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Theorem tglineeltr 28974

Description: Transitivity law for lines, one half of tglineelsb2 28975. (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 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝐺 ∈ TarskiG)
7		tglineelsb2.1	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
8	7	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑃 ∈ 𝐵)
9		tglineelsb2.2	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ 𝐵)
10	9	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑄 ∈ 𝐵)
11		simpllr 788	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑅 ∈ 𝐵)
12		eqid 2762	. . . . . . . 8 ⊢ (dist‘𝐺) = (dist‘𝐺)
13		tglineelsb2.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ 𝐵)
14	13	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑆 ∈ 𝐵)
15		simplr 781	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑅 ∈ (𝑃𝐼𝑆))
16		simpr 490	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑆 ∈ (𝑃𝐼𝑄))
17	2, 12, 4, 6, 8, 11, 14, 10, 15, 16	tgbtwnexch 28836	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑅 ∈ (𝑃𝐼𝑄))
18	2, 3, 4, 6, 8, 10, 11, 17	btwncolg1 28893	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
19	5	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝐺 ∈ TarskiG)
20	7	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ 𝐵)
21	9	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑄 ∈ 𝐵)
22		simpllr 788	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑅 ∈ 𝐵)
23	13	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑆 ∈ 𝐵)
24		simplr 781	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑅 ∈ (𝑃𝐼𝑆))
25	2, 12, 4, 19, 20, 22, 23, 24	tgbtwncom 28826	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑅 ∈ (𝑆𝐼𝑃))
26		simpr 490	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ (𝑆𝐼𝑄))
27	2, 12, 4, 19, 23, 22, 20, 21, 25, 26	tgbtwnexch3 28832	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ (𝑅𝐼𝑄))
28	2, 3, 4, 19, 20, 21, 22, 27	btwncolg2 28894	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
29	5	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝐺 ∈ TarskiG)
30	7	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑃 ∈ 𝐵)
31		simpllr 788	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑅 ∈ 𝐵)
32	9	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑄 ∈ 𝐵)
33	13	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑆 ∈ 𝐵)
34		simplr 781	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑅 ∈ (𝑃𝐼𝑆))
35		simpr 490	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑄 ∈ (𝑃𝐼𝑆))
36	2, 4, 29, 30, 31, 32, 33, 34, 35, 3	tgbtwnconnln3 28916	. . . . . 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 28891	. . . . . . . 8 ⊢ (𝜑 → (𝑆 ∈ (𝑃𝐿𝑄) ↔ (𝑆 ∈ (𝑃𝐼𝑄) ∨ 𝑃 ∈ (𝑆𝐼𝑄) ∨ 𝑄 ∈ (𝑃𝐼𝑆))))
40	37, 39	mpbid 235	. . . . . . 7 ⊢ (𝜑 → (𝑆 ∈ (𝑃𝐼𝑄) ∨ 𝑃 ∈ (𝑆𝐼𝑄) ∨ 𝑄 ∈ (𝑃𝐼𝑆)))
41	40	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) → (𝑆 ∈ (𝑃𝐼𝑄) ∨ 𝑃 ∈ (𝑆𝐼𝑄) ∨ 𝑄 ∈ (𝑃𝐼𝑆)))
42	18, 28, 36, 41	mpjao3dan 1459	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
43	42	an32s 665	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑅 ∈ 𝐵) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
44	1, 43	mpidan 702	. . 3 ⊢ ((𝜑 ∧ 𝑅 ∈ (𝑃𝐼𝑆)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
45	5	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝐺 ∈ TarskiG)
46	7	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑃 ∈ 𝐵)
47	9	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑄 ∈ 𝐵)
48		simpllr 788	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑅 ∈ 𝐵)
49	13	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑆 ∈ 𝐵)
50		tglineelsb2.5	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ≠ 𝑃)
51	50	necomd 3012	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ≠ 𝑆)
52	51	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑃 ≠ 𝑆)
53		simplr 781	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑃 ∈ (𝑅𝐼𝑆))
54		simpr 490	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑆 ∈ (𝑃𝐼𝑄))
55	2, 12, 4, 45, 48, 46, 49, 47, 52, 53, 54	tgbtwnouttr 28835	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑃 ∈ (𝑅𝐼𝑄))
56	2, 3, 4, 45, 46, 47, 48, 55	btwncolg2 28894	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
57	5	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝐺 ∈ TarskiG)
58	9	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑄 ∈ 𝐵)
59		simpllr 788	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑅 ∈ 𝐵)
60	7	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ 𝐵)
61	13	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑆 ∈ 𝐵)
62	50	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑆 ≠ 𝑃)
63		simpr 490	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ (𝑆𝐼𝑄))
64		simplr 781	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ (𝑅𝐼𝑆))
65	2, 12, 4, 57, 59, 60, 61, 64	tgbtwncom 28826	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ (𝑆𝐼𝑅))
66	2, 4, 57, 61, 60, 58, 59, 3, 62, 63, 65	tgbtwnconnln2 28919	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → (𝑃 ∈ (𝑄𝐿𝑅) ∨ 𝑄 = 𝑅))
67	2, 3, 4, 57, 58, 59, 60, 66	colrot2 28898	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
68	5	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝐺 ∈ TarskiG)
69	9	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑄 ∈ 𝐵)
70	7	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑃 ∈ 𝐵)
71		simpllr 788	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑅 ∈ 𝐵)
72	13	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑆 ∈ 𝐵)
73		simpr 490	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑄 ∈ (𝑃𝐼𝑆))
74		simplr 781	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑃 ∈ (𝑅𝐼𝑆))
75	2, 12, 4, 68, 69, 70, 71, 72, 73, 74	tgbtwnintr 28831	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑃 ∈ (𝑄𝐼𝑅))
76	2, 3, 4, 68, 69, 70, 71, 75	btwncolg3 28895	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → (𝑅 ∈ (𝑄𝐿𝑃) ∨ 𝑄 = 𝑃))
77	2, 3, 4, 68, 69, 70, 71, 76	colcom 28896	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
78	40	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) → (𝑆 ∈ (𝑃𝐼𝑄) ∨ 𝑃 ∈ (𝑆𝐼𝑄) ∨ 𝑄 ∈ (𝑃𝐼𝑆)))
79	56, 67, 77, 78	mpjao3dan 1459	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
80	79	an32s 665	. . . 4 ⊢ (((𝜑 ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑅 ∈ 𝐵) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
81	1, 80	mpidan 702	. . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ (𝑅𝐼𝑆)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
82	5	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝐺 ∈ TarskiG)
83	9	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑄 ∈ 𝐵)
84		simpllr 788	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑅 ∈ 𝐵)
85	7	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑃 ∈ 𝐵)
86	13	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑆 ∈ 𝐵)
87	51	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑃 ≠ 𝑆)
88		simpr 490	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑆 ∈ (𝑃𝐼𝑄))
89		simplr 781	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑆 ∈ (𝑃𝐼𝑅))
90	2, 4, 82, 85, 86, 83, 84, 3, 87, 88, 89	tgbtwnconnln1 28918	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → (𝑃 ∈ (𝑄𝐿𝑅) ∨ 𝑄 = 𝑅))
91	2, 3, 4, 82, 83, 84, 85, 90	colrot2 28898	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
92	5	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝐺 ∈ TarskiG)
93	7	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ 𝐵)
94	9	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑄 ∈ 𝐵)
95		simpllr 788	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑅 ∈ 𝐵)
96	13	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑆 ∈ 𝐵)
97	50	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑆 ≠ 𝑃)
98		simplr 781	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑆 ∈ (𝑃𝐼𝑅))
99	2, 12, 4, 92, 93, 96, 95, 98	tgbtwncom 28826	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑆 ∈ (𝑅𝐼𝑃))
100		simpr 490	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ (𝑆𝐼𝑄))
101	2, 12, 4, 92, 95, 96, 93, 94, 97, 99, 100	tgbtwnouttr2 28833	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ (𝑅𝐼𝑄))
102	2, 3, 4, 92, 93, 94, 95, 101	btwncolg2 28894	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
103	5	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝐺 ∈ TarskiG)
104	7	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑃 ∈ 𝐵)
105	9	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑄 ∈ 𝐵)
106		simpllr 788	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑅 ∈ 𝐵)
107	13	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑆 ∈ 𝐵)
108		simpr 490	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑄 ∈ (𝑃𝐼𝑆))
109		simplr 781	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑆 ∈ (𝑃𝐼𝑅))
110	2, 12, 4, 103, 104, 105, 107, 106, 108, 109	tgbtwnexch 28836	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑄 ∈ (𝑃𝐼𝑅))
111	2, 3, 4, 103, 104, 105, 106, 110	btwncolg3 28895	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
112	40	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) → (𝑆 ∈ (𝑃𝐼𝑄) ∨ 𝑃 ∈ (𝑆𝐼𝑄) ∨ 𝑄 ∈ (𝑃𝐼𝑆)))
113	91, 102, 111, 112	mpjao3dan 1459	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
114	113	an32s 665	. . . 4 ⊢ (((𝜑 ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑅 ∈ 𝐵) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
115	1, 114	mpidan 702	. . 3 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑃𝐼𝑅)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
116		id 23	. . . 4 ⊢ (𝜑 → 𝜑)
117		tglineeltr.8	. . . 4 ⊢ (𝜑 → 𝑅 ∈ (𝑃𝐿𝑆))
118	5	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ∈ 𝐵) → 𝐺 ∈ TarskiG)
119	7	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ∈ 𝐵) → 𝑃 ∈ 𝐵)
120	13	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ∈ 𝐵) → 𝑆 ∈ 𝐵)
121	51	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ∈ 𝐵) → 𝑃 ≠ 𝑆)
122		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ∈ 𝐵) → 𝑅 ∈ 𝐵)
123	2, 3, 4, 118, 119, 120, 121, 122	tgellng 28891	. . . . 5 ⊢ ((𝜑 ∧ 𝑅 ∈ 𝐵) → (𝑅 ∈ (𝑃𝐿𝑆) ↔ (𝑅 ∈ (𝑃𝐼𝑆) ∨ 𝑃 ∈ (𝑅𝐼𝑆) ∨ 𝑆 ∈ (𝑃𝐼𝑅))))
124	123	biimpa 482	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐿𝑆)) → (𝑅 ∈ (𝑃𝐼𝑆) ∨ 𝑃 ∈ (𝑅𝐼𝑆) ∨ 𝑆 ∈ (𝑃𝐼𝑅)))
125	116, 1, 117, 124	syl21anc 851	. . 3 ⊢ (𝜑 → (𝑅 ∈ (𝑃𝐼𝑆) ∨ 𝑃 ∈ (𝑅𝐼𝑆) ∨ 𝑆 ∈ (𝑃𝐼𝑅)))
126	44, 81, 115, 125	mpjao3dan 1459	. 2 ⊢ (𝜑 → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
127	38	neneqd 2962	. 2 ⊢ (𝜑 → ¬ 𝑃 = 𝑄)
128		pm5.61 1016	. . 3 ⊢ (((𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄) ∧ ¬ 𝑃 = 𝑄) ↔ (𝑅 ∈ (𝑃𝐿𝑄) ∧ ¬ 𝑃 = 𝑄))
129	128	simplbi 502	. 2 ⊢ (((𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄) ∧ ¬ 𝑃 = 𝑄) → 𝑅 ∈ (𝑃𝐿𝑄))
130	126, 127, 129	syl2anc 596	1 ⊢ (𝜑 → 𝑅 ∈ (𝑃𝐿𝑄))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∨ wo 861 ∨ w3o 1102 = wceq 1570 ∈ wcel 2145 ≠ wne 2957 ‘cfv 6537 (class class class)co 7416 Basecbs 17303 distcds 17353 TarskiGcstrkg 28764 Itvcitv 28770 LineGclng 28771

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7866 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-oadd 8462 df-er 8699 df-pm 8832 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-dju 9909 df-card 9947 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-nn 12259 df-2 12328 df-3 12329 df-n0 12530 df-xnn0 12603 df-z 12617 df-uz 12889 df-fz 13562 df-fzo 13710 df-hash 14395 df-word 14579 df-concat 14636 df-s1 14663 df-s2 14919 df-s3 14920 df-trkgc 28785 df-trkgb 28786 df-trkgcb 28787 df-trkg 28790 df-cgrg 28849

This theorem is used by: tglineelsb2 28975 colperpexlem3 29083 mideulem2 29085 angmndaddeu1 29250 morleylemrneab 35164

W3C validator