Theorem colline 28628

Description: Three points are colinear iff there is a line through all three of them. Theorem 6.23 of [Schwabhauser] p. 46. (Contributed by Thierry Arnoux, 28-May-2019.)

Hypotheses

Ref	Expression
tglineintmo.p	⊢ 𝑃 = (Base‘𝐺)
tglineintmo.i	⊢ 𝐼 = (Itv‘𝐺)
tglineintmo.l	⊢ 𝐿 = (LineG‘𝐺)
tglineintmo.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
colline.1	⊢ (𝜑 → 𝑋 ∈ 𝑃)
colline.2	⊢ (𝜑 → 𝑌 ∈ 𝑃)
colline.3	⊢ (𝜑 → 𝑍 ∈ 𝑃)
colline.4	⊢ (𝜑 → 2 ≤ (♯‘𝑃))

Assertion

Ref	Expression
colline	⊢ (𝜑 → ((𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍) ↔ ∃𝑎 ∈ ran 𝐿(𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎)))

Distinct variable groups:   𝐿,𝑎   𝑋,𝑎   𝑌,𝑎   𝑍,𝑎   𝜑,𝑎

Allowed substitution hints:   𝑃(𝑎)   𝐺(𝑎)   𝐼(𝑎)

Proof of Theorem colline

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tglineintmo.p	. . . . . . . 8 ⊢ 𝑃 = (Base‘𝐺)
2		tglineintmo.i	. . . . . . . 8 ⊢ 𝐼 = (Itv‘𝐺)
3		tglineintmo.l	. . . . . . . 8 ⊢ 𝐿 = (LineG‘𝐺)
4		tglineintmo.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad4antr 732	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥 ∈ 𝑃) ∧ 𝑋 ≠ 𝑥) → 𝐺 ∈ TarskiG)
6		colline.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
7	6	ad4antr 732	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥 ∈ 𝑃) ∧ 𝑋 ≠ 𝑥) → 𝑋 ∈ 𝑃)
8		simplr 768	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥 ∈ 𝑃) ∧ 𝑋 ≠ 𝑥) → 𝑥 ∈ 𝑃)
9		simpr 484	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥 ∈ 𝑃) ∧ 𝑋 ≠ 𝑥) → 𝑋 ≠ 𝑥)
10	1, 2, 3, 5, 7, 8, 9	tgelrnln 28609	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥 ∈ 𝑃) ∧ 𝑋 ≠ 𝑥) → (𝑋𝐿𝑥) ∈ ran 𝐿)
11	1, 2, 3, 5, 7, 8, 9	tglinerflx1 28612	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥 ∈ 𝑃) ∧ 𝑋 ≠ 𝑥) → 𝑋 ∈ (𝑋𝐿𝑥))
12		simp-4r 783	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥 ∈ 𝑃) ∧ 𝑋 ≠ 𝑥) → 𝑌 = 𝑍)
13		simpllr 775	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥 ∈ 𝑃) ∧ 𝑋 ≠ 𝑥) → 𝑋 = 𝑍)
14	13, 11	eqeltrrd 2834	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥 ∈ 𝑃) ∧ 𝑋 ≠ 𝑥) → 𝑍 ∈ (𝑋𝐿𝑥))
15	12, 14	eqeltrd 2833	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥 ∈ 𝑃) ∧ 𝑋 ≠ 𝑥) → 𝑌 ∈ (𝑋𝐿𝑥))
16		eleq2 2822	. . . . . . . . 9 ⊢ (𝑎 = (𝑋𝐿𝑥) → (𝑋 ∈ 𝑎 ↔ 𝑋 ∈ (𝑋𝐿𝑥)))
17		eleq2 2822	. . . . . . . . 9 ⊢ (𝑎 = (𝑋𝐿𝑥) → (𝑌 ∈ 𝑎 ↔ 𝑌 ∈ (𝑋𝐿𝑥)))
18		eleq2 2822	. . . . . . . . 9 ⊢ (𝑎 = (𝑋𝐿𝑥) → (𝑍 ∈ 𝑎 ↔ 𝑍 ∈ (𝑋𝐿𝑥)))
19	16, 17, 18	3anbi123d 1438	. . . . . . . 8 ⊢ (𝑎 = (𝑋𝐿𝑥) → ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎) ↔ (𝑋 ∈ (𝑋𝐿𝑥) ∧ 𝑌 ∈ (𝑋𝐿𝑥) ∧ 𝑍 ∈ (𝑋𝐿𝑥))))
20	19	rspcev 3573	. . . . . . 7 ⊢ (((𝑋𝐿𝑥) ∈ ran 𝐿 ∧ (𝑋 ∈ (𝑋𝐿𝑥) ∧ 𝑌 ∈ (𝑋𝐿𝑥) ∧ 𝑍 ∈ (𝑋𝐿𝑥))) → ∃𝑎 ∈ ran 𝐿(𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎))
21	10, 11, 15, 14, 20	syl13anc 1374	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 = 𝑍) ∧ 𝑥 ∈ 𝑃) ∧ 𝑋 ≠ 𝑥) → ∃𝑎 ∈ ran 𝐿(𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎))
22		eqid 2733	. . . . . . . 8 ⊢ (dist‘𝐺) = (dist‘𝐺)
23		colline.4	. . . . . . . 8 ⊢ (𝜑 → 2 ≤ (♯‘𝑃))
24	1, 22, 2, 4, 23, 6	tglowdim1i 28480	. . . . . . 7 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 𝑋 ≠ 𝑥)
25	24	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 = 𝑍) → ∃𝑥 ∈ 𝑃 𝑋 ≠ 𝑥)
26	21, 25	r19.29a 3141	. . . . 5 ⊢ (((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 = 𝑍) → ∃𝑎 ∈ ran 𝐿(𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎))
27	4	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 ≠ 𝑍) → 𝐺 ∈ TarskiG)
28	6	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 ≠ 𝑍) → 𝑋 ∈ 𝑃)
29		colline.3	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
30	29	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 ≠ 𝑍) → 𝑍 ∈ 𝑃)
31		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 ≠ 𝑍) → 𝑋 ≠ 𝑍)
32	1, 2, 3, 27, 28, 30, 31	tgelrnln 28609	. . . . . 6 ⊢ (((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 ≠ 𝑍) → (𝑋𝐿𝑍) ∈ ran 𝐿)
33	1, 2, 3, 27, 28, 30, 31	tglinerflx1 28612	. . . . . 6 ⊢ (((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 ≠ 𝑍) → 𝑋 ∈ (𝑋𝐿𝑍))
34		simplr 768	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 ≠ 𝑍) → 𝑌 = 𝑍)
35	1, 2, 3, 27, 28, 30, 31	tglinerflx2 28613	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 ≠ 𝑍) → 𝑍 ∈ (𝑋𝐿𝑍))
36	34, 35	eqeltrd 2833	. . . . . 6 ⊢ (((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 ≠ 𝑍) → 𝑌 ∈ (𝑋𝐿𝑍))
37		eleq2 2822	. . . . . . . 8 ⊢ (𝑎 = (𝑋𝐿𝑍) → (𝑋 ∈ 𝑎 ↔ 𝑋 ∈ (𝑋𝐿𝑍)))
38		eleq2 2822	. . . . . . . 8 ⊢ (𝑎 = (𝑋𝐿𝑍) → (𝑌 ∈ 𝑎 ↔ 𝑌 ∈ (𝑋𝐿𝑍)))
39		eleq2 2822	. . . . . . . 8 ⊢ (𝑎 = (𝑋𝐿𝑍) → (𝑍 ∈ 𝑎 ↔ 𝑍 ∈ (𝑋𝐿𝑍)))
40	37, 38, 39	3anbi123d 1438	. . . . . . 7 ⊢ (𝑎 = (𝑋𝐿𝑍) → ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎) ↔ (𝑋 ∈ (𝑋𝐿𝑍) ∧ 𝑌 ∈ (𝑋𝐿𝑍) ∧ 𝑍 ∈ (𝑋𝐿𝑍))))
41	40	rspcev 3573	. . . . . 6 ⊢ (((𝑋𝐿𝑍) ∈ ran 𝐿 ∧ (𝑋 ∈ (𝑋𝐿𝑍) ∧ 𝑌 ∈ (𝑋𝐿𝑍) ∧ 𝑍 ∈ (𝑋𝐿𝑍))) → ∃𝑎 ∈ ran 𝐿(𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎))
42	32, 33, 36, 35, 41	syl13anc 1374	. . . . 5 ⊢ (((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑋 ≠ 𝑍) → ∃𝑎 ∈ ran 𝐿(𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎))
43	26, 42	pm2.61dane 3016	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → ∃𝑎 ∈ ran 𝐿(𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎))
44	43	adantlr 715	. . 3 ⊢ (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌 = 𝑍) → ∃𝑎 ∈ ran 𝐿(𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎))
45		simpll 766	. . . . 5 ⊢ (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌 ≠ 𝑍) → 𝜑)
46		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌 ≠ 𝑍) → 𝑌 ≠ 𝑍)
47	46	neneqd 2934	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌 ≠ 𝑍) → ¬ 𝑌 = 𝑍)
48		simplr 768	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌 ≠ 𝑍) → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
49		orel2 890	. . . . . 6 ⊢ (¬ 𝑌 = 𝑍 → ((𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍) → 𝑋 ∈ (𝑌𝐿𝑍)))
50	47, 48, 49	sylc 65	. . . . 5 ⊢ (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌 ≠ 𝑍) → 𝑋 ∈ (𝑌𝐿𝑍))
51	4	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑌𝐿𝑍)) ∧ 𝑌 ≠ 𝑍) → 𝐺 ∈ TarskiG)
52		colline.2	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
53	52	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑌𝐿𝑍)) ∧ 𝑌 ≠ 𝑍) → 𝑌 ∈ 𝑃)
54	29	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑌𝐿𝑍)) ∧ 𝑌 ≠ 𝑍) → 𝑍 ∈ 𝑃)
55		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑌𝐿𝑍)) ∧ 𝑌 ≠ 𝑍) → 𝑌 ≠ 𝑍)
56	1, 2, 3, 51, 53, 54, 55	tgelrnln 28609	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑌𝐿𝑍)) ∧ 𝑌 ≠ 𝑍) → (𝑌𝐿𝑍) ∈ ran 𝐿)
57	45, 50, 46, 56	syl21anc 837	. . . 4 ⊢ (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌 ≠ 𝑍) → (𝑌𝐿𝑍) ∈ ran 𝐿)
58	1, 2, 3, 51, 53, 54, 55	tglinerflx1 28612	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑌𝐿𝑍)) ∧ 𝑌 ≠ 𝑍) → 𝑌 ∈ (𝑌𝐿𝑍))
59	45, 50, 46, 58	syl21anc 837	. . . 4 ⊢ (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌 ≠ 𝑍) → 𝑌 ∈ (𝑌𝐿𝑍))
60	1, 2, 3, 51, 53, 54, 55	tglinerflx2 28613	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑌𝐿𝑍)) ∧ 𝑌 ≠ 𝑍) → 𝑍 ∈ (𝑌𝐿𝑍))
61	45, 50, 46, 60	syl21anc 837	. . . 4 ⊢ (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌 ≠ 𝑍) → 𝑍 ∈ (𝑌𝐿𝑍))
62		eleq2 2822	. . . . . 6 ⊢ (𝑎 = (𝑌𝐿𝑍) → (𝑋 ∈ 𝑎 ↔ 𝑋 ∈ (𝑌𝐿𝑍)))
63		eleq2 2822	. . . . . 6 ⊢ (𝑎 = (𝑌𝐿𝑍) → (𝑌 ∈ 𝑎 ↔ 𝑌 ∈ (𝑌𝐿𝑍)))
64		eleq2 2822	. . . . . 6 ⊢ (𝑎 = (𝑌𝐿𝑍) → (𝑍 ∈ 𝑎 ↔ 𝑍 ∈ (𝑌𝐿𝑍)))
65	62, 63, 64	3anbi123d 1438	. . . . 5 ⊢ (𝑎 = (𝑌𝐿𝑍) → ((𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎) ↔ (𝑋 ∈ (𝑌𝐿𝑍) ∧ 𝑌 ∈ (𝑌𝐿𝑍) ∧ 𝑍 ∈ (𝑌𝐿𝑍))))
66	65	rspcev 3573	. . . 4 ⊢ (((𝑌𝐿𝑍) ∈ ran 𝐿 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∧ 𝑌 ∈ (𝑌𝐿𝑍) ∧ 𝑍 ∈ (𝑌𝐿𝑍))) → ∃𝑎 ∈ ran 𝐿(𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎))
67	57, 50, 59, 61, 66	syl13anc 1374	. . 3 ⊢ (((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) ∧ 𝑌 ≠ 𝑍) → ∃𝑎 ∈ ran 𝐿(𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎))
68	44, 67	pm2.61dane 3016	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) → ∃𝑎 ∈ ran 𝐿(𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎))
69		df-ne 2930	. . . . . 6 ⊢ (𝑌 ≠ 𝑍 ↔ ¬ 𝑌 = 𝑍)
70		simplr1 1216	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎)) ∧ 𝑌 ≠ 𝑍) → 𝑋 ∈ 𝑎)
71	4	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎)) ∧ 𝑌 ≠ 𝑍) → 𝐺 ∈ TarskiG)
72	52	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎)) ∧ 𝑌 ≠ 𝑍) → 𝑌 ∈ 𝑃)
73	29	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎)) ∧ 𝑌 ≠ 𝑍) → 𝑍 ∈ 𝑃)
74		simpr 484	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎)) ∧ 𝑌 ≠ 𝑍) → 𝑌 ≠ 𝑍)
75		simpllr 775	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎)) ∧ 𝑌 ≠ 𝑍) → 𝑎 ∈ ran 𝐿)
76		simplr2 1217	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎)) ∧ 𝑌 ≠ 𝑍) → 𝑌 ∈ 𝑎)
77		simplr3 1218	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎)) ∧ 𝑌 ≠ 𝑍) → 𝑍 ∈ 𝑎)
78	1, 2, 3, 71, 72, 73, 74, 74, 75, 76, 77	tglinethru 28615	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎)) ∧ 𝑌 ≠ 𝑍) → 𝑎 = (𝑌𝐿𝑍))
79	70, 78	eleqtrd 2835	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎)) ∧ 𝑌 ≠ 𝑍) → 𝑋 ∈ (𝑌𝐿𝑍))
80	79	ex 412	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎)) → (𝑌 ≠ 𝑍 → 𝑋 ∈ (𝑌𝐿𝑍)))
81	69, 80	biimtrrid 243	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎)) → (¬ 𝑌 = 𝑍 → 𝑋 ∈ (𝑌𝐿𝑍)))
82	81	orrd 863	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎)) → (𝑌 = 𝑍 ∨ 𝑋 ∈ (𝑌𝐿𝑍)))
83	82	orcomd 871	. . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ran 𝐿) ∧ (𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎)) → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
84	83	r19.29an 3137	. 2 ⊢ ((𝜑 ∧ ∃𝑎 ∈ ran 𝐿(𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎)) → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
85	68, 84	impbida 800	1 ⊢ (𝜑 → ((𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍) ↔ ∃𝑎 ∈ ran 𝐿(𝑋 ∈ 𝑎 ∧ 𝑌 ∈ 𝑎 ∧ 𝑍 ∈ 𝑎)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2929  ∃wrex 3057   class class class wbr 5093  ran crn 5620  ‘cfv 6486  (class class class)co 7352   ≤ cle 11154  2c2 12187  ♯chash 14239  Basecbs 17122  distcds 17172  TarskiGcstrkg 28406  Itvcitv 28412  LineGclng 28413

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-tp 4580  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  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 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-oadd 8395  df-er 8628  df-pm 8759  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-dju 9801  df-card 9839  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-nn 12133  df-2 12195  df-3 12196  df-n0 12389  df-xnn0 12462  df-z 12476  df-uz 12739  df-fz 13410  df-fzo 13557  df-hash 14240  df-word 14423  df-concat 14480  df-s1 14506  df-s2 14757  df-s3 14758  df-trkgc 28427  df-trkgb 28428  df-trkgcb 28429  df-trkg 28432  df-cgrg 28490

This theorem is referenced by: (None)