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Theorem tglineeltr 26416

Description: Transitivity law for lines, one half of tglineelsb2 26417. (Contributed by Thierry Arnoux, 25-May-2019.)

**Hypotheses**
Ref	Expression
tglineelsb2.p	⊢ 𝐵 = (Base‘𝐺)
tglineelsb2.i	⊢ 𝐼 = (Itv‘𝐺)
tglineelsb2.l	⊢ 𝐿 = (LineG‘𝐺)
tglineelsb2.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tglineelsb2.1	⊢ (𝜑 → 𝑃 ∈ 𝐵)
tglineelsb2.2	⊢ (𝜑 → 𝑄 ∈ 𝐵)
tglineelsb2.4	⊢ (𝜑 → 𝑃 ≠ 𝑄)
tglineelsb2.3	⊢ (𝜑 → 𝑆 ∈ 𝐵)
tglineelsb2.5	⊢ (𝜑 → 𝑆 ≠ 𝑃)
tglineelsb2.6	⊢ (𝜑 → 𝑆 ∈ (𝑃𝐿𝑄))
tglineeltr.7	⊢ (𝜑 → 𝑅 ∈ 𝐵)
tglineeltr.8	⊢ (𝜑 → 𝑅 ∈ (𝑃𝐿𝑆))

**Assertion**
Ref	Expression
tglineeltr	⊢ (𝜑 → 𝑅 ∈ (𝑃𝐿𝑄))

**Proof of Theorem tglineeltr**
Step	Hyp	Ref	Expression
1		tglineeltr.7	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝐵)
2		tglineelsb2.p	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐺)
3		tglineelsb2.l	. . . . . . 7 ⊢ 𝐿 = (LineG‘𝐺)
4		tglineelsb2.i	. . . . . . 7 ⊢ 𝐼 = (Itv‘𝐺)
5		tglineelsb2.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
6	5	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝐺 ∈ TarskiG)
7		tglineelsb2.1	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
8	7	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑃 ∈ 𝐵)
9		tglineelsb2.2	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ 𝐵)
10	9	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑄 ∈ 𝐵)
11		simpllr 774	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑅 ∈ 𝐵)
12		eqid 2821	. . . . . . . 8 ⊢ (dist‘𝐺) = (dist‘𝐺)
13		tglineelsb2.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ 𝐵)
14	13	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑆 ∈ 𝐵)
15		simplr 767	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑅 ∈ (𝑃𝐼𝑆))
16		simpr 487	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑆 ∈ (𝑃𝐼𝑄))
17	2, 12, 4, 6, 8, 11, 14, 10, 15, 16	tgbtwnexch 26283	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑅 ∈ (𝑃𝐼𝑄))
18	2, 3, 4, 6, 8, 10, 11, 17	btwncolg1 26340	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
19	5	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝐺 ∈ TarskiG)
20	7	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ 𝐵)
21	9	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑄 ∈ 𝐵)
22		simpllr 774	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑅 ∈ 𝐵)
23	13	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑆 ∈ 𝐵)
24		simplr 767	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑅 ∈ (𝑃𝐼𝑆))
25	2, 12, 4, 19, 20, 22, 23, 24	tgbtwncom 26273	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑅 ∈ (𝑆𝐼𝑃))
26		simpr 487	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ (𝑆𝐼𝑄))
27	2, 12, 4, 19, 23, 22, 20, 21, 25, 26	tgbtwnexch3 26279	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ (𝑅𝐼𝑄))
28	2, 3, 4, 19, 20, 21, 22, 27	btwncolg2 26341	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
29	5	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝐺 ∈ TarskiG)
30	7	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑃 ∈ 𝐵)
31		simpllr 774	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑅 ∈ 𝐵)
32	9	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑄 ∈ 𝐵)
33	13	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑆 ∈ 𝐵)
34		simplr 767	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑅 ∈ (𝑃𝐼𝑆))
35		simpr 487	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑄 ∈ (𝑃𝐼𝑆))
36	2, 4, 29, 30, 31, 32, 33, 34, 35, 3	tgbtwnconnln3 26363	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
37		tglineelsb2.6	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ (𝑃𝐿𝑄))
38		tglineelsb2.4	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
39	2, 3, 4, 5, 7, 9, 38, 13	tgellng 26338	. . . . . . . 8 ⊢ (𝜑 → (𝑆 ∈ (𝑃𝐿𝑄) ↔ (𝑆 ∈ (𝑃𝐼𝑄) ∨ 𝑃 ∈ (𝑆𝐼𝑄) ∨ 𝑄 ∈ (𝑃𝐼𝑆))))
40	37, 39	mpbid 234	. . . . . . 7 ⊢ (𝜑 → (𝑆 ∈ (𝑃𝐼𝑄) ∨ 𝑃 ∈ (𝑆𝐼𝑄) ∨ 𝑄 ∈ (𝑃𝐼𝑆)))
41	40	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) → (𝑆 ∈ (𝑃𝐼𝑄) ∨ 𝑃 ∈ (𝑆𝐼𝑄) ∨ 𝑄 ∈ (𝑃𝐼𝑆)))
42	18, 28, 36, 41	mpjao3dan 1427	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
43	42	an32s 650	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑅 ∈ 𝐵) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
44	1, 43	mpidan 687	. . 3 ⊢ ((𝜑 ∧ 𝑅 ∈ (𝑃𝐼𝑆)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
45	5	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝐺 ∈ TarskiG)
46	7	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑃 ∈ 𝐵)
47	9	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑄 ∈ 𝐵)
48		simpllr 774	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑅 ∈ 𝐵)
49	13	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑆 ∈ 𝐵)
50		tglineelsb2.5	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ≠ 𝑃)
51	50	necomd 3071	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ≠ 𝑆)
52	51	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑃 ≠ 𝑆)
53		simplr 767	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑃 ∈ (𝑅𝐼𝑆))
54		simpr 487	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑆 ∈ (𝑃𝐼𝑄))
55	2, 12, 4, 45, 48, 46, 49, 47, 52, 53, 54	tgbtwnouttr 26282	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑃 ∈ (𝑅𝐼𝑄))
56	2, 3, 4, 45, 46, 47, 48, 55	btwncolg2 26341	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
57	5	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝐺 ∈ TarskiG)
58	9	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑄 ∈ 𝐵)
59		simpllr 774	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑅 ∈ 𝐵)
60	7	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ 𝐵)
61	13	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑆 ∈ 𝐵)
62	50	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑆 ≠ 𝑃)
63		simpr 487	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ (𝑆𝐼𝑄))
64		simplr 767	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ (𝑅𝐼𝑆))
65	2, 12, 4, 57, 59, 60, 61, 64	tgbtwncom 26273	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ (𝑆𝐼𝑅))
66	2, 4, 57, 61, 60, 58, 59, 3, 62, 63, 65	tgbtwnconnln2 26366	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → (𝑃 ∈ (𝑄𝐿𝑅) ∨ 𝑄 = 𝑅))
67	2, 3, 4, 57, 58, 59, 60, 66	colrot2 26345	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
68	5	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝐺 ∈ TarskiG)
69	9	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑄 ∈ 𝐵)
70	7	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑃 ∈ 𝐵)
71		simpllr 774	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑅 ∈ 𝐵)
72	13	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑆 ∈ 𝐵)
73		simpr 487	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑄 ∈ (𝑃𝐼𝑆))
74		simplr 767	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑃 ∈ (𝑅𝐼𝑆))
75	2, 12, 4, 68, 69, 70, 71, 72, 73, 74	tgbtwnintr 26278	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑃 ∈ (𝑄𝐼𝑅))
76	2, 3, 4, 68, 69, 70, 71, 75	btwncolg3 26342	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → (𝑅 ∈ (𝑄𝐿𝑃) ∨ 𝑄 = 𝑃))
77	2, 3, 4, 68, 69, 70, 71, 76	colcom 26343	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
78	40	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) → (𝑆 ∈ (𝑃𝐼𝑄) ∨ 𝑃 ∈ (𝑆𝐼𝑄) ∨ 𝑄 ∈ (𝑃𝐼𝑆)))
79	56, 67, 77, 78	mpjao3dan 1427	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
80	79	an32s 650	. . . 4 ⊢ (((𝜑 ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑅 ∈ 𝐵) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
81	1, 80	mpidan 687	. . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ (𝑅𝐼𝑆)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
82	5	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝐺 ∈ TarskiG)
83	9	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑄 ∈ 𝐵)
84		simpllr 774	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑅 ∈ 𝐵)
85	7	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑃 ∈ 𝐵)
86	13	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑆 ∈ 𝐵)
87	51	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑃 ≠ 𝑆)
88		simpr 487	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑆 ∈ (𝑃𝐼𝑄))
89		simplr 767	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑆 ∈ (𝑃𝐼𝑅))
90	2, 4, 82, 85, 86, 83, 84, 3, 87, 88, 89	tgbtwnconnln1 26365	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → (𝑃 ∈ (𝑄𝐿𝑅) ∨ 𝑄 = 𝑅))
91	2, 3, 4, 82, 83, 84, 85, 90	colrot2 26345	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
92	5	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝐺 ∈ TarskiG)
93	7	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ 𝐵)
94	9	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑄 ∈ 𝐵)
95		simpllr 774	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑅 ∈ 𝐵)
96	13	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑆 ∈ 𝐵)
97	50	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑆 ≠ 𝑃)
98		simplr 767	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑆 ∈ (𝑃𝐼𝑅))
99	2, 12, 4, 92, 93, 96, 95, 98	tgbtwncom 26273	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑆 ∈ (𝑅𝐼𝑃))
100		simpr 487	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ (𝑆𝐼𝑄))
101	2, 12, 4, 92, 95, 96, 93, 94, 97, 99, 100	tgbtwnouttr2 26280	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ (𝑅𝐼𝑄))
102	2, 3, 4, 92, 93, 94, 95, 101	btwncolg2 26341	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
103	5	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝐺 ∈ TarskiG)
104	7	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑃 ∈ 𝐵)
105	9	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑄 ∈ 𝐵)
106		simpllr 774	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑅 ∈ 𝐵)
107	13	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑆 ∈ 𝐵)
108		simpr 487	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑄 ∈ (𝑃𝐼𝑆))
109		simplr 767	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑆 ∈ (𝑃𝐼𝑅))
110	2, 12, 4, 103, 104, 105, 107, 106, 108, 109	tgbtwnexch 26283	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑄 ∈ (𝑃𝐼𝑅))
111	2, 3, 4, 103, 104, 105, 106, 110	btwncolg3 26342	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
112	40	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) → (𝑆 ∈ (𝑃𝐼𝑄) ∨ 𝑃 ∈ (𝑆𝐼𝑄) ∨ 𝑄 ∈ (𝑃𝐼𝑆)))
113	91, 102, 111, 112	mpjao3dan 1427	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
114	113	an32s 650	. . . 4 ⊢ (((𝜑 ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑅 ∈ 𝐵) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
115	1, 114	mpidan 687	. . 3 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑃𝐼𝑅)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
116		id 22	. . . 4 ⊢ (𝜑 → 𝜑)
117		tglineeltr.8	. . . 4 ⊢ (𝜑 → 𝑅 ∈ (𝑃𝐿𝑆))
118	5	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ∈ 𝐵) → 𝐺 ∈ TarskiG)
119	7	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ∈ 𝐵) → 𝑃 ∈ 𝐵)
120	13	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ∈ 𝐵) → 𝑆 ∈ 𝐵)
121	51	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ∈ 𝐵) → 𝑃 ≠ 𝑆)
122		simpr 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ∈ 𝐵) → 𝑅 ∈ 𝐵)
123	2, 3, 4, 118, 119, 120, 121, 122	tgellng 26338	. . . . 5 ⊢ ((𝜑 ∧ 𝑅 ∈ 𝐵) → (𝑅 ∈ (𝑃𝐿𝑆) ↔ (𝑅 ∈ (𝑃𝐼𝑆) ∨ 𝑃 ∈ (𝑅𝐼𝑆) ∨ 𝑆 ∈ (𝑃𝐼𝑅))))
124	123	biimpa 479	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐿𝑆)) → (𝑅 ∈ (𝑃𝐼𝑆) ∨ 𝑃 ∈ (𝑅𝐼𝑆) ∨ 𝑆 ∈ (𝑃𝐼𝑅)))
125	116, 1, 117, 124	syl21anc 835	. . 3 ⊢ (𝜑 → (𝑅 ∈ (𝑃𝐼𝑆) ∨ 𝑃 ∈ (𝑅𝐼𝑆) ∨ 𝑆 ∈ (𝑃𝐼𝑅)))
126	44, 81, 115, 125	mpjao3dan 1427	. 2 ⊢ (𝜑 → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
127	38	neneqd 3021	. 2 ⊢ (𝜑 → ¬ 𝑃 = 𝑄)
128		pm5.61 997	. . 3 ⊢ (((𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄) ∧ ¬ 𝑃 = 𝑄) ↔ (𝑅 ∈ (𝑃𝐿𝑄) ∧ ¬ 𝑃 = 𝑄))
129	128	simplbi 500	. 2 ⊢ (((𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄) ∧ ¬ 𝑃 = 𝑄) → 𝑅 ∈ (𝑃𝐿𝑄))
130	126, 127, 129	syl2anc 586	1 ⊢ (𝜑 → 𝑅 ∈ (𝑃𝐿𝑄))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∨ wo 843 ∨ w3o 1082 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 distcds 16573 TarskiGcstrkg 26215 Itvcitv 26221 LineGclng 26222

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-pm 8408 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-dju 9329 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-n0 11897 df-xnn0 11967 df-z 11981 df-uz 12243 df-fz 12892 df-fzo 13033 df-hash 13690 df-word 13861 df-concat 13922 df-s1 13949 df-s2 14209 df-s3 14210 df-trkgc 26233 df-trkgb 26234 df-trkgcb 26235 df-trkg 26238 df-cgrg 26296

This theorem is referenced by: tglineelsb2 26417 colperpexlem3 26517 mideulem2 26519

W3C validator