch 26255	. . 3 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → ∃𝑞 ∈ 𝑃 (𝑞 ∈ (𝑟𝐼𝐶) ∧ 𝑞 ∈ (𝐵𝐼𝐸)))
20	5	ad2antrr 724	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞 ∈ 𝑃) ∧ 𝑞 ∈ (𝑟𝐼𝐶)) → 𝐺 ∈ TarskiG)
21		tgtrisegint.p	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
22	21	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝐹 ∈ 𝑃)
23	22	ad2antrr 724	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞 ∈ 𝑃) ∧ 𝑞 ∈ (𝑟𝐼𝐶)) → 𝐹 ∈ 𝑃)
24	12	ad2antrr 724	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞 ∈ 𝑃) ∧ 𝑞 ∈ (𝑟𝐼𝐶)) → 𝑟 ∈ 𝑃)
25		simplr 767	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞 ∈ 𝑃) ∧ 𝑞 ∈ (𝑟𝐼𝐶)) → 𝑞 ∈ 𝑃)
26	9	ad2antrr 724	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞 ∈ 𝑃) ∧ 𝑞 ∈ (𝑟𝐼𝐶)) → 𝐶 ∈ 𝑃)
27		simprr 771	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝑟 ∈ (𝐹𝐼𝐶))
28	27	ad2antrr 724	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞 ∈ 𝑃) ∧ 𝑞 ∈ (𝑟𝐼𝐶)) → 𝑟 ∈ (𝐹𝐼𝐶))
29		simpr 487	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞 ∈ 𝑃) ∧ 𝑞 ∈ (𝑟𝐼𝐶)) → 𝑞 ∈ (𝑟𝐼𝐶))
30	1, 2, 3, 20, 23, 24, 25, 26, 28, 29	tgbtwnexch2 26284	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞 ∈ 𝑃) ∧ 𝑞 ∈ (𝑟𝐼𝐶)) → 𝑞 ∈ (𝐹𝐼𝐶))
31	30	ex 415	. . . . 5 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞 ∈ 𝑃) → (𝑞 ∈ (𝑟𝐼𝐶) → 𝑞 ∈ (𝐹𝐼𝐶)))
32	31	anim1d 612	. . . 4 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞 ∈ 𝑃) → ((𝑞 ∈ (𝑟𝐼𝐶) ∧ 𝑞 ∈ (𝐵𝐼𝐸)) → (𝑞 ∈ (𝐹𝐼𝐶) ∧ 𝑞 ∈ (𝐵𝐼𝐸))))
33	32	reximdva 3276	. . 3 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → (∃𝑞 ∈ 𝑃 (𝑞 ∈ (𝑟𝐼𝐶) ∧ 𝑞 ∈ (𝐵𝐼𝐸)) → ∃𝑞 ∈ 𝑃 (𝑞 ∈ (𝐹𝐼𝐶) ∧ 𝑞 ∈ (𝐵𝐼𝐸))))
34	19, 33	mpd 15	. 2 ⊢ (((𝜑 ∧ 𝑟 ∈ 𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → ∃𝑞 ∈ 𝑃 (𝑞 ∈ (𝐹𝐼𝐶) ∧ 𝑞 ∈ (𝐵𝐼𝐸)))
35		tgbtwnintr.4	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
36		tgtrisegint.2	. . . 4 ⊢ (𝜑 → 𝐸 ∈ (𝐷𝐼𝐶))
37	1, 2, 3, 4, 35, 6, 8, 36	tgbtwncom 26276	. . 3 ⊢ (𝜑 → 𝐸 ∈ (𝐶𝐼𝐷))
38		tgtrisegint.3	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐴𝐼𝐷))
39	1, 2, 3, 4, 8, 10, 35, 6, 21, 37, 38	axtgpasch 26255	. 2 ⊢ (𝜑 → ∃𝑟 ∈ 𝑃 (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶)))
40	34, 39	r19.29a 3291	1 ⊢ (𝜑 → ∃𝑞 ∈ 𝑃 (𝑞 ∈ (𝐹𝐼𝐶) ∧ 𝑞 ∈ (𝐵𝐼𝐸)))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∃wrex 3141  ‘cfv 6357  (class class class)co 7158  Basecbs 16485  distcds 16576  TarskiGcstrkg 26218  Itvcitv 26224

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-nul 5212

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-iota 6316  df-fv 6365  df-ov 7161  df-trkgc 26236  df-trkgb 26237  df-trkgcb 26238  df-trkg 26241

This theorem is referenced by:  krippenlem  26478  colperpexlem3  26520