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Theorem hlpasch 28735

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 2735	. . . 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 2735	. . . . 5 ⊢ (dist‘𝐺) = (dist‘𝐺)
17		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐶 ∈ (𝐵𝐼𝐷))
18	1, 16, 2, 5, 13, 11, 7, 17	tgbtwncom 28467	. . . 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 28734	. . 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 28467	. . . . . . 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 7421	. . . . . . . . . . . 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 28445	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐵 = 𝐴)
37	36	eqcomd 2741	. . . . . . . . 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 2937	. . . . . . . . 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 2939	. . . . . . 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 28592	. . . . . 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 28467	. . . . . 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 3153	. . 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 5131	. . . . . 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 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 28583	. . . . . . . . 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 7420	. . . . . . . . . 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 28581	. . . . . . . . . . . 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 28590	. . . . . . . 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 28583	. . . . . 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 28466	. . . . . 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 3603	. . . 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 5131	. . . . . . . 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 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 28470	. . . . . . . 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 3603	. . . . 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 5131	. . . . . . 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 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 28585	. . . . . . . . 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 28473	. . . . . . . . 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 28556	. . . . . . . 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 28581	. . . . . . . . 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 28466	. . . . . . 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 3603	. . . . 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 2937	. . . . . . 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 7421	. . . . . . . . . . . . . . 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 28445	. . . . . . . . . . . . 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 2987	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) ∧ 𝑋 ≠ 𝐶) → 𝐶 ≠ 𝑋)
149	148	neneqd 2937	. . . . . . . . . . . . . 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 28601	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 ≠ 𝐵) → 𝑋 ∈ (𝐷(LineG‘𝐺)𝐵))
154	77	necomd 2987	. . . . . . . . . . . . . . . . . . . . . 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 28586	. . . . . . . . . . . . . . . . . . . . 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 28601	. . . . . . . . . . . . . . . . . . . 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 28626	. . . . . . . . . . . . . . . . . 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 28538	. . . . . . . . . . . . . . . . 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 28536	. . . . . . . . . . . . . 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 28626	. . . . . . . . . . . 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 3019	. . . . . . . . . . 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 28539	. . . . . . . . . 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 28537	. . . . . . . . 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 28594	. . . . 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 3019	. . 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 28446	. . . . 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 28467	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝑒 ∈ (𝐵𝐼𝐴))
192	38	necomd 2987	. . . . . . . . . 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 2987	. . . . . . . . . . . . . . . 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 2937	. . . . . . . . . . . . . 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 28542	. . . . . . . . . . . 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 28604	. . . . . . . . . . . . . 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 28598	. . . . . . . . . . . . 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 28612	. . . . . . . . . . . 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 2987	. . . . . . . . . . . . 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 28612	. . . . . . . . . . . 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 28613	. . . . . . . . . . . 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 28624	. . . . . . . . . . 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 2939	. . . . . . . . 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 28591	. . . . . . . 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 28583	. . . . . . 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 28467	. . . . . . . 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 3153	. . . 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 2108 ≠ wne 2932 ∃wrex 3060 class class class wbr 5119 ‘cfv 6531 (class class class)co 7405 Basecbs 17228 distcds 17280 TarskiGcstrkg 28406 Itvcitv 28412 LineGclng 28413 hlGchlg 28579

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-oadd 8484 df-er 8719 df-map 8842 df-pm 8843 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-dju 9915 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-nn 12241 df-2 12303 df-3 12304 df-n0 12502 df-xnn0 12575 df-z 12589 df-uz 12853 df-fz 13525 df-fzo 13672 df-hash 14349 df-word 14532 df-concat 14589 df-s1 14614 df-s2 14867 df-s3 14868 df-trkgc 28427 df-trkgb 28428 df-trkgcb 28429 df-trkgld 28431 df-trkg 28432 df-cgrg 28490 df-leg 28562 df-hlg 28580 df-mir 28632 df-rag 28673 df-perpg 28675

This theorem is referenced by: inaghl 28824

W3C validator