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Theorem tglineeltr 28802

Description: Transitivity law for lines, one half of tglineelsb2 28803. (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 740	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝐺 ∈ TarskiG)
7		tglineelsb2.1	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
8	7	ad3antrrr 740	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑃 ∈ 𝐵)
9		tglineelsb2.2	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ 𝐵)
10	9	ad3antrrr 740	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑄 ∈ 𝐵)
11		simpllr 785	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑅 ∈ 𝐵)
12		eqid 2764	. . . . . . . 8 ⊢ (dist‘𝐺) = (dist‘𝐺)
13		tglineelsb2.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ 𝐵)
14	13	ad3antrrr 740	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑆 ∈ 𝐵)
15		simplr 778	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑅 ∈ (𝑃𝐼𝑆))
16		simpr 488	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑆 ∈ (𝑃𝐼𝑄))
17	2, 12, 4, 6, 8, 11, 14, 10, 15, 16	tgbtwnexch 28669	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑅 ∈ (𝑃𝐼𝑄))
18	2, 3, 4, 6, 8, 10, 11, 17	btwncolg1 28726	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
19	5	ad3antrrr 740	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝐺 ∈ TarskiG)
20	7	ad3antrrr 740	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ 𝐵)
21	9	ad3antrrr 740	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑄 ∈ 𝐵)
22		simpllr 785	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑅 ∈ 𝐵)
23	13	ad3antrrr 740	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑆 ∈ 𝐵)
24		simplr 778	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑅 ∈ (𝑃𝐼𝑆))
25	2, 12, 4, 19, 20, 22, 23, 24	tgbtwncom 28659	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑅 ∈ (𝑆𝐼𝑃))
26		simpr 488	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ (𝑆𝐼𝑄))
27	2, 12, 4, 19, 23, 22, 20, 21, 25, 26	tgbtwnexch3 28665	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ (𝑅𝐼𝑄))
28	2, 3, 4, 19, 20, 21, 22, 27	btwncolg2 28727	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
29	5	ad3antrrr 740	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝐺 ∈ TarskiG)
30	7	ad3antrrr 740	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑃 ∈ 𝐵)
31		simpllr 785	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑅 ∈ 𝐵)
32	9	ad3antrrr 740	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑄 ∈ 𝐵)
33	13	ad3antrrr 740	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑆 ∈ 𝐵)
34		simplr 778	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑅 ∈ (𝑃𝐼𝑆))
35		simpr 488	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑄 ∈ (𝑃𝐼𝑆))
36	2, 4, 29, 30, 31, 32, 33, 34, 35, 3	tgbtwnconnln3 28749	. . . . . 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 28724	. . . . . . . 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 736	. . . . . 6 ⊢ (((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) → (𝑆 ∈ (𝑃𝐼𝑄) ∨ 𝑃 ∈ (𝑆𝐼𝑄) ∨ 𝑄 ∈ (𝑃𝐼𝑆)))
42	18, 28, 36, 41	mpjao3dan 1454	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
43	42	an32s 662	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑅 ∈ 𝐵) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
44	1, 43	mpidan 699	. . 3 ⊢ ((𝜑 ∧ 𝑅 ∈ (𝑃𝐼𝑆)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
45	5	ad3antrrr 740	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝐺 ∈ TarskiG)
46	7	ad3antrrr 740	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑃 ∈ 𝐵)
47	9	ad3antrrr 740	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑄 ∈ 𝐵)
48		simpllr 785	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑅 ∈ 𝐵)
49	13	ad3antrrr 740	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑆 ∈ 𝐵)
50		tglineelsb2.5	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ≠ 𝑃)
51	50	necomd 3014	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ≠ 𝑆)
52	51	ad3antrrr 740	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑃 ≠ 𝑆)
53		simplr 778	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑃 ∈ (𝑅𝐼𝑆))
54		simpr 488	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑆 ∈ (𝑃𝐼𝑄))
55	2, 12, 4, 45, 48, 46, 49, 47, 52, 53, 54	tgbtwnouttr 28668	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑃 ∈ (𝑅𝐼𝑄))
56	2, 3, 4, 45, 46, 47, 48, 55	btwncolg2 28727	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
57	5	ad3antrrr 740	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝐺 ∈ TarskiG)
58	9	ad3antrrr 740	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑄 ∈ 𝐵)
59		simpllr 785	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑅 ∈ 𝐵)
60	7	ad3antrrr 740	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ 𝐵)
61	13	ad3antrrr 740	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑆 ∈ 𝐵)
62	50	ad3antrrr 740	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑆 ≠ 𝑃)
63		simpr 488	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ (𝑆𝐼𝑄))
64		simplr 778	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ (𝑅𝐼𝑆))
65	2, 12, 4, 57, 59, 60, 61, 64	tgbtwncom 28659	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ (𝑆𝐼𝑅))
66	2, 4, 57, 61, 60, 58, 59, 3, 62, 63, 65	tgbtwnconnln2 28752	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → (𝑃 ∈ (𝑄𝐿𝑅) ∨ 𝑄 = 𝑅))
67	2, 3, 4, 57, 58, 59, 60, 66	colrot2 28731	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
68	5	ad3antrrr 740	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝐺 ∈ TarskiG)
69	9	ad3antrrr 740	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑄 ∈ 𝐵)
70	7	ad3antrrr 740	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑃 ∈ 𝐵)
71		simpllr 785	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑅 ∈ 𝐵)
72	13	ad3antrrr 740	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑆 ∈ 𝐵)
73		simpr 488	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑄 ∈ (𝑃𝐼𝑆))
74		simplr 778	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑃 ∈ (𝑅𝐼𝑆))
75	2, 12, 4, 68, 69, 70, 71, 72, 73, 74	tgbtwnintr 28664	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑃 ∈ (𝑄𝐼𝑅))
76	2, 3, 4, 68, 69, 70, 71, 75	btwncolg3 28728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → (𝑅 ∈ (𝑄𝐿𝑃) ∨ 𝑄 = 𝑃))
77	2, 3, 4, 68, 69, 70, 71, 76	colcom 28729	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
78	40	ad2antrr 736	. . . . . 6 ⊢ (((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) → (𝑆 ∈ (𝑃𝐼𝑄) ∨ 𝑃 ∈ (𝑆𝐼𝑄) ∨ 𝑄 ∈ (𝑃𝐼𝑆)))
79	56, 67, 77, 78	mpjao3dan 1454	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
80	79	an32s 662	. . . 4 ⊢ (((𝜑 ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑅 ∈ 𝐵) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
81	1, 80	mpidan 699	. . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ (𝑅𝐼𝑆)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
82	5	ad3antrrr 740	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝐺 ∈ TarskiG)
83	9	ad3antrrr 740	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑄 ∈ 𝐵)
84		simpllr 785	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑅 ∈ 𝐵)
85	7	ad3antrrr 740	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑃 ∈ 𝐵)
86	13	ad3antrrr 740	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑆 ∈ 𝐵)
87	51	ad3antrrr 740	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑃 ≠ 𝑆)
88		simpr 488	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑆 ∈ (𝑃𝐼𝑄))
89		simplr 778	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑆 ∈ (𝑃𝐼𝑅))
90	2, 4, 82, 85, 86, 83, 84, 3, 87, 88, 89	tgbtwnconnln1 28751	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → (𝑃 ∈ (𝑄𝐿𝑅) ∨ 𝑄 = 𝑅))
91	2, 3, 4, 82, 83, 84, 85, 90	colrot2 28731	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
92	5	ad3antrrr 740	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝐺 ∈ TarskiG)
93	7	ad3antrrr 740	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ 𝐵)
94	9	ad3antrrr 740	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑄 ∈ 𝐵)
95		simpllr 785	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑅 ∈ 𝐵)
96	13	ad3antrrr 740	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑆 ∈ 𝐵)
97	50	ad3antrrr 740	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑆 ≠ 𝑃)
98		simplr 778	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑆 ∈ (𝑃𝐼𝑅))
99	2, 12, 4, 92, 93, 96, 95, 98	tgbtwncom 28659	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑆 ∈ (𝑅𝐼𝑃))
100		simpr 488	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ (𝑆𝐼𝑄))
101	2, 12, 4, 92, 95, 96, 93, 94, 97, 99, 100	tgbtwnouttr2 28666	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ (𝑅𝐼𝑄))
102	2, 3, 4, 92, 93, 94, 95, 101	btwncolg2 28727	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
103	5	ad3antrrr 740	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝐺 ∈ TarskiG)
104	7	ad3antrrr 740	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑃 ∈ 𝐵)
105	9	ad3antrrr 740	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑄 ∈ 𝐵)
106		simpllr 785	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑅 ∈ 𝐵)
107	13	ad3antrrr 740	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑆 ∈ 𝐵)
108		simpr 488	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑄 ∈ (𝑃𝐼𝑆))
109		simplr 778	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑆 ∈ (𝑃𝐼𝑅))
110	2, 12, 4, 103, 104, 105, 107, 106, 108, 109	tgbtwnexch 28669	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑄 ∈ (𝑃𝐼𝑅))
111	2, 3, 4, 103, 104, 105, 106, 110	btwncolg3 28728	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
112	40	ad2antrr 736	. . . . . 6 ⊢ (((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) → (𝑆 ∈ (𝑃𝐼𝑄) ∨ 𝑃 ∈ (𝑆𝐼𝑄) ∨ 𝑄 ∈ (𝑃𝐼𝑆)))
113	91, 102, 111, 112	mpjao3dan 1454	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
114	113	an32s 662	. . . 4 ⊢ (((𝜑 ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑅 ∈ 𝐵) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
115	1, 114	mpidan 699	. . 3 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑃𝐼𝑅)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
116		id 22	. . . 4 ⊢ (𝜑 → 𝜑)
117		tglineeltr.8	. . . 4 ⊢ (𝜑 → 𝑅 ∈ (𝑃𝐿𝑆))
118	5	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ∈ 𝐵) → 𝐺 ∈ TarskiG)
119	7	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ∈ 𝐵) → 𝑃 ∈ 𝐵)
120	13	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ∈ 𝐵) → 𝑆 ∈ 𝐵)
121	51	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ∈ 𝐵) → 𝑃 ≠ 𝑆)
122		simpr 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ∈ 𝐵) → 𝑅 ∈ 𝐵)
123	2, 3, 4, 118, 119, 120, 121, 122	tgellng 28724	. . . . 5 ⊢ ((𝜑 ∧ 𝑅 ∈ 𝐵) → (𝑅 ∈ (𝑃𝐿𝑆) ↔ (𝑅 ∈ (𝑃𝐼𝑆) ∨ 𝑃 ∈ (𝑅𝐼𝑆) ∨ 𝑆 ∈ (𝑃𝐼𝑅))))
124	123	biimpa 480	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐿𝑆)) → (𝑅 ∈ (𝑃𝐼𝑆) ∨ 𝑃 ∈ (𝑅𝐼𝑆) ∨ 𝑆 ∈ (𝑃𝐼𝑅)))
125	116, 1, 117, 124	syl21anc 848	. . 3 ⊢ (𝜑 → (𝑅 ∈ (𝑃𝐼𝑆) ∨ 𝑃 ∈ (𝑅𝐼𝑆) ∨ 𝑆 ∈ (𝑃𝐼𝑅)))
126	44, 81, 115, 125	mpjao3dan 1454	. 2 ⊢ (𝜑 → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
127	38	neneqd 2964	. 2 ⊢ (𝜑 → ¬ 𝑃 = 𝑄)
128		pm5.61 1014	. . 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 593	1 ⊢ (𝜑 → 𝑅 ∈ (𝑃𝐿𝑄))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∨ wo 858 ∨ w3o 1098 = wceq 1562 ∈ wcel 2144 ≠ wne 2959 ‘cfv 6523 (class class class)co 7398 Basecbs 17247 distcds 17297 TarskiGcstrkg 28598 Itvcitv 28604 LineGclng 28605

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-rep 5229 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3064 df-ral 3079 df-rex 3089 df-reu 3370 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5544 df-eprel 5549 df-po 5557 df-so 5558 df-fr 5602 df-we 5604 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-pred 6290 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-riota 7355 df-ov 7401 df-oprab 7402 df-mpo 7403 df-om 7849 df-1st 7972 df-2nd 7973 df-frecs 8264 df-wrecs 8295 df-recs 8344 df-rdg 8383 df-1o 8439 df-oadd 8443 df-er 8680 df-pm 8813 df-en 8930 df-dom 8931 df-sdom 8932 df-fin 8933 df-dju 9861 df-card 9899 df-pnf 11220 df-mnf 11221 df-xr 11222 df-ltxr 11223 df-le 11224 df-sub 11418 df-neg 11419 df-nn 12213 df-2 12282 df-3 12283 df-n0 12484 df-xnn0 12557 df-z 12571 df-uz 12842 df-fz 13515 df-fzo 13662 df-hash 14346 df-word 14529 df-concat 14586 df-s1 14612 df-s2 14863 df-s3 14864 df-trkgc 28619 df-trkgb 28620 df-trkgcb 28621 df-trkg 28624 df-cgrg 28682

This theorem is referenced by: tglineelsb2 28803 colperpexlem3 28907 mideulem2 28909 morleylemrneab 34967

W3C validator