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Theorem tglineeltr 26425

Description: Transitivity law for lines, one half of tglineelsb2 26426. (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 729	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝐺 ∈ TarskiG)
7		tglineelsb2.1	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
8	7	ad3antrrr 729	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑃 ∈ 𝐵)
9		tglineelsb2.2	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ 𝐵)
10	9	ad3antrrr 729	. . . . . . 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 2798	. . . . . . . 8 ⊢ (dist‘𝐺) = (dist‘𝐺)
13		tglineelsb2.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ 𝐵)
14	13	ad3antrrr 729	. . . . . . . 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 488	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑆 ∈ (𝑃𝐼𝑄))
17	2, 12, 4, 6, 8, 11, 14, 10, 15, 16	tgbtwnexch 26292	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑅 ∈ (𝑃𝐼𝑄))
18	2, 3, 4, 6, 8, 10, 11, 17	btwncolg1 26349	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
19	5	ad3antrrr 729	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝐺 ∈ TarskiG)
20	7	ad3antrrr 729	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ 𝐵)
21	9	ad3antrrr 729	. . . . . . 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 729	. . . . . . . 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 26282	. . . . . . . 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 26288	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ (𝑅𝐼𝑄))
28	2, 3, 4, 19, 20, 21, 22, 27	btwncolg2 26350	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
29	5	ad3antrrr 729	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝐺 ∈ TarskiG)
30	7	ad3antrrr 729	. . . . . . 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 729	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑄 ∈ 𝐵)
33	13	ad3antrrr 729	. . . . . . 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 488	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑄 ∈ (𝑃𝐼𝑆))
36	2, 4, 29, 30, 31, 32, 33, 34, 35, 3	tgbtwnconnln3 26372	. . . . . 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 26347	. . . . . . . 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 725	. . . . . 6 ⊢ (((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) → (𝑆 ∈ (𝑃𝐼𝑄) ∨ 𝑃 ∈ (𝑆𝐼𝑄) ∨ 𝑄 ∈ (𝑃𝐼𝑆)))
42	18, 28, 36, 41	mpjao3dan 1428	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑅 ∈ (𝑃𝐼𝑆)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
43	42	an32s 651	. . . 4 ⊢ (((𝜑 ∧ 𝑅 ∈ (𝑃𝐼𝑆)) ∧ 𝑅 ∈ 𝐵) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
44	1, 43	mpidan 688	. . 3 ⊢ ((𝜑 ∧ 𝑅 ∈ (𝑃𝐼𝑆)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
45	5	ad3antrrr 729	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝐺 ∈ TarskiG)
46	7	ad3antrrr 729	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑃 ∈ 𝐵)
47	9	ad3antrrr 729	. . . . . . 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 729	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑆 ∈ 𝐵)
50		tglineelsb2.5	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ≠ 𝑃)
51	50	necomd 3042	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ≠ 𝑆)
52	51	ad3antrrr 729	. . . . . . . 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 488	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑆 ∈ (𝑃𝐼𝑄))
55	2, 12, 4, 45, 48, 46, 49, 47, 52, 53, 54	tgbtwnouttr 26291	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑃 ∈ (𝑅𝐼𝑄))
56	2, 3, 4, 45, 46, 47, 48, 55	btwncolg2 26350	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
57	5	ad3antrrr 729	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝐺 ∈ TarskiG)
58	9	ad3antrrr 729	. . . . . . 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 729	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ 𝐵)
61	13	ad3antrrr 729	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑆 ∈ 𝐵)
62	50	ad3antrrr 729	. . . . . . . 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 768	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ (𝑅𝐼𝑆))
65	2, 12, 4, 57, 59, 60, 61, 64	tgbtwncom 26282	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ (𝑆𝐼𝑅))
66	2, 4, 57, 61, 60, 58, 59, 3, 62, 63, 65	tgbtwnconnln2 26375	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → (𝑃 ∈ (𝑄𝐿𝑅) ∨ 𝑄 = 𝑅))
67	2, 3, 4, 57, 58, 59, 60, 66	colrot2 26354	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
68	5	ad3antrrr 729	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝐺 ∈ TarskiG)
69	9	ad3antrrr 729	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑄 ∈ 𝐵)
70	7	ad3antrrr 729	. . . . . . 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 729	. . . . . . . . 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 768	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑃 ∈ (𝑅𝐼𝑆))
75	2, 12, 4, 68, 69, 70, 71, 72, 73, 74	tgbtwnintr 26287	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑃 ∈ (𝑄𝐼𝑅))
76	2, 3, 4, 68, 69, 70, 71, 75	btwncolg3 26351	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → (𝑅 ∈ (𝑄𝐿𝑃) ∨ 𝑄 = 𝑃))
77	2, 3, 4, 68, 69, 70, 71, 76	colcom 26352	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
78	40	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) → (𝑆 ∈ (𝑃𝐼𝑄) ∨ 𝑃 ∈ (𝑆𝐼𝑄) ∨ 𝑄 ∈ (𝑃𝐼𝑆)))
79	56, 67, 77, 78	mpjao3dan 1428	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑃 ∈ (𝑅𝐼𝑆)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
80	79	an32s 651	. . . 4 ⊢ (((𝜑 ∧ 𝑃 ∈ (𝑅𝐼𝑆)) ∧ 𝑅 ∈ 𝐵) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
81	1, 80	mpidan 688	. . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ (𝑅𝐼𝑆)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
82	5	ad3antrrr 729	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝐺 ∈ TarskiG)
83	9	ad3antrrr 729	. . . . . . 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 729	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑃 ∈ 𝐵)
86	13	ad3antrrr 729	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑆 ∈ 𝐵)
87	51	ad3antrrr 729	. . . . . . . 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 768	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → 𝑆 ∈ (𝑃𝐼𝑅))
90	2, 4, 82, 85, 86, 83, 84, 3, 87, 88, 89	tgbtwnconnln1 26374	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → (𝑃 ∈ (𝑄𝐿𝑅) ∨ 𝑄 = 𝑅))
91	2, 3, 4, 82, 83, 84, 85, 90	colrot2 26354	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑆 ∈ (𝑃𝐼𝑄)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
92	5	ad3antrrr 729	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝐺 ∈ TarskiG)
93	7	ad3antrrr 729	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ 𝐵)
94	9	ad3antrrr 729	. . . . . . 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 729	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑆 ∈ 𝐵)
97	50	ad3antrrr 729	. . . . . . . 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 26282	. . . . . . . 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 26289	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → 𝑃 ∈ (𝑅𝐼𝑄))
102	2, 3, 4, 92, 93, 94, 95, 101	btwncolg2 26350	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑃 ∈ (𝑆𝐼𝑄)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
103	5	ad3antrrr 729	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝐺 ∈ TarskiG)
104	7	ad3antrrr 729	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑃 ∈ 𝐵)
105	9	ad3antrrr 729	. . . . . . 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 729	. . . . . . . 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 768	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑆 ∈ (𝑃𝐼𝑅))
110	2, 12, 4, 103, 104, 105, 107, 106, 108, 109	tgbtwnexch 26292	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → 𝑄 ∈ (𝑃𝐼𝑅))
111	2, 3, 4, 103, 104, 105, 106, 110	btwncolg3 26351	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑄 ∈ (𝑃𝐼𝑆)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
112	40	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) → (𝑆 ∈ (𝑃𝐼𝑄) ∨ 𝑃 ∈ (𝑆𝐼𝑄) ∨ 𝑄 ∈ (𝑃𝐼𝑆)))
113	91, 102, 111, 112	mpjao3dan 1428	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ∈ 𝐵) ∧ 𝑆 ∈ (𝑃𝐼𝑅)) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
114	113	an32s 651	. . . 4 ⊢ (((𝜑 ∧ 𝑆 ∈ (𝑃𝐼𝑅)) ∧ 𝑅 ∈ 𝐵) → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
115	1, 114	mpidan 688	. . 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 26347	. . . . 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 836	. . 3 ⊢ (𝜑 → (𝑅 ∈ (𝑃𝐼𝑆) ∨ 𝑃 ∈ (𝑅𝐼𝑆) ∨ 𝑆 ∈ (𝑃𝐼𝑅)))
126	44, 81, 115, 125	mpjao3dan 1428	. 2 ⊢ (𝜑 → (𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄))
127	38	neneqd 2992	. 2 ⊢ (𝜑 → ¬ 𝑃 = 𝑄)
128		pm5.61 998	. . 3 ⊢ (((𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄) ∧ ¬ 𝑃 = 𝑄) ↔ (𝑅 ∈ (𝑃𝐿𝑄) ∧ ¬ 𝑃 = 𝑄))
129	128	simplbi 501	. 2 ⊢ (((𝑅 ∈ (𝑃𝐿𝑄) ∨ 𝑃 = 𝑄) ∧ ¬ 𝑃 = 𝑄) → 𝑅 ∈ (𝑃𝐿𝑄))
130	126, 127, 129	syl2anc 587	1 ⊢ (𝜑 → 𝑅 ∈ (𝑃𝐿𝑄))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∨ wo 844 ∨ w3o 1083 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 distcds 16566 TarskiGcstrkg 26224 Itvcitv 26230 LineGclng 26231

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-hash 13687 df-word 13858 df-concat 13914 df-s1 13941 df-s2 14201 df-s3 14202 df-trkgc 26242 df-trkgb 26243 df-trkgcb 26244 df-trkg 26247 df-cgrg 26305

This theorem is referenced by: tglineelsb2 26426 colperpexlem3 26526 mideulem2 26528

W3C validator