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Theorem hlpasch 26544

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 2823	. . . 4 ⊢ (LineG‘𝐺) = (LineG‘𝐺)
4		hlpasch.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐺 ∈ TarskiG)
6		hlpasch.5	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
7	6	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐷 ∈ 𝑃)
8		hlpasch.4	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
9	8	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝑋 ∈ 𝑃)
10		hlpasch.3	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
11	10	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐶 ∈ 𝑃)
12		hlpasch.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
13	12	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐵 ∈ 𝑃)
14		hlpasch.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
15	14	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐴 ∈ 𝑃)
16		eqid 2823	. . . . 5 ⊢ (dist‘𝐺) = (dist‘𝐺)
17		simpr 487	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐶 ∈ (𝐵𝐼𝐷))
18	1, 16, 2, 5, 13, 11, 7, 17	tgbtwncom 26276	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐶 ∈ (𝐷𝐼𝐵))
19		hlpasch.8	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝑋𝐼𝐶))
20	19	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐴 ∈ (𝑋𝐼𝐶))
21	1, 2, 3, 5, 7, 9, 11, 13, 15, 18, 20	outpasch 26543	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → ∃𝑒 ∈ 𝑃 (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒)))
22		hlpasch.k	. . . . . . 7 ⊢ 𝐾 = (hlG‘𝐺)
23		simplr 767	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝑒 ∈ 𝑃)
24	13	ad2antrr 724	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐵 ∈ 𝑃)
25	15	ad2antrr 724	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐴 ∈ 𝑃)
26	5	ad2antrr 724	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐺 ∈ TarskiG)
27		simprr 771	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐴 ∈ (𝐵𝐼𝑒))
28	1, 16, 2, 26, 24, 25, 23, 27	tgbtwncom 26276	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐴 ∈ (𝑒𝐼𝐵))
29	26	adantr 483	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐺 ∈ TarskiG)
30	24	adantr 483	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐵 ∈ 𝑃)
31	25	adantr 483	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐴 ∈ 𝑃)
32		simplrr 776	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐴 ∈ (𝐵𝐼𝑒))
33		simpr 487	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝑒 = 𝐵)
34	33	oveq2d 7174	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → (𝐵𝐼𝑒) = (𝐵𝐼𝐵))
35	32, 34	eleqtrd 2917	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐴 ∈ (𝐵𝐼𝐵))
36	1, 16, 2, 29, 30, 31, 35	axtgbtwnid 26254	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐵 = 𝐴)
37	36	eqcomd 2829	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐴 = 𝐵)
38		hlpasch.6	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
39	38	ad3antrrr 728	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐴 ≠ 𝐵)
40	39	adantr 483	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐴 ≠ 𝐵)
41	40	neneqd 3023	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → ¬ 𝐴 = 𝐵)
42	37, 41	pm2.65da 815	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → ¬ 𝑒 = 𝐵)
43	42	neqned 3025	. . . . . . 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 26401	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐴(𝐾‘𝐵)𝑒)
45	7	ad2antrr 724	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐷 ∈ 𝑃)
46	9	ad2antrr 724	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝑋 ∈ 𝑃)
47		simprl 769	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝑒 ∈ (𝐷𝐼𝑋))
48	1, 16, 2, 26, 45, 23, 46, 47	tgbtwncom 26276	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝑒 ∈ (𝑋𝐼𝐷))
49	44, 48	jca 514	. . . . 5 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷)))
50	49	ex 415	. . . 4 ⊢ (((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) → ((𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒)) → (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷))))
51	50	reximdva 3276	. . 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 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝐷 ∈ 𝑃)
54	53	adantr 483	. . . . 5 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐷 ∈ 𝑃)
55		simpr 487	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) ∧ 𝑒 = 𝐷) → 𝑒 = 𝐷)
56	55	breq2d 5080	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) ∧ 𝑒 = 𝐷) → (𝐴(𝐾‘𝐵)𝑒 ↔ 𝐴(𝐾‘𝐵)𝐷))
57	55	eleq1d 2899	. . . . . 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 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝐴 ∈ 𝑃)
60	59	adantr 483	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐴 ∈ 𝑃)
61	12	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝐵 ∈ 𝑃)
62	61	adantr 483	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐵 ∈ 𝑃)
63	4	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝐺 ∈ TarskiG)
64	63	adantr 483	. . . . . . 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 26392	. . . . . . . . 9 ⊢ (𝜑 → 𝐷(𝐾‘𝐵)𝐶)
67	66	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐷(𝐾‘𝐵)𝐶)
68	10	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐶 ∈ 𝑃)
69	68	ad2antrr 724	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐶 ∈ 𝑃)
70	19	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐴 ∈ (𝑋𝐼𝐶))
71	70	ad2antrr 724	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐴 ∈ (𝑋𝐼𝐶))
72		simpr 487	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝑋 = 𝐵)
73	72	oveq1d 7173	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → (𝑋𝐼𝐶) = (𝐵𝐼𝐶))
74	71, 73	eleqtrd 2917	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐴 ∈ (𝐵𝐼𝐶))
75	1, 2, 22, 10, 6, 12, 4	ishlg 26390	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶(𝐾‘𝐵)𝐷 ↔ (𝐶 ≠ 𝐵 ∧ 𝐷 ≠ 𝐵 ∧ (𝐶 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐶)))))
76	65, 75	mpbid 234	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶 ≠ 𝐵 ∧ 𝐷 ≠ 𝐵 ∧ (𝐶 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐶))))
77	76	simp1d 1138	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
78	77	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐶 ≠ 𝐵)
79	38	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝐴 ≠ 𝐵)
80	79	adantr 483	. . . . . . . . 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 26399	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → (𝐷(𝐾‘𝐵)𝐶 ↔ 𝐷(𝐾‘𝐵)𝐴))
82	67, 81	mpbid 234	. . . . . . 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 26392	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐴(𝐾‘𝐵)𝐷)
84	8	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝑋 ∈ 𝑃)
85	84	adantr 483	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝑋 ∈ 𝑃)
86	1, 16, 2, 64, 85, 54	tgbtwntriv2 26275	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐷 ∈ (𝑋𝐼𝐷))
87	83, 86	jca 514	. . . . 5 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → (𝐴(𝐾‘𝐵)𝐷 ∧ 𝐷 ∈ (𝑋𝐼𝐷)))
88	54, 58, 87	rspcedvd 3628	. . . 4 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → ∃𝑒 ∈ 𝑃 (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷)))
89	84	ad2antrr 724	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐴(𝐾‘𝐵)𝑋) → 𝑋 ∈ 𝑃)
90		simpr 487	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 = 𝑋) → 𝑒 = 𝑋)
91	90	breq2d 5080	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 = 𝑋) → (𝐴(𝐾‘𝐵)𝑒 ↔ 𝐴(𝐾‘𝐵)𝑋))
92	90	eleq1d 2899	. . . . . . . 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 750	. . . . . 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 487	. . . . . . 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 26279	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝑋 ∈ (𝑋𝐼𝐷))
97	96	ad2antrr 724	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐴(𝐾‘𝐵)𝑋) → 𝑋 ∈ (𝑋𝐼𝐷))
98	95, 97	jca 514	. . . . . 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 3628	. . . . 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 724	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐷 ∈ 𝑃)
101		simpr 487	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) ∧ 𝑒 = 𝐷) → 𝑒 = 𝐷)
102	101	breq2d 5080	. . . . . . 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 2899	. . . . . . 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 724	. . . . . . . 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 26394	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ≠ 𝐵)
107	106	ad4antr 730	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐷 ≠ 𝐵)
108	63	ad2antrr 724	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐺 ∈ TarskiG)
109	61	ad2antrr 724	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐵 ∈ 𝑃)
110	59	ad2antrr 724	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐴 ∈ 𝑃)
111	68	ad2antrr 724	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝐶 ∈ 𝑃)
112	111	adantr 483	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐶 ∈ 𝑃)
113	84	ad2antrr 724	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝑋 ∈ 𝑃)
114		simpr 487	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐵 ∈ (𝑋𝐼𝐴))
115	70	ad2antrr 724	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝐴 ∈ (𝑋𝐼𝐶))
116	115	adantr 483	. . . . . . . . . 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 26282	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐴 ∈ (𝐵𝐼𝐶))
118		simp-4r 782	. . . . . . . . 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 26365	. . . . . . . 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 26390	. . . . . . . . 9 ⊢ (𝜑 → (𝐴(𝐾‘𝐵)𝐷 ↔ (𝐴 ≠ 𝐵 ∧ 𝐷 ≠ 𝐵 ∧ (𝐴 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐴)))))
121	120	ad4antr 730	. . . . . . . 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 1338	. . . . . . 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 26275	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐷 ∈ (𝑋𝐼𝐷))
124	122, 123	jca 514	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → (𝐴(𝐾‘𝐵)𝐷 ∧ 𝐷 ∈ (𝑋𝐼𝐷)))
125	100, 104, 124	rspcedvd 3628	. . . . 5 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → ∃𝑒 ∈ 𝑃 (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷)))
126	8	ad3antrrr 728	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝑋 ∈ 𝑃)
127	12	ad3antrrr 728	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝐵 ∈ 𝑃)
128	14	ad3antrrr 728	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝐴 ∈ 𝑃)
129	4	ad3antrrr 728	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝐺 ∈ TarskiG)
130		simpr 487	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝑋 ≠ 𝐵)
131	130	neneqd 3023	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → ¬ 𝑋 = 𝐵)
132	63	adantr 483	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝐺 ∈ TarskiG)
133	132	adantr 483	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 = 𝐶) → 𝐺 ∈ TarskiG)
134	126	adantr 483	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 = 𝐶) → 𝑋 ∈ 𝑃)
135	128	adantr 483	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 = 𝐶) → 𝐴 ∈ 𝑃)
136	115	adantr 483	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 = 𝐶) → 𝐴 ∈ (𝑋𝐼𝐶))
137		simpr 487	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 = 𝐶) → 𝑋 = 𝐶)
138	137	oveq2d 7174	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 = 𝐶) → (𝑋𝐼𝑋) = (𝑋𝐼𝐶))
139	136, 138	eleqtrrd 2918	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 = 𝐶) → 𝐴 ∈ (𝑋𝐼𝑋))
140	1, 16, 2, 133, 134, 135, 139	axtgbtwnid 26254	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 = 𝐶) → 𝑋 = 𝐴)
141	140	olcd 870	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 = 𝐶) → (𝐵 ∈ (𝑋(LineG‘𝐺)𝐴) ∨ 𝑋 = 𝐴))
142	132	adantr 483	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 ≠ 𝐶) → 𝐺 ∈ TarskiG)
143	127	adantr 483	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 ≠ 𝐶) → 𝐵 ∈ 𝑃)
144	111	adantr 483	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 ≠ 𝐶) → 𝐶 ∈ 𝑃)
145	126	adantr 483	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 ≠ 𝐶) → 𝑋 ∈ 𝑃)
146	128	adantr 483	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 ≠ 𝐶) → 𝐴 ∈ 𝑃)
147		simpr 487	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 ≠ 𝐶) → 𝑋 ≠ 𝐶)
148	147	necomd 3073	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 ≠ 𝐶) → 𝐶 ≠ 𝑋)
149	148	neneqd 3023	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 ≠ 𝐶) → ¬ 𝐶 = 𝑋)
150	53	adantr 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝐷 ∈ 𝑃)
151	106	ad3antrrr 728	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝐷 ≠ 𝐵)
152		simplr 767	. . . . . . . . . . . . . . . . . . . 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 26410	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝑋 ∈ (𝐷(LineG‘𝐺)𝐵))
154	77	necomd 3073	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
155	154	ad3antrrr 728	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝐵 ≠ 𝐶)
156	66	ad3antrrr 728	. . . . . . . . . . . . . . . . . . . . . 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 26395	. . . . . . . . . . . . . . . . . . . . 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 26410	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝐷 ∈ (𝐵(LineG‘𝐺)𝐶))
159	158	orcd 869	. . . . . . . . . . . . . . . . . . 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 26435	. . . . . . . . . . . . . . . . . 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 26347	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → (𝐵 ∈ (𝐶(LineG‘𝐺)𝑋) ∨ 𝐶 = 𝑋))
162	161	orcomd 867	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → (𝐶 = 𝑋 ∨ 𝐵 ∈ (𝐶(LineG‘𝐺)𝑋)))
163	162	adantr 483	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 ≠ 𝐶) → (𝐶 = 𝑋 ∨ 𝐵 ∈ (𝐶(LineG‘𝐺)𝑋)))
164	163	ord 860	. . . . . . . . . . . . . 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 26345	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → (𝐶 ∈ (𝑋(LineG‘𝐺)𝐴) ∨ 𝑋 = 𝐴))
167	166	adantr 483	. . . . . . . . . . . . 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 26435	. . . . . . . . . . . 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 3106	. . . . . . . . . . 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 26348	. . . . . . . . . 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 26346	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → (𝐴 ∈ (𝑋(LineG‘𝐺)𝐵) ∨ 𝑋 = 𝐵))
172	171	orcomd 867	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → (𝑋 = 𝐵 ∨ 𝐴 ∈ (𝑋(LineG‘𝐺)𝐵)))
173	172	ord 860	. . . . . . 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 26403	. . . . 5 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → (𝐴(𝐾‘𝐵)𝑋 ∨ 𝐵 ∈ (𝑋𝐼𝐴)))
176	99, 125, 175	mpjaodan 955	. . . 4 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → ∃𝑒 ∈ 𝑃 (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷)))
177	88, 176	pm2.61dane 3106	. . 3 ⊢ (((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → ∃𝑒 ∈ 𝑃 (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷)))
178	4	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐺 ∈ TarskiG)
179	8	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝑋 ∈ 𝑃)
180	12	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐵 ∈ 𝑃)
181	14	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐴 ∈ 𝑃)
182	6	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐷 ∈ 𝑃)
183		simpr 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐷 ∈ (𝐵𝐼𝐶))
184	1, 16, 2, 178, 179, 180, 68, 181, 182, 70, 183	axtgpasch 26255	. . . . 5 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → ∃𝑒 ∈ 𝑃 (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋)))
185	184	adantr 483	. . . 4 ⊢ (((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → ∃𝑒 ∈ 𝑃 (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋)))
186		simplr 767	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝑒 ∈ 𝑃)
187	181	ad3antrrr 728	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝐴 ∈ 𝑃)
188	180	ad3antrrr 728	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝐵 ∈ 𝑃)
189	178	ad3antrrr 728	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝐺 ∈ TarskiG)
190		simprl 769	. . . . . . . . . 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 26276	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝑒 ∈ (𝐵𝐼𝐴))
192	38	necomd 3073	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ≠ 𝐴)
193	192	ad4antr 730	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝐵 ≠ 𝐴)
194	189	adantr 483	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐺 ∈ TarskiG)
195	6	ad5antr 732	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐷 ∈ 𝑃)
196	8	ad5antr 732	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝑋 ∈ 𝑃)
197	188	adantr 483	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐵 ∈ 𝑃)
198		simp-4r 782	. . . . . . . . . . . . . 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 3073	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 ≠ 𝐷)
200	199	ad5antr 732	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐵 ≠ 𝐷)
201	200	neneqd 3023	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → ¬ 𝐵 = 𝐷)
202		ioran 980	. . . . . . . . . . . . . 14 ⊢ (¬ (𝑋 ∈ (𝐵(LineG‘𝐺)𝐷) ∨ 𝐵 = 𝐷) ↔ (¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷) ∧ ¬ 𝐵 = 𝐷))
203	198, 201, 202	sylanbrc 585	. . . . . . . . . . . . 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 26351	. . . . . . . . . . . 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 487	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝑒 = 𝐵)
206	186	adantr 483	. . . . . . . . . . . . . 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 26413	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐷 ≠ 𝑋)
208		simplrr 776	. . . . . . . . . . . . . 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 26407	. . . . . . . . . . . . 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 2916	. . . . . . . . . . . 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 26421	. . . . . . . . . . . 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 732	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐷 ≠ 𝐵)
213	212	necomd 3073	. . . . . . . . . . . . 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 26421	. . . . . . . . . . . 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 26422	. . . . . . . . . . . 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 26433	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐵 = 𝐷)
217	216, 201	pm2.65da 815	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → ¬ 𝑒 = 𝐵)
218	217	neqned 3025	. . . . . . . . 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 26400	. . . . . . . 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 26392	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝐴(𝐾‘𝐵)𝑒)
221	178	ad3antrrr 728	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → 𝐺 ∈ TarskiG)
222	182	ad3antrrr 728	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → 𝐷 ∈ 𝑃)
223		simplr 767	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → 𝑒 ∈ 𝑃)
224	179	ad3antrrr 728	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → 𝑋 ∈ 𝑃)
225		simpr 487	. . . . . . . . 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 26276	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → 𝑒 ∈ (𝑋𝐼𝐷))
227	226	adantrl 714	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝑒 ∈ (𝑋𝐼𝐷))
228	220, 227	jca 514	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷)))
229	228	ex 415	. . . . 5 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) → ((𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷))))
230	229	reximdva 3276	. . . 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 811	. 2 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → ∃𝑒 ∈ 𝑃 (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷)))
233	76	simp3d 1140	. 2 ⊢ (𝜑 → (𝐶 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐶)))
234	52, 232, 233	mpjaodan 955	1 ⊢ (𝜑 → ∃𝑒 ∈ 𝑃 (𝐴(𝐾‘𝐵)𝑒 ∧ 𝑒 ∈ (𝑋𝐼𝐷)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∃wrex 3141 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 distcds 16576 TarskiGcstrkg 26218 Itvcitv 26224 LineGclng 26225 hlGchlg 26388

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-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 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-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-dju 9332 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-concat 13925 df-s1 13952 df-s2 14212 df-s3 14213 df-trkgc 26236 df-trkgb 26237 df-trkgcb 26238 df-trkgld 26240 df-trkg 26241 df-cgrg 26299 df-leg 26371 df-hlg 26389 df-mir 26441 df-rag 26482 df-perpg 26484

This theorem is referenced by: inaghl 26633

W3C validator