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Theorem hlpasch 27997

Description: An application of the axiom of Pasch for half-lines. (Contributed by Thierry Arnoux, 15-Sep-2020.)

**Hypotheses**
Ref	Expression
hlpasch.p	⊢ 𝑃 = (Base‘𝐺)
hlpasch.i	⊢ 𝐼 = (Itv‘𝐺)
hlpasch.k	⊢ 𝐾 = (hlG‘𝐺)
hlpasch.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
hlpasch.1	⊢ (𝜑 → 𝐴 ∈ 𝑃)
hlpasch.2	⊢ (𝜑 → 𝐵 ∈ 𝑃)
hlpasch.3	⊢ (𝜑 → 𝐶 ∈ 𝑃)
hlpasch.4	⊢ (𝜑 → 𝑋 ∈ 𝑃)
hlpasch.5	⊢ (𝜑 → 𝐷 ∈ 𝑃)
hlpasch.6	⊢ (𝜑 → 𝐴 ≠ 𝐵)
hlpasch.7	⊢ (𝜑 → 𝐶(𝐾‘𝐵)𝐷)
hlpasch.8	⊢ (𝜑 → 𝐴 ∈ (𝑋𝐼𝐶))

**Assertion**
Ref	Expression
hlpasch	⊢ (𝜑 → ∃𝑒 ∈ 𝑃 (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷)))

Distinct variable groups: 𝐴,𝑒 𝐵,𝑒 𝐶,𝑒 𝐷,𝑒 𝑒,𝐺 𝑒,𝐼 𝑒,𝐾 𝑃,𝑒 𝑒,𝑋 𝜑,𝑒

**Proof of Theorem hlpasch**
Step	Hyp	Ref	Expression
1		hlpasch.p	. . . 4 ⊢ 𝑃 = (Base‘𝐺)
2		hlpasch.i	. . . 4 ⊢ 𝐼 = (Itv‘𝐺)
3		eqid 2733	. . . 4 ⊢ (LineG‘𝐺) = (LineG‘𝐺)
4		hlpasch.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐺 ∈ TarskiG)
6		hlpasch.5	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
7	6	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐷 ∈ 𝑃)
8		hlpasch.4	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
9	8	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝑋 ∈ 𝑃)
10		hlpasch.3	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
11	10	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐶 ∈ 𝑃)
12		hlpasch.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
13	12	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐵 ∈ 𝑃)
14		hlpasch.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
15	14	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐴 ∈ 𝑃)
16		eqid 2733	. . . . 5 ⊢ (dist‘𝐺) = (dist‘𝐺)
17		simpr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐶 ∈ (𝐵𝐼𝐷))
18	1, 16, 2, 5, 13, 11, 7, 17	tgbtwncom 27729	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐶 ∈ (𝐷𝐼𝐵))
19		hlpasch.8	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝑋𝐼𝐶))
20	19	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐴 ∈ (𝑋𝐼𝐶))
21	1, 2, 3, 5, 7, 9, 11, 13, 15, 18, 20	outpasch 27996	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → ∃𝑒 ∈ 𝑃 (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒)))
22		hlpasch.k	. . . . . . 7 ⊢ 𝐾 = (hlG‘𝐺)
23		simplr 768	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝑒 ∈ 𝑃)
24	13	ad2antrr 725	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐵 ∈ 𝑃)
25	15	ad2antrr 725	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐴 ∈ 𝑃)
26	5	ad2antrr 725	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐺 ∈ TarskiG)
27		simprr 772	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐴 ∈ (𝐵𝐼𝑒))
28	1, 16, 2, 26, 24, 25, 23, 27	tgbtwncom 27729	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐴 ∈ (𝑒𝐼𝐵))
29	26	adantr 482	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐺 ∈ TarskiG)
30	24	adantr 482	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐵 ∈ 𝑃)
31	25	adantr 482	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐴 ∈ 𝑃)
32		simplrr 777	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐴 ∈ (𝐵𝐼𝑒))
33		simpr 486	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝑒 = 𝐵)
34	33	oveq2d 7422	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → (𝐵𝐼𝑒) = (𝐵𝐼𝐵))
35	32, 34	eleqtrd 2836	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐴 ∈ (𝐵𝐼𝐵))
36	1, 16, 2, 29, 30, 31, 35	axtgbtwnid 27707	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐵 = 𝐴)
37	36	eqcomd 2739	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐴 = 𝐵)
38		hlpasch.6	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
39	38	ad3antrrr 729	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐴 ≠ 𝐵)
40	39	adantr 482	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐴 ≠ 𝐵)
41	40	neneqd 2946	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → ¬ 𝐴 = 𝐵)
42	37, 41	pm2.65da 816	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → ¬ 𝑒 = 𝐵)
43	42	neqned 2948	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝑒 ≠ 𝐵)
44	1, 2, 22, 23, 24, 25, 26, 25, 28, 43, 39	btwnhl2 27854	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐴(𝐾‘𝐵)𝑒)
45	7	ad2antrr 725	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐷 ∈ 𝑃)
46	9	ad2antrr 725	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝑋 ∈ 𝑃)
47		simprl 770	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝑒 ∈ (𝐷𝐼𝑋))
48	1, 16, 2, 26, 45, 23, 46, 47	tgbtwncom 27729	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝑒 ∈ (𝑋𝐼𝐷))
49	44, 48	jca 513	. . . . 5 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷)))
50	49	ex 414	. . . 4 ⊢ (((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) → ((𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒)) → (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷))))
51	50	reximdva 3169	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → (∃𝑒 ∈ 𝑃 (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒)) → ∃𝑒 ∈ 𝑃 (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷))))
52	21, 51	mpd 15	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → ∃𝑒 ∈ 𝑃 (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷)))
53	6	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝐷 ∈ 𝑃)
54	53	adantr 482	. . . . 5 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐷 ∈ 𝑃)
55		simpr 486	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) ∧ 𝑒 = 𝐷) → 𝑒 = 𝐷)
56	55	breq2d 5160	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) ∧ 𝑒 = 𝐷) → (𝐴(𝐾‘𝐵)𝑒 ↔ 𝐴(𝐾‘𝐵)𝐷))
57	55	eleq1d 2819	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) ∧ 𝑒 = 𝐷) → (𝑒 ∈ (𝑋𝐼𝐷) ↔ 𝐷 ∈ (𝑋𝐼𝐷)))
58	56, 57	anbi12d 632	. . . . 5 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) ∧ 𝑒 = 𝐷) → ((𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷)) ↔ (𝐴(𝐾‘𝐵)𝐷 ∧ 𝐷 ∈ (𝑋𝐼𝐷))))
59	14	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝐴 ∈ 𝑃)
60	59	adantr 482	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐴 ∈ 𝑃)
61	12	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝐵 ∈ 𝑃)
62	61	adantr 482	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐵 ∈ 𝑃)
63	4	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝐺 ∈ TarskiG)
64	63	adantr 482	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐺 ∈ TarskiG)
65		hlpasch.7	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶(𝐾‘𝐵)𝐷)
66	1, 2, 22, 10, 6, 12, 4, 65	hlcomd 27845	. . . . . . . . 9 ⊢ (𝜑 → 𝐷(𝐾‘𝐵)𝐶)
67	66	ad3antrrr 729	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐷(𝐾‘𝐵)𝐶)
68	10	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐶 ∈ 𝑃)
69	68	ad2antrr 725	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐶 ∈ 𝑃)
70	19	adantr 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐴 ∈ (𝑋𝐼𝐶))
71	70	ad2antrr 725	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐴 ∈ (𝑋𝐼𝐶))
72		simpr 486	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝑋 = 𝐵)
73	72	oveq1d 7421	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → (𝑋𝐼𝐶) = (𝐵𝐼𝐶))
74	71, 73	eleqtrd 2836	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐴 ∈ (𝐵𝐼𝐶))
75	1, 2, 22, 10, 6, 12, 4	ishlg 27843	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶(𝐾‘𝐵)𝐷 ↔ (𝐶 ≠ 𝐵 ∧ 𝐷 ≠ 𝐵 ∧ (𝐶 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐶)))))
76	65, 75	mpbid 231	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶 ≠ 𝐵 ∧ 𝐷 ≠ 𝐵 ∧ (𝐶 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐶))))
77	76	simp1d 1143	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
78	77	ad3antrrr 729	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐶 ≠ 𝐵)
79	38	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝐴 ≠ 𝐵)
80	79	adantr 482	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐴 ≠ 𝐵)
81	1, 2, 22, 54, 69, 62, 64, 60, 74, 78, 80	hlbtwn 27852	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → (𝐷(𝐾‘𝐵)𝐶 ↔ 𝐷(𝐾‘𝐵)𝐴))
82	67, 81	mpbid 231	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐷(𝐾‘𝐵)𝐴)
83	1, 2, 22, 54, 60, 62, 64, 82	hlcomd 27845	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐴(𝐾‘𝐵)𝐷)
84	8	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝑋 ∈ 𝑃)
85	84	adantr 482	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝑋 ∈ 𝑃)
86	1, 16, 2, 64, 85, 54	tgbtwntriv2 27728	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐷 ∈ (𝑋𝐼𝐷))
87	83, 86	jca 513	. . . . 5 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → (𝐴(𝐾‘𝐵)𝐷 ∧ 𝐷 ∈ (𝑋𝐼𝐷)))
88	54, 58, 87	rspcedvd 3615	. . . 4 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → ∃𝑒 ∈ 𝑃 (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷)))
89	84	ad2antrr 725	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐴(𝐾‘𝐵)𝑋) → 𝑋 ∈ 𝑃)
90		simpr 486	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 = 𝑋) → 𝑒 = 𝑋)
91	90	breq2d 5160	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 = 𝑋) → (𝐴(𝐾‘𝐵)𝑒 ↔ 𝐴(𝐾‘𝐵)𝑋))
92	90	eleq1d 2819	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 = 𝑋) → (𝑒 ∈ (𝑋𝐼𝐷) ↔ 𝑋 ∈ (𝑋𝐼𝐷)))
93	91, 92	anbi12d 632	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 = 𝑋) → ((𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷)) ↔ (𝐴(𝐾‘𝐵)𝑋 ∧ 𝑋 ∈ (𝑋𝐼𝐷))))
94	93	ad4ant14 751	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐴(𝐾‘𝐵)𝑋) ∧ 𝑒 = 𝑋) → ((𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷)) ↔ (𝐴(𝐾‘𝐵)𝑋 ∧ 𝑋 ∈ (𝑋𝐼𝐷))))
95		simpr 486	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐴(𝐾‘𝐵)𝑋) → 𝐴(𝐾‘𝐵)𝑋)
96	1, 16, 2, 63, 84, 53	tgbtwntriv1 27732	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝑋 ∈ (𝑋𝐼𝐷))
97	96	ad2antrr 725	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐴(𝐾‘𝐵)𝑋) → 𝑋 ∈ (𝑋𝐼𝐷))
98	95, 97	jca 513	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐴(𝐾‘𝐵)𝑋) → (𝐴(𝐾‘𝐵)𝑋 ∧ 𝑋 ∈ (𝑋𝐼𝐷)))
99	89, 94, 98	rspcedvd 3615	. . . . 5 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐴(𝐾‘𝐵)𝑋) → ∃𝑒 ∈ 𝑃 (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷)))
100	53	ad2antrr 725	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐷 ∈ 𝑃)
101		simpr 486	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) ∧ 𝑒 = 𝐷) → 𝑒 = 𝐷)
102	101	breq2d 5160	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) ∧ 𝑒 = 𝐷) → (𝐴(𝐾‘𝐵)𝑒 ↔ 𝐴(𝐾‘𝐵)𝐷))
103	101	eleq1d 2819	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) ∧ 𝑒 = 𝐷) → (𝑒 ∈ (𝑋𝐼𝐷) ↔ 𝐷 ∈ (𝑋𝐼𝐷)))
104	102, 103	anbi12d 632	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) ∧ 𝑒 = 𝐷) → ((𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷)) ↔ (𝐴(𝐾‘𝐵)𝐷 ∧ 𝐷 ∈ (𝑋𝐼𝐷))))
105	79	ad2antrr 725	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐴 ≠ 𝐵)
106	1, 2, 22, 10, 6, 12, 4, 65	hlne2 27847	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ≠ 𝐵)
107	106	ad4antr 731	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐷 ≠ 𝐵)
108	63	ad2antrr 725	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐺 ∈ TarskiG)
109	61	ad2antrr 725	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐵 ∈ 𝑃)
110	59	ad2antrr 725	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐴 ∈ 𝑃)
111	68	ad2antrr 725	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝐶 ∈ 𝑃)
112	111	adantr 482	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐶 ∈ 𝑃)
113	84	ad2antrr 725	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝑋 ∈ 𝑃)
114		simpr 486	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐵 ∈ (𝑋𝐼𝐴))
115	70	ad2antrr 725	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝐴 ∈ (𝑋𝐼𝐶))
116	115	adantr 482	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐴 ∈ (𝑋𝐼𝐶))
117	1, 16, 2, 108, 113, 109, 110, 112, 114, 116	tgbtwnexch3 27735	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐴 ∈ (𝐵𝐼𝐶))
118		simp-4r 783	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐷 ∈ (𝐵𝐼𝐶))
119	1, 2, 108, 109, 110, 100, 112, 117, 118	tgbtwnconn3 27818	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → (𝐴 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐴)))
120	1, 2, 22, 14, 6, 12, 4	ishlg 27843	. . . . . . . . 9 ⊢ (𝜑 → (𝐴(𝐾‘𝐵)𝐷 ↔ (𝐴 ≠ 𝐵 ∧ 𝐷 ≠ 𝐵 ∧ (𝐴 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐴)))))
121	120	ad4antr 731	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → (𝐴(𝐾‘𝐵)𝐷 ↔ (𝐴 ≠ 𝐵 ∧ 𝐷 ≠ 𝐵 ∧ (𝐴 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐴)))))
122	105, 107, 119, 121	mpbir3and 1343	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐴(𝐾‘𝐵)𝐷)
123	1, 16, 2, 108, 113, 100	tgbtwntriv2 27728	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐷 ∈ (𝑋𝐼𝐷))
124	122, 123	jca 513	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → (𝐴(𝐾‘𝐵)𝐷 ∧ 𝐷 ∈ (𝑋𝐼𝐷)))
125	100, 104, 124	rspcedvd 3615	. . . . 5 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → ∃𝑒 ∈ 𝑃 (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷)))
126	8	ad3antrrr 729	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝑋 ∈ 𝑃)
127	12	ad3antrrr 729	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝐵 ∈ 𝑃)
128	14	ad3antrrr 729	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝐴 ∈ 𝑃)
129	4	ad3antrrr 729	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝐺 ∈ TarskiG)
130		simpr 486	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝑋 ≠ 𝐵)
131	130	neneqd 2946	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → ¬ 𝑋 = 𝐵)
132	63	adantr 482	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝐺 ∈ TarskiG)
133	132	adantr 482	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 = 𝐶) → 𝐺 ∈ TarskiG)
134	126	adantr 482	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 = 𝐶) → 𝑋 ∈ 𝑃)
135	128	adantr 482	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 = 𝐶) → 𝐴 ∈ 𝑃)
136	115	adantr 482	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 = 𝐶) → 𝐴 ∈ (𝑋𝐼𝐶))
137		simpr 486	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 = 𝐶) → 𝑋 = 𝐶)
138	137	oveq2d 7422	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 = 𝐶) → (𝑋𝐼𝑋) = (𝑋𝐼𝐶))
139	136, 138	eleqtrrd 2837	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 = 𝐶) → 𝐴 ∈ (𝑋𝐼𝑋))
140	1, 16, 2, 133, 134, 135, 139	axtgbtwnid 27707	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 = 𝐶) → 𝑋 = 𝐴)
141	140	olcd 873	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 = 𝐶) → (𝐵 ∈ (𝑋(LineG‘𝐺)𝐴) ∨ 𝑋 = 𝐴))
142	132	adantr 482	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 ≠ 𝐶) → 𝐺 ∈ TarskiG)
143	127	adantr 482	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 ≠ 𝐶) → 𝐵 ∈ 𝑃)
144	111	adantr 482	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 ≠ 𝐶) → 𝐶 ∈ 𝑃)
145	126	adantr 482	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 ≠ 𝐶) → 𝑋 ∈ 𝑃)
146	128	adantr 482	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 ≠ 𝐶) → 𝐴 ∈ 𝑃)
147		simpr 486	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 ≠ 𝐶) → 𝑋 ≠ 𝐶)
148	147	necomd 2997	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 ≠ 𝐶) → 𝐶 ≠ 𝑋)
149	148	neneqd 2946	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 ≠ 𝐶) → ¬ 𝐶 = 𝑋)
150	53	adantr 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝐷 ∈ 𝑃)
151	106	ad3antrrr 729	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝐷 ≠ 𝐵)
152		simplr 768	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷))
153	1, 2, 3, 132, 150, 127, 126, 151, 152	lncom 27863	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝑋 ∈ (𝐷(LineG‘𝐺)𝐵))
154	77	necomd 2997	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
155	154	ad3antrrr 729	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝐵 ≠ 𝐶)
156	66	ad3antrrr 729	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝐷(𝐾‘𝐵)𝐶)
157	1, 2, 22, 150, 111, 127, 132, 3, 156	hlln 27848	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝐷 ∈ (𝐶(LineG‘𝐺)𝐵))
158	1, 2, 3, 132, 127, 111, 150, 155, 157	lncom 27863	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝐷 ∈ (𝐵(LineG‘𝐺)𝐶))
159	158	orcd 872	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → (𝐷 ∈ (𝐵(LineG‘𝐺)𝐶) ∨ 𝐵 = 𝐶))
160	1, 2, 3, 132, 126, 150, 127, 111, 153, 159	coltr 27888	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → (𝑋 ∈ (𝐵(LineG‘𝐺)𝐶) ∨ 𝐵 = 𝐶))
161	1, 3, 2, 132, 127, 111, 126, 160	colrot1 27800	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → (𝐵 ∈ (𝐶(LineG‘𝐺)𝑋) ∨ 𝐶 = 𝑋))
162	161	orcomd 870	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → (𝐶 = 𝑋 ∨ 𝐵 ∈ (𝐶(LineG‘𝐺)𝑋)))
163	162	adantr 482	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 ≠ 𝐶) → (𝐶 = 𝑋 ∨ 𝐵 ∈ (𝐶(LineG‘𝐺)𝑋)))
164	163	ord 863	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 ≠ 𝐶) → (¬ 𝐶 = 𝑋 → 𝐵 ∈ (𝐶(LineG‘𝐺)𝑋)))
165	149, 164	mpd 15	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 ≠ 𝐶) → 𝐵 ∈ (𝐶(LineG‘𝐺)𝑋))
166	1, 3, 2, 132, 126, 128, 111, 115	btwncolg3 27798	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → (𝐶 ∈ (𝑋(LineG‘𝐺)𝐴) ∨ 𝑋 = 𝐴))
167	166	adantr 482	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 ≠ 𝐶) → (𝐶 ∈ (𝑋(LineG‘𝐺)𝐴) ∨ 𝑋 = 𝐴))
168	1, 2, 3, 142, 143, 144, 145, 146, 165, 167	coltr 27888	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 ≠ 𝐶) → (𝐵 ∈ (𝑋(LineG‘𝐺)𝐴) ∨ 𝑋 = 𝐴))
169	141, 168	pm2.61dane 3030	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → (𝐵 ∈ (𝑋(LineG‘𝐺)𝐴) ∨ 𝑋 = 𝐴))
170	1, 3, 2, 132, 126, 128, 127, 169	colrot2 27801	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → (𝐴 ∈ (𝐵(LineG‘𝐺)𝑋) ∨ 𝐵 = 𝑋))
171	1, 3, 2, 132, 127, 126, 128, 170	colcom 27799	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → (𝐴 ∈ (𝑋(LineG‘𝐺)𝐵) ∨ 𝑋 = 𝐵))
172	171	orcomd 870	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → (𝑋 = 𝐵 ∨ 𝐴 ∈ (𝑋(LineG‘𝐺)𝐵)))
173	172	ord 863	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → (¬ 𝑋 = 𝐵 → 𝐴 ∈ (𝑋(LineG‘𝐺)𝐵)))
174	131, 173	mpd 15	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝐴 ∈ (𝑋(LineG‘𝐺)𝐵))
175	1, 2, 22, 126, 127, 128, 129, 128, 3, 174	lnhl 27856	. . . . 5 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → (𝐴(𝐾‘𝐵)𝑋 ∨ 𝐵 ∈ (𝑋𝐼𝐴)))
176	99, 125, 175	mpjaodan 958	. . . 4 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → ∃𝑒 ∈ 𝑃 (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷)))
177	88, 176	pm2.61dane 3030	. . 3 ⊢ (((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → ∃𝑒 ∈ 𝑃 (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷)))
178	4	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐺 ∈ TarskiG)
179	8	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝑋 ∈ 𝑃)
180	12	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐵 ∈ 𝑃)
181	14	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐴 ∈ 𝑃)
182	6	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐷 ∈ 𝑃)
183		simpr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐷 ∈ (𝐵𝐼𝐶))
184	1, 16, 2, 178, 179, 180, 68, 181, 182, 70, 183	axtgpasch 27708	. . . . 5 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → ∃𝑒 ∈ 𝑃 (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋)))
185	184	adantr 482	. . . 4 ⊢ (((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → ∃𝑒 ∈ 𝑃 (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋)))
186		simplr 768	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝑒 ∈ 𝑃)
187	181	ad3antrrr 729	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝐴 ∈ 𝑃)
188	180	ad3antrrr 729	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝐵 ∈ 𝑃)
189	178	ad3antrrr 729	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝐺 ∈ TarskiG)
190		simprl 770	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝑒 ∈ (𝐴𝐼𝐵))
191	1, 16, 2, 189, 187, 186, 188, 190	tgbtwncom 27729	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝑒 ∈ (𝐵𝐼𝐴))
192	38	necomd 2997	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ≠ 𝐴)
193	192	ad4antr 731	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝐵 ≠ 𝐴)
194	189	adantr 482	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐺 ∈ TarskiG)
195	6	ad5antr 733	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐷 ∈ 𝑃)
196	8	ad5antr 733	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝑋 ∈ 𝑃)
197	188	adantr 482	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐵 ∈ 𝑃)
198		simp-4r 783	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷))
199	106	necomd 2997	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 ≠ 𝐷)
200	199	ad5antr 733	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐵 ≠ 𝐷)
201	200	neneqd 2946	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → ¬ 𝐵 = 𝐷)
202		ioran 983	. . . . . . . . . . . . . 14 ⊢ (¬ (𝑋 ∈ (𝐵(LineG‘𝐺)𝐷) ∨ 𝐵 = 𝐷) ↔ (¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷) ∧ ¬ 𝐵 = 𝐷))
203	198, 201, 202	sylanbrc 584	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → ¬ (𝑋 ∈ (𝐵(LineG‘𝐺)𝐷) ∨ 𝐵 = 𝐷))
204	1, 3, 2, 194, 197, 195, 196, 203	ncolrot2 27804	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → ¬ (𝐷 ∈ (𝑋(LineG‘𝐺)𝐵) ∨ 𝑋 = 𝐵))
205		simpr 486	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝑒 = 𝐵)
206	186	adantr 482	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝑒 ∈ 𝑃)
207	1, 2, 3, 194, 195, 196, 197, 204	ncolne1 27866	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐷 ≠ 𝑋)
208		simplrr 777	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝑒 ∈ (𝐷𝐼𝑋))
209	1, 2, 3, 194, 195, 196, 206, 207, 208	btwnlng1 27860	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝑒 ∈ (𝐷(LineG‘𝐺)𝑋))
210	205, 209	eqeltrrd 2835	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐵 ∈ (𝐷(LineG‘𝐺)𝑋))
211	1, 2, 3, 194, 195, 196, 207	tglinerflx1 27874	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐷 ∈ (𝐷(LineG‘𝐺)𝑋))
212	106	ad5antr 733	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐷 ≠ 𝐵)
213	212	necomd 2997	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐵 ≠ 𝐷)
214	1, 2, 3, 194, 197, 195, 213	tglinerflx1 27874	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐵 ∈ (𝐵(LineG‘𝐺)𝐷))
215	1, 2, 3, 194, 197, 195, 213	tglinerflx2 27875	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐷 ∈ (𝐵(LineG‘𝐺)𝐷))
216	1, 2, 3, 194, 195, 196, 197, 195, 204, 210, 211, 214, 215	tglineinteq 27886	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐵 = 𝐷)
217	216, 201	pm2.65da 816	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → ¬ 𝑒 = 𝐵)
218	217	neqned 2948	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝑒 ≠ 𝐵)
219	1, 2, 22, 188, 187, 186, 189, 187, 191, 193, 218	btwnhl1 27853	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝑒(𝐾‘𝐵)𝐴)
220	1, 2, 22, 186, 187, 188, 189, 219	hlcomd 27845	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝐴(𝐾‘𝐵)𝑒)
221	178	ad3antrrr 729	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → 𝐺 ∈ TarskiG)
222	182	ad3antrrr 729	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → 𝐷 ∈ 𝑃)
223		simplr 768	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → 𝑒 ∈ 𝑃)
224	179	ad3antrrr 729	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → 𝑋 ∈ 𝑃)
225		simpr 486	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → 𝑒 ∈ (𝐷𝐼𝑋))
226	1, 16, 2, 221, 222, 223, 224, 225	tgbtwncom 27729	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → 𝑒 ∈ (𝑋𝐼𝐷))
227	226	adantrl 715	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝑒 ∈ (𝑋𝐼𝐷))
228	220, 227	jca 513	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷)))
229	228	ex 414	. . . . 5 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) → ((𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷))))
230	229	reximdva 3169	. . . 4 ⊢ (((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → (∃𝑒 ∈ 𝑃 (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → ∃𝑒 ∈ 𝑃 (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷))))
231	185, 230	mpd 15	. . 3 ⊢ (((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → ∃𝑒 ∈ 𝑃 (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷)))
232	177, 231	pm2.61dan 812	. 2 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → ∃𝑒 ∈ 𝑃 (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷)))
233	76	simp3d 1145	. 2 ⊢ (𝜑 → (𝐶 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐶)))
234	52, 232, 233	mpjaodan 958	1 ⊢ (𝜑 → ∃𝑒 ∈ 𝑃 (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∃wrex 3071 class class class wbr 5148 ‘cfv 6541 (class class class)co 7406 Basecbs 17141 distcds 17203 TarskiGcstrkg 27668 Itvcitv 27674 LineGclng 27675 hlGchlg 27841

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-oadd 8467 df-er 8700 df-map 8819 df-pm 8820 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-dju 9893 df-card 9931 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-n0 12470 df-xnn0 12542 df-z 12556 df-uz 12820 df-fz 13482 df-fzo 13625 df-hash 14288 df-word 14462 df-concat 14518 df-s1 14543 df-s2 14796 df-s3 14797 df-trkgc 27689 df-trkgb 27690 df-trkgcb 27691 df-trkgld 27693 df-trkg 27694 df-cgrg 27752 df-leg 27824 df-hlg 27842 df-mir 27894 df-rag 27935 df-perpg 27937

This theorem is referenced by: inaghl 28086

W3C validator