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Theorem hlpasch 28719

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 2729	. . . 4 ⊢ (LineG‘𝐺) = (LineG‘𝐺)
4		hlpasch.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐺 ∈ TarskiG)
6		hlpasch.5	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐷 ∈ 𝑃)
8		hlpasch.4	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
9	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝑋 ∈ 𝑃)
10		hlpasch.3	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
11	10	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐶 ∈ 𝑃)
12		hlpasch.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
13	12	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐵 ∈ 𝑃)
14		hlpasch.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
15	14	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐴 ∈ 𝑃)
16		eqid 2729	. . . . 5 ⊢ (dist‘𝐺) = (dist‘𝐺)
17		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐶 ∈ (𝐵𝐼𝐷))
18	1, 16, 2, 5, 13, 11, 7, 17	tgbtwncom 28451	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐶 ∈ (𝐷𝐼𝐵))
19		hlpasch.8	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝑋𝐼𝐶))
20	19	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐴 ∈ (𝑋𝐼𝐶))
21	1, 2, 3, 5, 7, 9, 11, 13, 15, 18, 20	outpasch 28718	. . 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 726	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐵 ∈ 𝑃)
25	15	ad2antrr 726	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐴 ∈ 𝑃)
26	5	ad2antrr 726	. . . . . . 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 28451	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐴 ∈ (𝑒𝐼𝐵))
29	26	adantr 480	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐺 ∈ TarskiG)
30	24	adantr 480	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐵 ∈ 𝑃)
31	25	adantr 480	. . . . . . . . . . 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 484	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝑒 = 𝐵)
34	33	oveq2d 7369	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → (𝐵𝐼𝑒) = (𝐵𝐼𝐵))
35	32, 34	eleqtrd 2830	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐴 ∈ (𝐵𝐼𝐵))
36	1, 16, 2, 29, 30, 31, 35	axtgbtwnid 28429	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐵 = 𝐴)
37	36	eqcomd 2735	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐴 = 𝐵)
38		hlpasch.6	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
39	38	ad3antrrr 730	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐴 ≠ 𝐵)
40	39	adantr 480	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐴 ≠ 𝐵)
41	40	neneqd 2930	. . . . . . . . 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 2932	. . . . . . 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 28576	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐴(𝐾‘𝐵)𝑒)
45	7	ad2antrr 726	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐷 ∈ 𝑃)
46	9	ad2antrr 726	. . . . . . 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 28451	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝑒 ∈ (𝑋𝐼𝐷))
49	44, 48	jca 511	. . . . 5 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷)))
50	49	ex 412	. . . 4 ⊢ (((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) → ((𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒)) → (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷))))
51	50	reximdva 3142	. . 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 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝐷 ∈ 𝑃)
54	53	adantr 480	. . . . 5 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐷 ∈ 𝑃)
55		simpr 484	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) ∧ 𝑒 = 𝐷) → 𝑒 = 𝐷)
56	55	breq2d 5107	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) ∧ 𝑒 = 𝐷) → (𝐴(𝐾‘𝐵)𝑒 ↔ 𝐴(𝐾‘𝐵)𝐷))
57	55	eleq1d 2813	. . . . . 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 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝐴 ∈ 𝑃)
60	59	adantr 480	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐴 ∈ 𝑃)
61	12	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝐵 ∈ 𝑃)
62	61	adantr 480	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐵 ∈ 𝑃)
63	4	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝐺 ∈ TarskiG)
64	63	adantr 480	. . . . . . 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 28567	. . . . . . . . 9 ⊢ (𝜑 → 𝐷(𝐾‘𝐵)𝐶)
67	66	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐷(𝐾‘𝐵)𝐶)
68	10	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐶 ∈ 𝑃)
69	68	ad2antrr 726	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐶 ∈ 𝑃)
70	19	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐴 ∈ (𝑋𝐼𝐶))
71	70	ad2antrr 726	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐴 ∈ (𝑋𝐼𝐶))
72		simpr 484	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝑋 = 𝐵)
73	72	oveq1d 7368	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → (𝑋𝐼𝐶) = (𝐵𝐼𝐶))
74	71, 73	eleqtrd 2830	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐴 ∈ (𝐵𝐼𝐶))
75	1, 2, 22, 10, 6, 12, 4	ishlg 28565	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶(𝐾‘𝐵)𝐷 ↔ (𝐶 ≠ 𝐵 ∧ 𝐷 ≠ 𝐵 ∧ (𝐶 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐶)))))
76	65, 75	mpbid 232	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶 ≠ 𝐵 ∧ 𝐷 ≠ 𝐵 ∧ (𝐶 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐶))))
77	76	simp1d 1142	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
78	77	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐶 ≠ 𝐵)
79	38	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝐴 ≠ 𝐵)
80	79	adantr 480	. . . . . . . . 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 28574	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → (𝐷(𝐾‘𝐵)𝐶 ↔ 𝐷(𝐾‘𝐵)𝐴))
82	67, 81	mpbid 232	. . . . . . 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 28567	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐴(𝐾‘𝐵)𝐷)
84	8	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝑋 ∈ 𝑃)
85	84	adantr 480	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝑋 ∈ 𝑃)
86	1, 16, 2, 64, 85, 54	tgbtwntriv2 28450	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐷 ∈ (𝑋𝐼𝐷))
87	83, 86	jca 511	. . . . 5 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → (𝐴(𝐾‘𝐵)𝐷 ∧ 𝐷 ∈ (𝑋𝐼𝐷)))
88	54, 58, 87	rspcedvd 3581	. . . 4 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → ∃𝑒 ∈ 𝑃 (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷)))
89	84	ad2antrr 726	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐴(𝐾‘𝐵)𝑋) → 𝑋 ∈ 𝑃)
90		simpr 484	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 = 𝑋) → 𝑒 = 𝑋)
91	90	breq2d 5107	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 = 𝑋) → (𝐴(𝐾‘𝐵)𝑒 ↔ 𝐴(𝐾‘𝐵)𝑋))
92	90	eleq1d 2813	. . . . . . . 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 752	. . . . . 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 484	. . . . . . 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 28454	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝑋 ∈ (𝑋𝐼𝐷))
97	96	ad2antrr 726	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐴(𝐾‘𝐵)𝑋) → 𝑋 ∈ (𝑋𝐼𝐷))
98	95, 97	jca 511	. . . . . 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 3581	. . . . 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 726	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐷 ∈ 𝑃)
101		simpr 484	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) ∧ 𝑒 = 𝐷) → 𝑒 = 𝐷)
102	101	breq2d 5107	. . . . . . 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 2813	. . . . . . 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 726	. . . . . . . 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 28569	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ≠ 𝐵)
107	106	ad4antr 732	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐷 ≠ 𝐵)
108	63	ad2antrr 726	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐺 ∈ TarskiG)
109	61	ad2antrr 726	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐵 ∈ 𝑃)
110	59	ad2antrr 726	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐴 ∈ 𝑃)
111	68	ad2antrr 726	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝐶 ∈ 𝑃)
112	111	adantr 480	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐶 ∈ 𝑃)
113	84	ad2antrr 726	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝑋 ∈ 𝑃)
114		simpr 484	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐵 ∈ (𝑋𝐼𝐴))
115	70	ad2antrr 726	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝐴 ∈ (𝑋𝐼𝐶))
116	115	adantr 480	. . . . . . . . . 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 28457	. . . . . . . . 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 28540	. . . . . . . 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 28565	. . . . . . . . 9 ⊢ (𝜑 → (𝐴(𝐾‘𝐵)𝐷 ↔ (𝐴 ≠ 𝐵 ∧ 𝐷 ≠ 𝐵 ∧ (𝐴 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐴)))))
121	120	ad4antr 732	. . . . . . . 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 28450	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐷 ∈ (𝑋𝐼𝐷))
124	122, 123	jca 511	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → (𝐴(𝐾‘𝐵)𝐷 ∧ 𝐷 ∈ (𝑋𝐼𝐷)))
125	100, 104, 124	rspcedvd 3581	. . . . 5 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → ∃𝑒 ∈ 𝑃 (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷)))
126	8	ad3antrrr 730	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝑋 ∈ 𝑃)
127	12	ad3antrrr 730	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝐵 ∈ 𝑃)
128	14	ad3antrrr 730	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝐴 ∈ 𝑃)
129	4	ad3antrrr 730	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝐺 ∈ TarskiG)
130		simpr 484	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝑋 ≠ 𝐵)
131	130	neneqd 2930	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → ¬ 𝑋 = 𝐵)
132	63	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝐺 ∈ TarskiG)
133	132	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 = 𝐶) → 𝐺 ∈ TarskiG)
134	126	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 = 𝐶) → 𝑋 ∈ 𝑃)
135	128	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 = 𝐶) → 𝐴 ∈ 𝑃)
136	115	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 = 𝐶) → 𝐴 ∈ (𝑋𝐼𝐶))
137		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 = 𝐶) → 𝑋 = 𝐶)
138	137	oveq2d 7369	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 = 𝐶) → (𝑋𝐼𝑋) = (𝑋𝐼𝐶))
139	136, 138	eleqtrrd 2831	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 = 𝐶) → 𝐴 ∈ (𝑋𝐼𝑋))
140	1, 16, 2, 133, 134, 135, 139	axtgbtwnid 28429	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 = 𝐶) → 𝑋 = 𝐴)
141	140	olcd 874	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 = 𝐶) → (𝐵 ∈ (𝑋(LineG‘𝐺)𝐴) ∨ 𝑋 = 𝐴))
142	132	adantr 480	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 ≠ 𝐶) → 𝐺 ∈ TarskiG)
143	127	adantr 480	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 ≠ 𝐶) → 𝐵 ∈ 𝑃)
144	111	adantr 480	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 ≠ 𝐶) → 𝐶 ∈ 𝑃)
145	126	adantr 480	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 ≠ 𝐶) → 𝑋 ∈ 𝑃)
146	128	adantr 480	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 ≠ 𝐶) → 𝐴 ∈ 𝑃)
147		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 ≠ 𝐶) → 𝑋 ≠ 𝐶)
148	147	necomd 2980	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 ≠ 𝐶) → 𝐶 ≠ 𝑋)
149	148	neneqd 2930	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 ≠ 𝐶) → ¬ 𝐶 = 𝑋)
150	53	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝐷 ∈ 𝑃)
151	106	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . 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 28585	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝑋 ∈ (𝐷(LineG‘𝐺)𝐵))
154	77	necomd 2980	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
155	154	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝐵 ≠ 𝐶)
156	66	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . 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 28570	. . . . . . . . . . . . . . . . . . . . 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 28585	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝐷 ∈ (𝐵(LineG‘𝐺)𝐶))
159	158	orcd 873	. . . . . . . . . . . . . . . . . . 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 28610	. . . . . . . . . . . . . . . . . 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 28522	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → (𝐵 ∈ (𝐶(LineG‘𝐺)𝑋) ∨ 𝐶 = 𝑋))
162	161	orcomd 871	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → (𝐶 = 𝑋 ∨ 𝐵 ∈ (𝐶(LineG‘𝐺)𝑋)))
163	162	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 ≠ 𝐶) → (𝐶 = 𝑋 ∨ 𝐵 ∈ (𝐶(LineG‘𝐺)𝑋)))
164	163	ord 864	. . . . . . . . . . . . . 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 28520	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → (𝐶 ∈ (𝑋(LineG‘𝐺)𝐴) ∨ 𝑋 = 𝐴))
167	166	adantr 480	. . . . . . . . . . . . 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 28610	. . . . . . . . . . . 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 3012	. . . . . . . . . . 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 28523	. . . . . . . . . 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 28521	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → (𝐴 ∈ (𝑋(LineG‘𝐺)𝐵) ∨ 𝑋 = 𝐵))
172	171	orcomd 871	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → (𝑋 = 𝐵 ∨ 𝐴 ∈ (𝑋(LineG‘𝐺)𝐵)))
173	172	ord 864	. . . . . . 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 28578	. . . . 5 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → (𝐴(𝐾‘𝐵)𝑋 ∨ 𝐵 ∈ (𝑋𝐼𝐴)))
176	99, 125, 175	mpjaodan 960	. . . 4 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → ∃𝑒 ∈ 𝑃 (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷)))
177	88, 176	pm2.61dane 3012	. . 3 ⊢ (((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → ∃𝑒 ∈ 𝑃 (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷)))
178	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐺 ∈ TarskiG)
179	8	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝑋 ∈ 𝑃)
180	12	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐵 ∈ 𝑃)
181	14	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐴 ∈ 𝑃)
182	6	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐷 ∈ 𝑃)
183		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐷 ∈ (𝐵𝐼𝐶))
184	1, 16, 2, 178, 179, 180, 68, 181, 182, 70, 183	axtgpasch 28430	. . . . 5 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → ∃𝑒 ∈ 𝑃 (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋)))
185	184	adantr 480	. . . 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 730	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝐴 ∈ 𝑃)
188	180	ad3antrrr 730	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝐵 ∈ 𝑃)
189	178	ad3antrrr 730	. . . . . . . 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 28451	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝑒 ∈ (𝐵𝐼𝐴))
192	38	necomd 2980	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ≠ 𝐴)
193	192	ad4antr 732	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝐵 ≠ 𝐴)
194	189	adantr 480	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐺 ∈ TarskiG)
195	6	ad5antr 734	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐷 ∈ 𝑃)
196	8	ad5antr 734	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝑋 ∈ 𝑃)
197	188	adantr 480	. . . . . . . . . . . 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 2980	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 ≠ 𝐷)
200	199	ad5antr 734	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐵 ≠ 𝐷)
201	200	neneqd 2930	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → ¬ 𝐵 = 𝐷)
202		ioran 985	. . . . . . . . . . . . . 14 ⊢ (¬ (𝑋 ∈ (𝐵(LineG‘𝐺)𝐷) ∨ 𝐵 = 𝐷) ↔ (¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷) ∧ ¬ 𝐵 = 𝐷))
203	198, 201, 202	sylanbrc 583	. . . . . . . . . . . . 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 28526	. . . . . . . . . . . 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 484	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝑒 = 𝐵)
206	186	adantr 480	. . . . . . . . . . . . . 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 28588	. . . . . . . . . . . . . 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 28582	. . . . . . . . . . . . 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 2829	. . . . . . . . . . . 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 28596	. . . . . . . . . . . 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 734	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐷 ≠ 𝐵)
213	212	necomd 2980	. . . . . . . . . . . . 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 28596	. . . . . . . . . . . 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 28597	. . . . . . . . . . . 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 28608	. . . . . . . . . . 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 2932	. . . . . . . . 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 28575	. . . . . . . 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 28567	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝐴(𝐾‘𝐵)𝑒)
221	178	ad3antrrr 730	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → 𝐺 ∈ TarskiG)
222	182	ad3antrrr 730	. . . . . . . . 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 730	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → 𝑋 ∈ 𝑃)
225		simpr 484	. . . . . . . . 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 28451	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → 𝑒 ∈ (𝑋𝐼𝐷))
227	226	adantrl 716	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝑒 ∈ (𝑋𝐼𝐷))
228	220, 227	jca 511	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷)))
229	228	ex 412	. . . . 5 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) → ((𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷))))
230	229	reximdva 3142	. . . 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 1144	. 2 ⊢ (𝜑 → (𝐶 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐶)))
234	52, 232, 233	mpjaodan 960	1 ⊢ (𝜑 → ∃𝑒 ∈ 𝑃 (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∃wrex 3053 class class class wbr 5095 ‘cfv 6486 (class class class)co 7353 Basecbs 17138 distcds 17188 TarskiGcstrkg 28390 Itvcitv 28396 LineGclng 28397 hlGchlg 28563

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-oadd 8399 df-er 8632 df-map 8762 df-pm 8763 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-dju 9816 df-card 9854 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-nn 12147 df-2 12209 df-3 12210 df-n0 12403 df-xnn0 12476 df-z 12490 df-uz 12754 df-fz 13429 df-fzo 13576 df-hash 14256 df-word 14439 df-concat 14496 df-s1 14521 df-s2 14773 df-s3 14774 df-trkgc 28411 df-trkgb 28412 df-trkgcb 28413 df-trkgld 28415 df-trkg 28416 df-cgrg 28474 df-leg 28546 df-hlg 28564 df-mir 28616 df-rag 28657 df-perpg 28659

This theorem is referenced by: inaghl 28808

W3C validator