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Theorem colhp 26118

Description: Half-plane relation for colinear points. Theorem 9.19 of [Schwabhauser] p. 73. (Contributed by Thierry Arnoux, 3-Aug-2020.)

**Hypotheses**
Ref	Expression
hpgid.p	⊢ 𝑃 = (Base‘𝐺)
hpgid.i	⊢ 𝐼 = (Itv‘𝐺)
hpgid.l	⊢ 𝐿 = (LineG‘𝐺)
hpgid.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
hpgid.d	⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
hpgid.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
hpgid.o	⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
colopp.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
colopp.p	⊢ (𝜑 → 𝐶 ∈ 𝐷)
colopp.1	⊢ (𝜑 → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
colhp.k	⊢ 𝐾 = (hlG‘𝐺)

**Assertion**
Ref	Expression
colhp	⊢ (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ (𝐴(𝐾‘𝐶)𝐵 ∧ ¬ 𝐴 ∈ 𝐷)))

Distinct variable groups: 𝑡,𝐴 𝑡,𝐵 𝐷,𝑎,𝑏,𝑡 𝐺,𝑎,𝑏,𝑡 𝐼,𝑎,𝑏,𝑡 𝑂,𝑎,𝑏,𝑡 𝑃,𝑎,𝑏,𝑡 𝜑,𝑡 𝑡,𝐶 𝐿,𝑎,𝑏,𝑡 𝐴,𝑎,𝑏 𝐶,𝑎,𝑏 Allowed substitution hints: 𝜑(𝑎,𝑏) 𝐵(𝑎,𝑏) 𝐾(𝑡,𝑎,𝑏)

**Proof of Theorem colhp**
Step	Hyp	Ref	Expression
1		ancom 454	. . 3 ⊢ ((𝐴(𝐾‘𝐶)𝐵 ∧ ¬ 𝐴 ∈ 𝐷) ↔ (¬ 𝐴 ∈ 𝐷 ∧ 𝐴(𝐾‘𝐶)𝐵))
2	1	a1i 11	. 2 ⊢ (𝜑 → ((𝐴(𝐾‘𝐶)𝐵 ∧ ¬ 𝐴 ∈ 𝐷) ↔ (¬ 𝐴 ∈ 𝐷 ∧ 𝐴(𝐾‘𝐶)𝐵)))
3		hpgid.p	. . . . 5 ⊢ 𝑃 = (Base‘𝐺)
4		hpgid.i	. . . . 5 ⊢ 𝐼 = (Itv‘𝐺)
5		hpgid.l	. . . . 5 ⊢ 𝐿 = (LineG‘𝐺)
6		hpgid.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7	6	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → 𝐺 ∈ TarskiG)
8		hpgid.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
9	8	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → 𝐷 ∈ ran 𝐿)
10		colopp.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
11	10	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → 𝐵 ∈ 𝑃)
12		hpgid.o	. . . . 5 ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
13		eqid 2778	. . . . . . 7 ⊢ (dist‘𝐺) = (dist‘𝐺)
14		eqid 2778	. . . . . . 7 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
15		colopp.p	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝐷)
16	3, 5, 4, 6, 8, 15	tglnpt 25900	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
17		eqid 2778	. . . . . . 7 ⊢ ((pInvG‘𝐺)‘𝐶) = ((pInvG‘𝐺)‘𝐶)
18		hpgid.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
19	3, 13, 4, 5, 14, 6, 16, 17, 18	mircl 26012	. . . . . 6 ⊢ (𝜑 → (((pInvG‘𝐺)‘𝐶)‘𝐴) ∈ 𝑃)
20	19	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → (((pInvG‘𝐺)‘𝐶)‘𝐴) ∈ 𝑃)
21	15	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → 𝐶 ∈ 𝐷)
22	16	adantr 474	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → 𝐶 ∈ 𝑃)
23	18	adantr 474	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → 𝐴 ∈ 𝑃)
24		nelne2 3068	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ 𝐷 ∧ ¬ 𝐴 ∈ 𝐷) → 𝐶 ≠ 𝐴)
25	15, 24	sylan 575	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → 𝐶 ≠ 𝐴)
26	25	necomd 3024	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → 𝐴 ≠ 𝐶)
27	26	neneqd 2974	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → ¬ 𝐴 = 𝐶)
28		simpr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → ¬ 𝐴 ∈ 𝐷)
29	3, 13, 4, 5, 14, 6, 16, 17, 18	mirmir 26013	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (((pInvG‘𝐺)‘𝐶)‘(((pInvG‘𝐺)‘𝐶)‘𝐴)) = 𝐴)
30	29	adantr 474	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (((pInvG‘𝐺)‘𝐶)‘𝐴) ∈ 𝐷) → (((pInvG‘𝐺)‘𝐶)‘(((pInvG‘𝐺)‘𝐶)‘𝐴)) = 𝐴)
31	6	adantr 474	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (((pInvG‘𝐺)‘𝐶)‘𝐴) ∈ 𝐷) → 𝐺 ∈ TarskiG)
32	8	adantr 474	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (((pInvG‘𝐺)‘𝐶)‘𝐴) ∈ 𝐷) → 𝐷 ∈ ran 𝐿)
33	15	adantr 474	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (((pInvG‘𝐺)‘𝐶)‘𝐴) ∈ 𝐷) → 𝐶 ∈ 𝐷)
34		simpr 479	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (((pInvG‘𝐺)‘𝐶)‘𝐴) ∈ 𝐷) → (((pInvG‘𝐺)‘𝐶)‘𝐴) ∈ 𝐷)
35	3, 13, 4, 5, 14, 31, 17, 32, 33, 34	mirln 26027	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (((pInvG‘𝐺)‘𝐶)‘𝐴) ∈ 𝐷) → (((pInvG‘𝐺)‘𝐶)‘(((pInvG‘𝐺)‘𝐶)‘𝐴)) ∈ 𝐷)
36	30, 35	eqeltrrd 2860	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (((pInvG‘𝐺)‘𝐶)‘𝐴) ∈ 𝐷) → 𝐴 ∈ 𝐷)
37	36	stoic1a 1816	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → ¬ (((pInvG‘𝐺)‘𝐶)‘𝐴) ∈ 𝐷)
38		simpr 479	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑡 = 𝐶) → 𝑡 = 𝐶)
39		eqidd 2779	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑡 = 𝐶) → (𝐴𝐼(((pInvG‘𝐺)‘𝐶)‘𝐴)) = (𝐴𝐼(((pInvG‘𝐺)‘𝐶)‘𝐴)))
40	38, 39	eleq12d 2853	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑡 = 𝐶) → (𝑡 ∈ (𝐴𝐼(((pInvG‘𝐺)‘𝐶)‘𝐴)) ↔ 𝐶 ∈ (𝐴𝐼(((pInvG‘𝐺)‘𝐶)‘𝐴))))
41	3, 13, 4, 5, 14, 6, 16, 17, 18	mirbtwn 26009	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐶 ∈ ((((pInvG‘𝐺)‘𝐶)‘𝐴)𝐼𝐴))
42	3, 13, 4, 6, 19, 16, 18, 41	tgbtwncom 25839	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼(((pInvG‘𝐺)‘𝐶)‘𝐴)))
43	15, 40, 42	rspcedvd 3518	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼(((pInvG‘𝐺)‘𝐶)‘𝐴)))
44	43	adantr 474	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼(((pInvG‘𝐺)‘𝐶)‘𝐴)))
45	28, 37, 44	jca31 510	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → ((¬ 𝐴 ∈ 𝐷 ∧ ¬ (((pInvG‘𝐺)‘𝐶)‘𝐴) ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼(((pInvG‘𝐺)‘𝐶)‘𝐴))))
46	3, 13, 4, 12, 23, 20	islnopp 26087	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → (𝐴𝑂(((pInvG‘𝐺)‘𝐶)‘𝐴) ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ (((pInvG‘𝐺)‘𝐶)‘𝐴) ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼(((pInvG‘𝐺)‘𝐶)‘𝐴)))))
47	45, 46	mpbird 249	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → 𝐴𝑂(((pInvG‘𝐺)‘𝐶)‘𝐴))
48	3, 13, 4, 12, 5, 9, 7, 23, 20, 47	oppne3 26091	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → 𝐴 ≠ (((pInvG‘𝐺)‘𝐶)‘𝐴))
49	42	adantr 474	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → 𝐶 ∈ (𝐴𝐼(((pInvG‘𝐺)‘𝐶)‘𝐴)))
50	3, 4, 5, 7, 23, 20, 22, 48, 49	btwnlng1 25970	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → 𝐶 ∈ (𝐴𝐿(((pInvG‘𝐺)‘𝐶)‘𝐴)))
51	50	orcd 862	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → (𝐶 ∈ (𝐴𝐿(((pInvG‘𝐺)‘𝐶)‘𝐴)) ∨ 𝐴 = (((pInvG‘𝐺)‘𝐶)‘𝐴)))
52	3, 5, 4, 7, 23, 20, 22, 51	colcom 25909	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → (𝐶 ∈ ((((pInvG‘𝐺)‘𝐶)‘𝐴)𝐿𝐴) ∨ (((pInvG‘𝐺)‘𝐶)‘𝐴) = 𝐴))
53	3, 5, 4, 7, 20, 23, 22, 52	colrot1 25910	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → ((((pInvG‘𝐺)‘𝐶)‘𝐴) ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
54	53	orcomd 860	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → (𝐴 = 𝐶 ∨ (((pInvG‘𝐺)‘𝐶)‘𝐴) ∈ (𝐴𝐿𝐶)))
55	54	ord 853	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → (¬ 𝐴 = 𝐶 → (((pInvG‘𝐺)‘𝐶)‘𝐴) ∈ (𝐴𝐿𝐶)))
56	27, 55	mpd 15	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → (((pInvG‘𝐺)‘𝐶)‘𝐴) ∈ (𝐴𝐿𝐶))
57		colopp.1	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
58	3, 5, 4, 6, 18, 10, 16, 57	colrot1 25910	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
59	3, 5, 4, 6, 10, 16, 18, 58	colcom 25909	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∈ (𝐶𝐿𝐵) ∨ 𝐶 = 𝐵))
60	59	adantr 474	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → (𝐴 ∈ (𝐶𝐿𝐵) ∨ 𝐶 = 𝐵))
61	3, 4, 5, 7, 20, 23, 22, 11, 56, 60	coltr 25998	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → ((((pInvG‘𝐺)‘𝐶)‘𝐴) ∈ (𝐶𝐿𝐵) ∨ 𝐶 = 𝐵))
62	3, 5, 4, 7, 22, 11, 20, 61	colrot1 25910	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → (𝐶 ∈ (𝐵𝐿(((pInvG‘𝐺)‘𝐶)‘𝐴)) ∨ 𝐵 = (((pInvG‘𝐺)‘𝐶)‘𝐴)))
63	3, 4, 5, 7, 9, 11, 12, 20, 21, 62	colopp 26117	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → (𝐵𝑂(((pInvG‘𝐺)‘𝐶)‘𝐴) ↔ (𝐶 ∈ (𝐵𝐼(((pInvG‘𝐺)‘𝐶)‘𝐴)) ∧ ¬ 𝐵 ∈ 𝐷 ∧ ¬ (((pInvG‘𝐺)‘𝐶)‘𝐴) ∈ 𝐷)))
64	3, 4, 5, 12, 7, 9, 23, 11, 20, 47	lnopp2hpgb 26111	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → (𝐵𝑂(((pInvG‘𝐺)‘𝐶)‘𝐴) ↔ 𝐴((hpG‘𝐺)‘𝐷)𝐵))
65		colhp.k	. . . . . . . . 9 ⊢ 𝐾 = (hlG‘𝐺)
66		simpll 757	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ (𝐶 ∈ (𝐵𝐼(((pInvG‘𝐺)‘𝐶)‘𝐴)) ∧ ¬ 𝐵 ∈ 𝐷)) → 𝜑)
67	66, 10	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ (𝐶 ∈ (𝐵𝐼(((pInvG‘𝐺)‘𝐶)‘𝐴)) ∧ ¬ 𝐵 ∈ 𝐷)) → 𝐵 ∈ 𝑃)
68	66, 18	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ (𝐶 ∈ (𝐵𝐼(((pInvG‘𝐺)‘𝐶)‘𝐴)) ∧ ¬ 𝐵 ∈ 𝐷)) → 𝐴 ∈ 𝑃)
69	66, 16	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ (𝐶 ∈ (𝐵𝐼(((pInvG‘𝐺)‘𝐶)‘𝐴)) ∧ ¬ 𝐵 ∈ 𝐷)) → 𝐶 ∈ 𝑃)
70	66, 6	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ (𝐶 ∈ (𝐵𝐼(((pInvG‘𝐺)‘𝐶)‘𝐴)) ∧ ¬ 𝐵 ∈ 𝐷)) → 𝐺 ∈ TarskiG)
71	66, 15	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ (𝐶 ∈ (𝐵𝐼(((pInvG‘𝐺)‘𝐶)‘𝐴)) ∧ ¬ 𝐵 ∈ 𝐷)) → 𝐶 ∈ 𝐷)
72		simprr 763	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ (𝐶 ∈ (𝐵𝐼(((pInvG‘𝐺)‘𝐶)‘𝐴)) ∧ ¬ 𝐵 ∈ 𝐷)) → ¬ 𝐵 ∈ 𝐷)
73		nelne2 3068	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷) → 𝐶 ≠ 𝐵)
74	73	necomd 3024	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷) → 𝐵 ≠ 𝐶)
75	71, 72, 74	syl2anc 579	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ (𝐶 ∈ (𝐵𝐼(((pInvG‘𝐺)‘𝐶)‘𝐴)) ∧ ¬ 𝐵 ∈ 𝐷)) → 𝐵 ≠ 𝐶)
76	26	adantr 474	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ (𝐶 ∈ (𝐵𝐼(((pInvG‘𝐺)‘𝐶)‘𝐴)) ∧ ¬ 𝐵 ∈ 𝐷)) → 𝐴 ≠ 𝐶)
77		simprl 761	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ (𝐶 ∈ (𝐵𝐼(((pInvG‘𝐺)‘𝐶)‘𝐴)) ∧ ¬ 𝐵 ∈ 𝐷)) → 𝐶 ∈ (𝐵𝐼(((pInvG‘𝐺)‘𝐶)‘𝐴)))
78	3, 13, 4, 5, 14, 70, 17, 65, 69, 67, 68, 68, 75, 76, 77	mirhl2 26032	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ (𝐶 ∈ (𝐵𝐼(((pInvG‘𝐺)‘𝐶)‘𝐴)) ∧ ¬ 𝐵 ∈ 𝐷)) → 𝐵(𝐾‘𝐶)𝐴)
79	3, 4, 65, 67, 68, 69, 70, 78	hlcomd 25955	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ (𝐶 ∈ (𝐵𝐼(((pInvG‘𝐺)‘𝐶)‘𝐴)) ∧ ¬ 𝐵 ∈ 𝐷)) → 𝐴(𝐾‘𝐶)𝐵)
80	28	adantr 474	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ (𝐶 ∈ (𝐵𝐼(((pInvG‘𝐺)‘𝐶)‘𝐴)) ∧ ¬ 𝐵 ∈ 𝐷)) → ¬ 𝐴 ∈ 𝐷)
81	79, 80	jca 507	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ (𝐶 ∈ (𝐵𝐼(((pInvG‘𝐺)‘𝐶)‘𝐴)) ∧ ¬ 𝐵 ∈ 𝐷)) → (𝐴(𝐾‘𝐶)𝐵 ∧ ¬ 𝐴 ∈ 𝐷))
82	81	3adantr3 1173	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ (𝐶 ∈ (𝐵𝐼(((pInvG‘𝐺)‘𝐶)‘𝐴)) ∧ ¬ 𝐵 ∈ 𝐷 ∧ ¬ (((pInvG‘𝐺)‘𝐶)‘𝐴) ∈ 𝐷)) → (𝐴(𝐾‘𝐶)𝐵 ∧ ¬ 𝐴 ∈ 𝐷))
83	82	simpld 490	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ (𝐶 ∈ (𝐵𝐼(((pInvG‘𝐺)‘𝐶)‘𝐴)) ∧ ¬ 𝐵 ∈ 𝐷 ∧ ¬ (((pInvG‘𝐺)‘𝐶)‘𝐴) ∈ 𝐷)) → 𝐴(𝐾‘𝐶)𝐵)
84	23	adantr 474	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝐴(𝐾‘𝐶)𝐵) → 𝐴 ∈ 𝑃)
85	11	adantr 474	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝐴(𝐾‘𝐶)𝐵) → 𝐵 ∈ 𝑃)
86	20	adantr 474	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝐴(𝐾‘𝐶)𝐵) → (((pInvG‘𝐺)‘𝐶)‘𝐴) ∈ 𝑃)
87	7	adantr 474	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝐴(𝐾‘𝐶)𝐵) → 𝐺 ∈ TarskiG)
88	16	ad2antrr 716	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝐴(𝐾‘𝐶)𝐵) → 𝐶 ∈ 𝑃)
89		simpr 479	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝐴(𝐾‘𝐶)𝐵) → 𝐴(𝐾‘𝐶)𝐵)
90	42	ad2antrr 716	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝐴(𝐾‘𝐶)𝐵) → 𝐶 ∈ (𝐴𝐼(((pInvG‘𝐺)‘𝐶)‘𝐴)))
91	3, 4, 65, 84, 85, 86, 87, 88, 89, 90	btwnhl 25965	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝐴(𝐾‘𝐶)𝐵) → 𝐶 ∈ (𝐵𝐼(((pInvG‘𝐺)‘𝐶)‘𝐴)))
92	3, 4, 65, 84, 85, 88, 87, 5, 89	hlln 25958	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝐴(𝐾‘𝐶)𝐵) → 𝐴 ∈ (𝐵𝐿𝐶))
93	92	adantr 474	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝐴(𝐾‘𝐶)𝐵) ∧ 𝐵 ∈ 𝐷) → 𝐴 ∈ (𝐵𝐿𝐶))
94	87	adantr 474	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝐴(𝐾‘𝐶)𝐵) ∧ 𝐵 ∈ 𝐷) → 𝐺 ∈ TarskiG)
95	85	adantr 474	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝐴(𝐾‘𝐶)𝐵) ∧ 𝐵 ∈ 𝐷) → 𝐵 ∈ 𝑃)
96	88	adantr 474	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝐴(𝐾‘𝐶)𝐵) ∧ 𝐵 ∈ 𝐷) → 𝐶 ∈ 𝑃)
97	84	adantr 474	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝐴(𝐾‘𝐶)𝐵) ∧ 𝐵 ∈ 𝐷) → 𝐴 ∈ 𝑃)
98	89	adantr 474	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝐴(𝐾‘𝐶)𝐵) ∧ 𝐵 ∈ 𝐷) → 𝐴(𝐾‘𝐶)𝐵)
99	3, 4, 65, 97, 95, 96, 94, 98	hlne2 25957	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝐴(𝐾‘𝐶)𝐵) ∧ 𝐵 ∈ 𝐷) → 𝐵 ≠ 𝐶)
100	9	ad2antrr 716	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝐴(𝐾‘𝐶)𝐵) ∧ 𝐵 ∈ 𝐷) → 𝐷 ∈ ran 𝐿)
101		simpr 479	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝐴(𝐾‘𝐶)𝐵) ∧ 𝐵 ∈ 𝐷) → 𝐵 ∈ 𝐷)
102	15	ad3antrrr 720	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝐴(𝐾‘𝐶)𝐵) ∧ 𝐵 ∈ 𝐷) → 𝐶 ∈ 𝐷)
103	3, 4, 5, 94, 95, 96, 99, 99, 100, 101, 102	tglinethru 25987	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝐴(𝐾‘𝐶)𝐵) ∧ 𝐵 ∈ 𝐷) → 𝐷 = (𝐵𝐿𝐶))
104	93, 103	eleqtrrd 2862	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝐴(𝐾‘𝐶)𝐵) ∧ 𝐵 ∈ 𝐷) → 𝐴 ∈ 𝐷)
105	28	ad2antrr 716	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝐴(𝐾‘𝐶)𝐵) ∧ 𝐵 ∈ 𝐷) → ¬ 𝐴 ∈ 𝐷)
106	104, 105	pm2.65da 807	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝐴(𝐾‘𝐶)𝐵) → ¬ 𝐵 ∈ 𝐷)
107	37	adantr 474	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝐴(𝐾‘𝐶)𝐵) → ¬ (((pInvG‘𝐺)‘𝐶)‘𝐴) ∈ 𝐷)
108	91, 106, 107	3jca 1119	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) ∧ 𝐴(𝐾‘𝐶)𝐵) → (𝐶 ∈ (𝐵𝐼(((pInvG‘𝐺)‘𝐶)‘𝐴)) ∧ ¬ 𝐵 ∈ 𝐷 ∧ ¬ (((pInvG‘𝐺)‘𝐶)‘𝐴) ∈ 𝐷))
109	83, 108	impbida 791	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → ((𝐶 ∈ (𝐵𝐼(((pInvG‘𝐺)‘𝐶)‘𝐴)) ∧ ¬ 𝐵 ∈ 𝐷 ∧ ¬ (((pInvG‘𝐺)‘𝐶)‘𝐴) ∈ 𝐷) ↔ 𝐴(𝐾‘𝐶)𝐵))
110	63, 64, 109	3bitr3d 301	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ 𝐷) → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ 𝐴(𝐾‘𝐶)𝐵))
111	110	pm5.32da 574	. 2 ⊢ (𝜑 → ((¬ 𝐴 ∈ 𝐷 ∧ 𝐴((hpG‘𝐺)‘𝐷)𝐵) ↔ (¬ 𝐴 ∈ 𝐷 ∧ 𝐴(𝐾‘𝐶)𝐵)))
112		simpr 479	. . . 4 ⊢ ((𝜑 ∧ 𝐴((hpG‘𝐺)‘𝐷)𝐵) → 𝐴((hpG‘𝐺)‘𝐷)𝐵)
113	112	adantrl 706	. . 3 ⊢ ((𝜑 ∧ (¬ 𝐴 ∈ 𝐷 ∧ 𝐴((hpG‘𝐺)‘𝐷)𝐵)) → 𝐴((hpG‘𝐺)‘𝐷)𝐵)
114	6	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝐴((hpG‘𝐺)‘𝐷)𝐵) → 𝐺 ∈ TarskiG)
115	8	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝐴((hpG‘𝐺)‘𝐷)𝐵) → 𝐷 ∈ ran 𝐿)
116	18	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝐴((hpG‘𝐺)‘𝐷)𝐵) → 𝐴 ∈ 𝑃)
117	10	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝐴((hpG‘𝐺)‘𝐷)𝐵) → 𝐵 ∈ 𝑃)
118	3, 4, 5, 12, 114, 115, 116, 117, 112	hpgne1 26109	. . . 4 ⊢ ((𝜑 ∧ 𝐴((hpG‘𝐺)‘𝐷)𝐵) → ¬ 𝐴 ∈ 𝐷)
119	118, 112	jca 507	. . 3 ⊢ ((𝜑 ∧ 𝐴((hpG‘𝐺)‘𝐷)𝐵) → (¬ 𝐴 ∈ 𝐷 ∧ 𝐴((hpG‘𝐺)‘𝐷)𝐵))
120	113, 119	impbida 791	. 2 ⊢ (𝜑 → ((¬ 𝐴 ∈ 𝐷 ∧ 𝐴((hpG‘𝐺)‘𝐷)𝐵) ↔ 𝐴((hpG‘𝐺)‘𝐷)𝐵))
121	2, 111, 120	3bitr2rd 300	1 ⊢ (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ (𝐴(𝐾‘𝐶)𝐵 ∧ ¬ 𝐴 ∈ 𝐷)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 ∨ wo 836 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ∃wrex 3091 ∖ cdif 3789 class class class wbr 4886 {copab 4948 ran crn 5356 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 distcds 16347 TarskiGcstrkg 25781 Itvcitv 25787 LineGclng 25788 hlGchlg 25951 pInvGcmir 26003 hpGchpg 26105

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-card 9098 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-n0 11643 df-xnn0 11715 df-z 11729 df-uz 11993 df-fz 12644 df-fzo 12785 df-hash 13436 df-word 13600 df-concat 13661 df-s1 13686 df-s2 13999 df-s3 14000 df-trkgc 25799 df-trkgb 25800 df-trkgcb 25801 df-trkgld 25803 df-trkg 25804 df-cgrg 25862 df-leg 25934 df-hlg 25952 df-mir 26004 df-rag 26045 df-perpg 26047 df-hpg 26106

This theorem is referenced by: hphl 26119 trgcopy 26152

W3C validator