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Theorem lnext 27798

Description: Extend a line with a missing point. Theorem 4.14 of [Schwabhauser] p. 37. (Contributed by Thierry Arnoux, 27-Apr-2019.)

**Hypotheses**
Ref	Expression
tglngval.p	⊢ 𝑃 = (Base‘𝐺)
tglngval.l	⊢ 𝐿 = (LineG‘𝐺)
tglngval.i	⊢ 𝐼 = (Itv‘𝐺)
tglngval.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tglngval.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
tglngval.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
tgcolg.z	⊢ (𝜑 → 𝑍 ∈ 𝑃)
lnxfr.r	⊢ ∼ = (cgrG‘𝐺)
lnxfr.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
lnxfr.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
lnxfr.d	⊢ − = (dist‘𝐺)
lnext.1	⊢ (𝜑 → (𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍))
lnext.2	⊢ (𝜑 → (𝑋 − 𝑌) = (𝐴 − 𝐵))

**Assertion**
Ref	Expression
lnext	⊢ (𝜑 → ∃𝑐 ∈ 𝑃 ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝐴𝐵𝑐”⟩)

Distinct variable groups: − ,𝑐 ∼ ,𝑐 𝐴,𝑐 𝐵,𝑐 𝐼,𝑐 𝑃,𝑐 𝑋,𝑐 𝑌,𝑐 𝑍,𝑐 𝜑,𝑐 Allowed substitution hints: 𝐺(𝑐) 𝐿(𝑐)

**Proof of Theorem lnext**
Step	Hyp	Ref	Expression
1		tglngval.p	. . . . 5 ⊢ 𝑃 = (Base‘𝐺)
2		lnxfr.d	. . . . 5 ⊢ − = (dist‘𝐺)
3		tglngval.i	. . . . 5 ⊢ 𝐼 = (Itv‘𝐺)
4		tglngval.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5		lnxfr.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
6		lnxfr.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
7		tglngval.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
8		tgcolg.z	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
9	1, 2, 3, 4, 5, 6, 7, 8	axtgsegcon 27695	. . . 4 ⊢ (𝜑 → ∃𝑐 ∈ 𝑃 (𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 − 𝑐) = (𝑌 − 𝑍)))
10	9	adantr 482	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → ∃𝑐 ∈ 𝑃 (𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 − 𝑐) = (𝑌 − 𝑍)))
11		lnxfr.r	. . . . . 6 ⊢ ∼ = (cgrG‘𝐺)
12	4	ad3antrrr 729	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 − 𝑐) = (𝑌 − 𝑍))) → 𝐺 ∈ TarskiG)
13		tglngval.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
14	13	ad3antrrr 729	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 − 𝑐) = (𝑌 − 𝑍))) → 𝑋 ∈ 𝑃)
15	7	ad3antrrr 729	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 − 𝑐) = (𝑌 − 𝑍))) → 𝑌 ∈ 𝑃)
16	8	ad3antrrr 729	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 − 𝑐) = (𝑌 − 𝑍))) → 𝑍 ∈ 𝑃)
17	5	ad3antrrr 729	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 − 𝑐) = (𝑌 − 𝑍))) → 𝐴 ∈ 𝑃)
18	6	ad3antrrr 729	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 − 𝑐) = (𝑌 − 𝑍))) → 𝐵 ∈ 𝑃)
19		simplr 768	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 − 𝑐) = (𝑌 − 𝑍))) → 𝑐 ∈ 𝑃)
20		lnext.2	. . . . . . 7 ⊢ (𝜑 → (𝑋 − 𝑌) = (𝐴 − 𝐵))
21	20	ad3antrrr 729	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 − 𝑐) = (𝑌 − 𝑍))) → (𝑋 − 𝑌) = (𝐴 − 𝐵))
22		simprr 772	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 − 𝑐) = (𝑌 − 𝑍))) → (𝐵 − 𝑐) = (𝑌 − 𝑍))
23	22	eqcomd 2739	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 − 𝑐) = (𝑌 − 𝑍))) → (𝑌 − 𝑍) = (𝐵 − 𝑐))
24		simpllr 775	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 − 𝑐) = (𝑌 − 𝑍))) → 𝑌 ∈ (𝑋𝐼𝑍))
25		simprl 770	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 − 𝑐) = (𝑌 − 𝑍))) → 𝐵 ∈ (𝐴𝐼𝑐))
26	1, 2, 3, 12, 14, 15, 16, 17, 18, 19, 24, 25, 21, 23	tgcgrextend 27716	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 − 𝑐) = (𝑌 − 𝑍))) → (𝑋 − 𝑍) = (𝐴 − 𝑐))
27	1, 2, 3, 12, 14, 16, 17, 19, 26	tgcgrcomlr 27711	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 − 𝑐) = (𝑌 − 𝑍))) → (𝑍 − 𝑋) = (𝑐 − 𝐴))
28	1, 2, 11, 12, 14, 15, 16, 17, 18, 19, 21, 23, 27	trgcgr 27747	. . . . 5 ⊢ ((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 − 𝑐) = (𝑌 − 𝑍))) → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝐴𝐵𝑐”⟩)
29	28	ex 414	. . . 4 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) → ((𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 − 𝑐) = (𝑌 − 𝑍)) → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝐴𝐵𝑐”⟩))
30	29	reximdva 3169	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → (∃𝑐 ∈ 𝑃 (𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 − 𝑐) = (𝑌 − 𝑍)) → ∃𝑐 ∈ 𝑃 ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝐴𝐵𝑐”⟩))
31	10, 30	mpd 15	. 2 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → ∃𝑐 ∈ 𝑃 ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝐴𝐵𝑐”⟩)
32	1, 2, 3, 4, 6, 5, 13, 8	axtgsegcon 27695	. . . 4 ⊢ (𝜑 → ∃𝑐 ∈ 𝑃 (𝐴 ∈ (𝐵𝐼𝑐) ∧ (𝐴 − 𝑐) = (𝑋 − 𝑍)))
33	32	adantr 482	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → ∃𝑐 ∈ 𝑃 (𝐴 ∈ (𝐵𝐼𝑐) ∧ (𝐴 − 𝑐) = (𝑋 − 𝑍)))
34	4	ad3antrrr 729	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑐) ∧ (𝐴 − 𝑐) = (𝑋 − 𝑍))) → 𝐺 ∈ TarskiG)
35	13	ad3antrrr 729	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑐) ∧ (𝐴 − 𝑐) = (𝑋 − 𝑍))) → 𝑋 ∈ 𝑃)
36	7	ad3antrrr 729	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑐) ∧ (𝐴 − 𝑐) = (𝑋 − 𝑍))) → 𝑌 ∈ 𝑃)
37	8	ad3antrrr 729	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑐) ∧ (𝐴 − 𝑐) = (𝑋 − 𝑍))) → 𝑍 ∈ 𝑃)
38	5	ad3antrrr 729	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑐) ∧ (𝐴 − 𝑐) = (𝑋 − 𝑍))) → 𝐴 ∈ 𝑃)
39	6	ad3antrrr 729	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑐) ∧ (𝐴 − 𝑐) = (𝑋 − 𝑍))) → 𝐵 ∈ 𝑃)
40		simplr 768	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑐) ∧ (𝐴 − 𝑐) = (𝑋 − 𝑍))) → 𝑐 ∈ 𝑃)
41	20	ad3antrrr 729	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑐) ∧ (𝐴 − 𝑐) = (𝑋 − 𝑍))) → (𝑋 − 𝑌) = (𝐴 − 𝐵))
42		simpllr 775	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑐) ∧ (𝐴 − 𝑐) = (𝑋 − 𝑍))) → 𝑋 ∈ (𝑌𝐼𝑍))
43		simprl 770	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑐) ∧ (𝐴 − 𝑐) = (𝑋 − 𝑍))) → 𝐴 ∈ (𝐵𝐼𝑐))
44	1, 2, 3, 34, 35, 36, 38, 39, 41	tgcgrcomlr 27711	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑐) ∧ (𝐴 − 𝑐) = (𝑋 − 𝑍))) → (𝑌 − 𝑋) = (𝐵 − 𝐴))
45		simprr 772	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑐) ∧ (𝐴 − 𝑐) = (𝑋 − 𝑍))) → (𝐴 − 𝑐) = (𝑋 − 𝑍))
46	45	eqcomd 2739	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑐) ∧ (𝐴 − 𝑐) = (𝑋 − 𝑍))) → (𝑋 − 𝑍) = (𝐴 − 𝑐))
47	1, 2, 3, 34, 36, 35, 37, 39, 38, 40, 42, 43, 44, 46	tgcgrextend 27716	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑐) ∧ (𝐴 − 𝑐) = (𝑋 − 𝑍))) → (𝑌 − 𝑍) = (𝐵 − 𝑐))
48	1, 2, 3, 34, 35, 37, 38, 40, 46	tgcgrcomlr 27711	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑐) ∧ (𝐴 − 𝑐) = (𝑋 − 𝑍))) → (𝑍 − 𝑋) = (𝑐 − 𝐴))
49	1, 2, 11, 34, 35, 36, 37, 38, 39, 40, 41, 47, 48	trgcgr 27747	. . . . 5 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑐) ∧ (𝐴 − 𝑐) = (𝑋 − 𝑍))) → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝐴𝐵𝑐”⟩)
50	49	ex 414	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) → ((𝐴 ∈ (𝐵𝐼𝑐) ∧ (𝐴 − 𝑐) = (𝑋 − 𝑍)) → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝐴𝐵𝑐”⟩))
51	50	reximdva 3169	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → (∃𝑐 ∈ 𝑃 (𝐴 ∈ (𝐵𝐼𝑐) ∧ (𝐴 − 𝑐) = (𝑋 − 𝑍)) → ∃𝑐 ∈ 𝑃 ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝐴𝐵𝑐”⟩))
52	33, 51	mpd 15	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → ∃𝑐 ∈ 𝑃 ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝐴𝐵𝑐”⟩)
53	4	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝐺 ∈ TarskiG)
54	13	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑋 ∈ 𝑃)
55	8	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑍 ∈ 𝑃)
56	7	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑌 ∈ 𝑃)
57	5	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝐴 ∈ 𝑃)
58	6	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝐵 ∈ 𝑃)
59		simpr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑍 ∈ (𝑋𝐼𝑌))
60	20	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → (𝑋 − 𝑌) = (𝐴 − 𝐵))
61	1, 2, 3, 11, 53, 54, 55, 56, 57, 58, 59, 60	tgcgrxfr 27749	. . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → ∃𝑐 ∈ 𝑃 (𝑐 ∈ (𝐴𝐼𝐵) ∧ ⟨“𝑋𝑍𝑌”⟩ ∼ ⟨“𝐴𝑐𝐵”⟩))
62	4	ad3antrrr 729	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) ∧ 𝑐 ∈ 𝑃) ∧ (𝑐 ∈ (𝐴𝐼𝐵) ∧ ⟨“𝑋𝑍𝑌”⟩ ∼ ⟨“𝐴𝑐𝐵”⟩)) → 𝐺 ∈ TarskiG)
63	13	ad3antrrr 729	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) ∧ 𝑐 ∈ 𝑃) ∧ (𝑐 ∈ (𝐴𝐼𝐵) ∧ ⟨“𝑋𝑍𝑌”⟩ ∼ ⟨“𝐴𝑐𝐵”⟩)) → 𝑋 ∈ 𝑃)
64	8	ad3antrrr 729	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) ∧ 𝑐 ∈ 𝑃) ∧ (𝑐 ∈ (𝐴𝐼𝐵) ∧ ⟨“𝑋𝑍𝑌”⟩ ∼ ⟨“𝐴𝑐𝐵”⟩)) → 𝑍 ∈ 𝑃)
65	7	ad3antrrr 729	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) ∧ 𝑐 ∈ 𝑃) ∧ (𝑐 ∈ (𝐴𝐼𝐵) ∧ ⟨“𝑋𝑍𝑌”⟩ ∼ ⟨“𝐴𝑐𝐵”⟩)) → 𝑌 ∈ 𝑃)
66	5	ad3antrrr 729	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) ∧ 𝑐 ∈ 𝑃) ∧ (𝑐 ∈ (𝐴𝐼𝐵) ∧ ⟨“𝑋𝑍𝑌”⟩ ∼ ⟨“𝐴𝑐𝐵”⟩)) → 𝐴 ∈ 𝑃)
67		simplr 768	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) ∧ 𝑐 ∈ 𝑃) ∧ (𝑐 ∈ (𝐴𝐼𝐵) ∧ ⟨“𝑋𝑍𝑌”⟩ ∼ ⟨“𝐴𝑐𝐵”⟩)) → 𝑐 ∈ 𝑃)
68	6	ad3antrrr 729	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) ∧ 𝑐 ∈ 𝑃) ∧ (𝑐 ∈ (𝐴𝐼𝐵) ∧ ⟨“𝑋𝑍𝑌”⟩ ∼ ⟨“𝐴𝑐𝐵”⟩)) → 𝐵 ∈ 𝑃)
69		simprr 772	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) ∧ 𝑐 ∈ 𝑃) ∧ (𝑐 ∈ (𝐴𝐼𝐵) ∧ ⟨“𝑋𝑍𝑌”⟩ ∼ ⟨“𝐴𝑐𝐵”⟩)) → ⟨“𝑋𝑍𝑌”⟩ ∼ ⟨“𝐴𝑐𝐵”⟩)
70	1, 2, 3, 11, 62, 63, 64, 65, 66, 67, 68, 69	cgr3swap23 27755	. . . . 5 ⊢ ((((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) ∧ 𝑐 ∈ 𝑃) ∧ (𝑐 ∈ (𝐴𝐼𝐵) ∧ ⟨“𝑋𝑍𝑌”⟩ ∼ ⟨“𝐴𝑐𝐵”⟩)) → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝐴𝐵𝑐”⟩)
71	70	ex 414	. . . 4 ⊢ (((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) ∧ 𝑐 ∈ 𝑃) → ((𝑐 ∈ (𝐴𝐼𝐵) ∧ ⟨“𝑋𝑍𝑌”⟩ ∼ ⟨“𝐴𝑐𝐵”⟩) → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝐴𝐵𝑐”⟩))
72	71	reximdva 3169	. . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → (∃𝑐 ∈ 𝑃 (𝑐 ∈ (𝐴𝐼𝐵) ∧ ⟨“𝑋𝑍𝑌”⟩ ∼ ⟨“𝐴𝑐𝐵”⟩) → ∃𝑐 ∈ 𝑃 ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝐴𝐵𝑐”⟩))
73	61, 72	mpd 15	. 2 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → ∃𝑐 ∈ 𝑃 ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝐴𝐵𝑐”⟩)
74		lnext.1	. . 3 ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍))
75		tglngval.l	. . . 4 ⊢ 𝐿 = (LineG‘𝐺)
76	1, 75, 3, 4, 13, 8, 7	tgcolg 27785	. . 3 ⊢ (𝜑 → ((𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍) ↔ (𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑋 ∈ (𝑌𝐼𝑍) ∨ 𝑍 ∈ (𝑋𝐼𝑌))))
77	74, 76	mpbid 231	. 2 ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑋 ∈ (𝑌𝐼𝑍) ∨ 𝑍 ∈ (𝑋𝐼𝑌)))
78	31, 52, 73, 77	mpjao3dan 1432	1 ⊢ (𝜑 → ∃𝑐 ∈ 𝑃 ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝐴𝐵𝑐”⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 ∨ wo 846 ∨ w3o 1087 = wceq 1542 ∈ wcel 2107 ∃wrex 3071 class class class wbr 5147 ‘cfv 6540 (class class class)co 7404 ⟨“cs3 14789 Basecbs 17140 distcds 17202 TarskiGcstrkg 27658 Itvcitv 27664 LineGclng 27665 cgrGccgrg 27741

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-oadd 8465 df-er 8699 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-xnn0 12541 df-z 12555 df-uz 12819 df-fz 13481 df-fzo 13624 df-hash 14287 df-word 14461 df-concat 14517 df-s1 14542 df-s2 14795 df-s3 14796 df-trkgc 27679 df-trkgb 27680 df-trkgcb 27681 df-trkg 27684 df-cgrg 27742

This theorem is referenced by: legov 27816 legov2 27817 legtrd 27820 symquadlem 27920 trgcopy 28035 cgrg3col4 28084

W3C validator