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Theorem tglineeltr 28576

Description: Transitivity law for lines, one half of tglineelsb2 28577. (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 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝐺 ∈ TarskiG)
7		tglineelsb2.1	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
8	7	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑃 ∈ 𝐵)
9		tglineelsb2.2	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ 𝐵)
10	9	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑄 ∈ 𝐵)
11		simpllr 775	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑅 ∈ 𝐵)
12		eqid 2729	. . . . . . . 8 ⊢ (dist‘𝐺) = (dist‘𝐺)
13		tglineelsb2.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ 𝐵)
14	13	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑆 ∈ 𝐵)
15		simplr 768	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑅 ∈ (𝑃𝐼𝑆))
16		simpr 484	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑆 ∈ (𝑃𝐼𝑄))
17	2, 12, 4, 6, 8, 11, 14, 10, 15, 16	tgbtwnexch 28443	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑅 ∈ (𝑃𝐼𝑄))
18	2, 3, 4, 6, 8, 10, 11, 17	btwncolg1 28500	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
19	5	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝐺 ∈ TarskiG)
20	7	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ 𝐵)
21	9	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑄 ∈ 𝐵)
22		simpllr 775	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑅 ∈ 𝐵)
23	13	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑆 ∈ 𝐵)
24		simplr 768	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑅 ∈ (𝑃𝐼𝑆))
25	2, 12, 4, 19, 20, 22, 23, 24	tgbtwncom 28433	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑅 ∈ (𝑆𝐼𝑃))
26		simpr 484	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ (𝑆𝐼𝑄))
27	2, 12, 4, 19, 23, 22, 20, 21, 25, 26	tgbtwnexch3 28439	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ (𝑅𝐼𝑄))
28	2, 3, 4, 19, 20, 21, 22, 27	btwncolg2 28501	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
29	5	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝐺 ∈ TarskiG)
30	7	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑃 ∈ 𝐵)
31		simpllr 775	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑅 ∈ 𝐵)
32	9	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑄 ∈ 𝐵)
33	13	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑆 ∈ 𝐵)
34		simplr 768	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑅 ∈ (𝑃𝐼𝑆))
35		simpr 484	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑄 ∈ (𝑃𝐼𝑆))
36	2, 4, 29, 30, 31, 32, 33, 34, 35, 3	tgbtwnconnln3 28523	. . . . . 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 28498	. . . . . . . 8 ⊢ (𝜑 → (𝑆 ∈ (𝑃𝐿𝑄) ↔ (𝑆 ∈ (𝑃𝐼𝑄) ∨ 𝑃 ∈ (𝑆𝐼𝑄) ∨ 𝑄 ∈ (𝑃𝐼𝑆))))
40	37, 39	mpbid 232	. . . . . . 7 ⊢ (𝜑 → (𝑆 ∈ (𝑃𝐼𝑄) ∨ 𝑃 ∈ (𝑆𝐼𝑄) ∨ 𝑄 ∈ (𝑃𝐼𝑆)))
41	40	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) → (𝑆 ∈ (𝑃𝐼𝑄) ∨ 𝑃 ∈ (𝑆𝐼𝑄) ∨ 𝑄 ∈ (𝑃𝐼𝑆)))
42	18, 28, 36, 41	mpjao3dan 1434	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
43	42	an32s 652	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑅 ∈ 𝐵) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
44	1, 43	mpidan 689	. . 3 ⊢ ((𝜑 ∧ 𝑅 ∈ (𝑃𝐼𝑆)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
45	5	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝐺 ∈ TarskiG)
46	7	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑃 ∈ 𝐵)
47	9	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑄 ∈ 𝐵)
48		simpllr 775	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑅 ∈ 𝐵)
49	13	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑆 ∈ 𝐵)
50		tglineelsb2.5	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ≠ 𝑃)
51	50	necomd 2980	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ≠ 𝑆)
52	51	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑃 ≠ 𝑆)
53		simplr 768	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑃 ∈ (𝑅𝐼𝑆))
54		simpr 484	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑆 ∈ (𝑃𝐼𝑄))
55	2, 12, 4, 45, 48, 46, 49, 47, 52, 53, 54	tgbtwnouttr 28442	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑃 ∈ (𝑅𝐼𝑄))
56	2, 3, 4, 45, 46, 47, 48, 55	btwncolg2 28501	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
57	5	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝐺 ∈ TarskiG)
58	9	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑄 ∈ 𝐵)
59		simpllr 775	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑅 ∈ 𝐵)
60	7	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ 𝐵)
61	13	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑆 ∈ 𝐵)
62	50	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑆 ≠ 𝑃)
63		simpr 484	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ (𝑆𝐼𝑄))
64		simplr 768	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ (𝑅𝐼𝑆))
65	2, 12, 4, 57, 59, 60, 61, 64	tgbtwncom 28433	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ (𝑆𝐼𝑅))
66	2, 4, 57, 61, 60, 58, 59, 3, 62, 63, 65	tgbtwnconnln2 28526	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → (𝑃 ∈ (𝑄𝐿𝑅) ∨ 𝑄 = 𝑅))
67	2, 3, 4, 57, 58, 59, 60, 66	colrot2 28505	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
68	5	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝐺 ∈ TarskiG)
69	9	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑄 ∈ 𝐵)
70	7	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑃 ∈ 𝐵)
71		simpllr 775	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑅 ∈ 𝐵)
72	13	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑆 ∈ 𝐵)
73		simpr 484	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑄 ∈ (𝑃𝐼𝑆))
74		simplr 768	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑃 ∈ (𝑅𝐼𝑆))
75	2, 12, 4, 68, 69, 70, 71, 72, 73, 74	tgbtwnintr 28438	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑃 ∈ (𝑄𝐼𝑅))
76	2, 3, 4, 68, 69, 70, 71, 75	btwncolg3 28502	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → (𝑅 ∈ (𝑄𝐿𝑃) ∨ 𝑄 = 𝑃))
77	2, 3, 4, 68, 69, 70, 71, 76	colcom 28503	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
78	40	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) → (𝑆 ∈ (𝑃𝐼𝑄) ∨ 𝑃 ∈ (𝑆𝐼𝑄) ∨ 𝑄 ∈ (𝑃𝐼𝑆)))
79	56, 67, 77, 78	mpjao3dan 1434	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
80	79	an32s 652	. . . 4 ⊢ (((𝜑 ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑅 ∈ 𝐵) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
81	1, 80	mpidan 689	. . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ (𝑅𝐼𝑆)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
82	5	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝐺 ∈ TarskiG)
83	9	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑄 ∈ 𝐵)
84		simpllr 775	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑅 ∈ 𝐵)
85	7	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑃 ∈ 𝐵)
86	13	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑆 ∈ 𝐵)
87	51	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑃 ≠ 𝑆)
88		simpr 484	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑆 ∈ (𝑃𝐼𝑄))
89		simplr 768	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑆 ∈ (𝑃𝐼𝑅))
90	2, 4, 82, 85, 86, 83, 84, 3, 87, 88, 89	tgbtwnconnln1 28525	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → (𝑃 ∈ (𝑄𝐿𝑅) ∨ 𝑄 = 𝑅))
91	2, 3, 4, 82, 83, 84, 85, 90	colrot2 28505	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
92	5	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝐺 ∈ TarskiG)
93	7	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ 𝐵)
94	9	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑄 ∈ 𝐵)
95		simpllr 775	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑅 ∈ 𝐵)
96	13	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑆 ∈ 𝐵)
97	50	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑆 ≠ 𝑃)
98		simplr 768	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑆 ∈ (𝑃𝐼𝑅))
99	2, 12, 4, 92, 93, 96, 95, 98	tgbtwncom 28433	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑆 ∈ (𝑅𝐼𝑃))
100		simpr 484	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ (𝑆𝐼𝑄))
101	2, 12, 4, 92, 95, 96, 93, 94, 97, 99, 100	tgbtwnouttr2 28440	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ (𝑅𝐼𝑄))
102	2, 3, 4, 92, 93, 94, 95, 101	btwncolg2 28501	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
103	5	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝐺 ∈ TarskiG)
104	7	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑃 ∈ 𝐵)
105	9	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑄 ∈ 𝐵)
106		simpllr 775	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑅 ∈ 𝐵)
107	13	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑆 ∈ 𝐵)
108		simpr 484	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑄 ∈ (𝑃𝐼𝑆))
109		simplr 768	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑆 ∈ (𝑃𝐼𝑅))
110	2, 12, 4, 103, 104, 105, 107, 106, 108, 109	tgbtwnexch 28443	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑄 ∈ (𝑃𝐼𝑅))
111	2, 3, 4, 103, 104, 105, 106, 110	btwncolg3 28502	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
112	40	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) → (𝑆 ∈ (𝑃𝐼𝑄) ∨ 𝑃 ∈ (𝑆𝐼𝑄) ∨ 𝑄 ∈ (𝑃𝐼𝑆)))
113	91, 102, 111, 112	mpjao3dan 1434	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
114	113	an32s 652	. . . 4 ⊢ (((𝜑 ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑅 ∈ 𝐵) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
115	1, 114	mpidan 689	. . 3 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑃𝐼𝑅)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
116		id 22	. . . 4 ⊢ (𝜑 → 𝜑)
117		tglineeltr.8	. . . 4 ⊢ (𝜑 → 𝑅 ∈ (𝑃𝐿𝑆))
118	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ∈ 𝐵) → 𝐺 ∈ TarskiG)
119	7	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ∈ 𝐵) → 𝑃 ∈ 𝐵)
120	13	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ∈ 𝐵) → 𝑆 ∈ 𝐵)
121	51	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ∈ 𝐵) → 𝑃 ≠ 𝑆)
122		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ∈ 𝐵) → 𝑅 ∈ 𝐵)
123	2, 3, 4, 118, 119, 120, 121, 122	tgellng 28498	. . . . 5 ⊢ ((𝜑 ∧ 𝑅 ∈ 𝐵) → (𝑅 ∈ (𝑃𝐿𝑆) ↔ (𝑅 ∈ (𝑃𝐼𝑆) ∨ 𝑃 ∈ (𝑅𝐼𝑆) ∨ 𝑆 ∈ (𝑃𝐼𝑅))))
124	123	biimpa 476	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐿𝑆)) → (𝑅 ∈ (𝑃𝐼𝑆) ∨ 𝑃 ∈ (𝑅𝐼𝑆) ∨ 𝑆 ∈ (𝑃𝐼𝑅)))
125	116, 1, 117, 124	syl21anc 837	. . 3 ⊢ (𝜑 → (𝑅 ∈ (𝑃𝐼𝑆) ∨ 𝑃 ∈ (𝑅𝐼𝑆) ∨ 𝑆 ∈ (𝑃𝐼𝑅)))
126	44, 81, 115, 125	mpjao3dan 1434	. 2 ⊢ (𝜑 → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
127	38	neneqd 2930	. 2 ⊢ (𝜑 → ¬ 𝑃 = 𝑄)
128		pm5.61 1002	. . 3 ⊢ (((𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄) ∧ ¬ 𝑃 = 𝑄) ↔ (𝑅 ∈ (𝑃𝐿𝑄) ∧ ¬ 𝑃 = 𝑄))
129	128	simplbi 497	. 2 ⊢ (((𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄) ∧ ¬ 𝑃 = 𝑄) → 𝑅 ∈ (𝑃𝐿𝑄))
130	126, 127, 129	syl2anc 584	1 ⊢ (𝜑 → 𝑅 ∈ (𝑃𝐿𝑄))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 847 ∨ w3o 1085 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ‘cfv 6482 (class class class)co 7349 Basecbs 17120 distcds 17170 TarskiGcstrkg 28372 Itvcitv 28378 LineGclng 28379

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086

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 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 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 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-oadd 8392 df-er 8625 df-pm 8756 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-dju 9797 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-nn 12129 df-2 12191 df-3 12192 df-n0 12385 df-xnn0 12458 df-z 12472 df-uz 12736 df-fz 13411 df-fzo 13558 df-hash 14238 df-word 14421 df-concat 14478 df-s1 14503 df-s2 14755 df-s3 14756 df-trkgc 28393 df-trkgb 28394 df-trkgcb 28395 df-trkg 28398 df-cgrg 28456

This theorem is referenced by: tglineelsb2 28577 colperpexlem3 28677 mideulem2 28679

W3C validator