lnext - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > lnext		Structured version Visualization version GIF version

Theorem lnext 28873

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 28770	. . . 4 ⊢ (𝜑 → ∃𝑐 ∈ 𝑃 (𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 − 𝑐) = (𝑌 − 𝑍)))
10	9	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → ∃𝑐 ∈ 𝑃 (𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 − 𝑐) = (𝑌 − 𝑍)))
11		lnxfr.r	. . . . . 6 ⊢ ∼ = (cgrG‘𝐺)
12	4	ad3antrrr 743	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 − 𝑐) = (𝑌 − 𝑍))) → 𝐺 ∈ TarskiG)
13		tglngval.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
14	13	ad3antrrr 743	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 − 𝑐) = (𝑌 − 𝑍))) → 𝑋 ∈ 𝑃)
15	7	ad3antrrr 743	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 − 𝑐) = (𝑌 − 𝑍))) → 𝑌 ∈ 𝑃)
16	8	ad3antrrr 743	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 − 𝑐) = (𝑌 − 𝑍))) → 𝑍 ∈ 𝑃)
17	5	ad3antrrr 743	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 − 𝑐) = (𝑌 − 𝑍))) → 𝐴 ∈ 𝑃)
18	6	ad3antrrr 743	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 − 𝑐) = (𝑌 − 𝑍))) → 𝐵 ∈ 𝑃)
19		simplr 781	. . . . . 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 743	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 − 𝑐) = (𝑌 − 𝑍))) → (𝑋 − 𝑌) = (𝐴 − 𝐵))
22		simprr 785	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 − 𝑐) = (𝑌 − 𝑍))) → (𝐵 − 𝑐) = (𝑌 − 𝑍))
23	22	eqcomd 2772	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 − 𝑐) = (𝑌 − 𝑍))) → (𝑌 − 𝑍) = (𝐵 − 𝑐))
24		simpllr 788	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 − 𝑐) = (𝑌 − 𝑍))) → 𝑌 ∈ (𝑋𝐼𝑍))
25		simprl 783	. . . . . . . 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 28791	. . . . . . 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 28786	. . . . . 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 28822	. . . . 5 ⊢ ((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 − 𝑐) = (𝑌 − 𝑍))) → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝐴𝐵𝑐”⟩)
29	28	ex 418	. . . 4 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) → ((𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 − 𝑐) = (𝑌 − 𝑍)) → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝐴𝐵𝑐”⟩))
30	29	reximdva 3181	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → (∃𝑐 ∈ 𝑃 (𝐵 ∈ (𝐴𝐼𝑐) ∧ (𝐵 − 𝑐) = (𝑌 − 𝑍)) → ∃𝑐 ∈ 𝑃 ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝐴𝐵𝑐”⟩))
31	10, 30	mpd 16	. 2 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → ∃𝑐 ∈ 𝑃 ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝐴𝐵𝑐”⟩)
32	1, 2, 3, 4, 6, 5, 13, 8	axtgsegcon 28770	. . . 4 ⊢ (𝜑 → ∃𝑐 ∈ 𝑃 (𝐴 ∈ (𝐵𝐼𝑐) ∧ (𝐴 − 𝑐) = (𝑋 − 𝑍)))
33	32	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → ∃𝑐 ∈ 𝑃 (𝐴 ∈ (𝐵𝐼𝑐) ∧ (𝐴 − 𝑐) = (𝑋 − 𝑍)))
34	4	ad3antrrr 743	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑐) ∧ (𝐴 − 𝑐) = (𝑋 − 𝑍))) → 𝐺 ∈ TarskiG)
35	13	ad3antrrr 743	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑐) ∧ (𝐴 − 𝑐) = (𝑋 − 𝑍))) → 𝑋 ∈ 𝑃)
36	7	ad3antrrr 743	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑐) ∧ (𝐴 − 𝑐) = (𝑋 − 𝑍))) → 𝑌 ∈ 𝑃)
37	8	ad3antrrr 743	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑐) ∧ (𝐴 − 𝑐) = (𝑋 − 𝑍))) → 𝑍 ∈ 𝑃)
38	5	ad3antrrr 743	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑐) ∧ (𝐴 − 𝑐) = (𝑋 − 𝑍))) → 𝐴 ∈ 𝑃)
39	6	ad3antrrr 743	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑐) ∧ (𝐴 − 𝑐) = (𝑋 − 𝑍))) → 𝐵 ∈ 𝑃)
40		simplr 781	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑐) ∧ (𝐴 − 𝑐) = (𝑋 − 𝑍))) → 𝑐 ∈ 𝑃)
41	20	ad3antrrr 743	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑐) ∧ (𝐴 − 𝑐) = (𝑋 − 𝑍))) → (𝑋 − 𝑌) = (𝐴 − 𝐵))
42		simpllr 788	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑐) ∧ (𝐴 − 𝑐) = (𝑋 − 𝑍))) → 𝑋 ∈ (𝑌𝐼𝑍))
43		simprl 783	. . . . . . 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 28786	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑐) ∧ (𝐴 − 𝑐) = (𝑋 − 𝑍))) → (𝑌 − 𝑋) = (𝐵 − 𝐴))
45		simprr 785	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑐) ∧ (𝐴 − 𝑐) = (𝑋 − 𝑍))) → (𝐴 − 𝑐) = (𝑋 − 𝑍))
46	45	eqcomd 2772	. . . . . . 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 28791	. . . . . 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 28786	. . . . . 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 28822	. . . . 5 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑐) ∧ (𝐴 − 𝑐) = (𝑋 − 𝑍))) → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝐴𝐵𝑐”⟩)
50	49	ex 418	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) ∧ 𝑐 ∈ 𝑃) → ((𝐴 ∈ (𝐵𝐼𝑐) ∧ (𝐴 − 𝑐) = (𝑋 − 𝑍)) → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝐴𝐵𝑐”⟩))
51	50	reximdva 3181	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → (∃𝑐 ∈ 𝑃 (𝐴 ∈ (𝐵𝐼𝑐) ∧ (𝐴 − 𝑐) = (𝑋 − 𝑍)) → ∃𝑐 ∈ 𝑃 ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝐴𝐵𝑐”⟩))
52	33, 51	mpd 16	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → ∃𝑐 ∈ 𝑃 ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝐴𝐵𝑐”⟩)
53	4	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝐺 ∈ TarskiG)
54	13	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑋 ∈ 𝑃)
55	8	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑍 ∈ 𝑃)
56	7	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑌 ∈ 𝑃)
57	5	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝐴 ∈ 𝑃)
58	6	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝐵 ∈ 𝑃)
59		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑍 ∈ (𝑋𝐼𝑌))
60	20	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → (𝑋 − 𝑌) = (𝐴 − 𝐵))
61	1, 2, 3, 11, 53, 54, 55, 56, 57, 58, 59, 60	tgcgrxfr 28824	. . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → ∃𝑐 ∈ 𝑃 (𝑐 ∈ (𝐴𝐼𝐵) ∧ ⟨“𝑋𝑍𝑌”⟩ ∼ ⟨“𝐴𝑐𝐵”⟩))
62	4	ad3antrrr 743	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) ∧ 𝑐 ∈ 𝑃) ∧ (𝑐 ∈ (𝐴𝐼𝐵) ∧ ⟨“𝑋𝑍𝑌”⟩ ∼ ⟨“𝐴𝑐𝐵”⟩)) → 𝐺 ∈ TarskiG)
63	13	ad3antrrr 743	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) ∧ 𝑐 ∈ 𝑃) ∧ (𝑐 ∈ (𝐴𝐼𝐵) ∧ ⟨“𝑋𝑍𝑌”⟩ ∼ ⟨“𝐴𝑐𝐵”⟩)) → 𝑋 ∈ 𝑃)
64	8	ad3antrrr 743	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) ∧ 𝑐 ∈ 𝑃) ∧ (𝑐 ∈ (𝐴𝐼𝐵) ∧ ⟨“𝑋𝑍𝑌”⟩ ∼ ⟨“𝐴𝑐𝐵”⟩)) → 𝑍 ∈ 𝑃)
65	7	ad3antrrr 743	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) ∧ 𝑐 ∈ 𝑃) ∧ (𝑐 ∈ (𝐴𝐼𝐵) ∧ ⟨“𝑋𝑍𝑌”⟩ ∼ ⟨“𝐴𝑐𝐵”⟩)) → 𝑌 ∈ 𝑃)
66	5	ad3antrrr 743	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) ∧ 𝑐 ∈ 𝑃) ∧ (𝑐 ∈ (𝐴𝐼𝐵) ∧ ⟨“𝑋𝑍𝑌”⟩ ∼ ⟨“𝐴𝑐𝐵”⟩)) → 𝐴 ∈ 𝑃)
67		simplr 781	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) ∧ 𝑐 ∈ 𝑃) ∧ (𝑐 ∈ (𝐴𝐼𝐵) ∧ ⟨“𝑋𝑍𝑌”⟩ ∼ ⟨“𝐴𝑐𝐵”⟩)) → 𝑐 ∈ 𝑃)
68	6	ad3antrrr 743	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) ∧ 𝑐 ∈ 𝑃) ∧ (𝑐 ∈ (𝐴𝐼𝐵) ∧ ⟨“𝑋𝑍𝑌”⟩ ∼ ⟨“𝐴𝑐𝐵”⟩)) → 𝐵 ∈ 𝑃)
69		simprr 785	. . . . . 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 28830	. . . . 5 ⊢ ((((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) ∧ 𝑐 ∈ 𝑃) ∧ (𝑐 ∈ (𝐴𝐼𝐵) ∧ ⟨“𝑋𝑍𝑌”⟩ ∼ ⟨“𝐴𝑐𝐵”⟩)) → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝐴𝐵𝑐”⟩)
71	70	ex 418	. . . 4 ⊢ (((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) ∧ 𝑐 ∈ 𝑃) → ((𝑐 ∈ (𝐴𝐼𝐵) ∧ ⟨“𝑋𝑍𝑌”⟩ ∼ ⟨“𝐴𝑐𝐵”⟩) → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝐴𝐵𝑐”⟩))
72	71	reximdva 3181	. . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → (∃𝑐 ∈ 𝑃 (𝑐 ∈ (𝐴𝐼𝐵) ∧ ⟨“𝑋𝑍𝑌”⟩ ∼ ⟨“𝐴𝑐𝐵”⟩) → ∃𝑐 ∈ 𝑃 ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝐴𝐵𝑐”⟩))
73	61, 72	mpd 16	. 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 28860	. . 3 ⊢ (𝜑 → ((𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍) ↔ (𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑋 ∈ (𝑌𝐼𝑍) ∨ 𝑍 ∈ (𝑋𝐼𝑌))))
77	74, 76	mpbid 235	. 2 ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑋 ∈ (𝑌𝐼𝑍) ∨ 𝑍 ∈ (𝑋𝐼𝑌)))
78	31, 52, 73, 77	mpjao3dan 1459	1 ⊢ (𝜑 → ∃𝑐 ∈ 𝑃 ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝐴𝐵𝑐”⟩)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∨ wo 861 ∨ w3o 1102 = wceq 1570 ∈ wcel 2146 ∃wrex 3092 class class class wbr 5114 ‘cfv 6543 (class class class)co 7423 ⟨“cs3 14905 Basecbs 17294 distcds 17344 TarskiGcstrkg 28733 Itvcitv 28739 LineGclng 28740 cgrGccgrg 28816

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195

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 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7872 df-1st 7995 df-2nd 7996 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-oadd 8466 df-er 8703 df-pm 8836 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-dju 9906 df-card 9944 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-nn 12252 df-2 12321 df-3 12322 df-n0 12523 df-xnn0 12596 df-z 12610 df-uz 12881 df-fz 13554 df-fzo 13702 df-hash 14387 df-word 14571 df-concat 14628 df-s1 14655 df-s2 14911 df-s3 14912 df-trkgc 28754 df-trkgb 28755 df-trkgcb 28756 df-trkg 28759 df-cgrg 28817

This theorem is used by: legov 28891 legov2 28892 legtrd 28895 symquadlem 29003 trgcopy 29152 cgrg3col4 29207

W3C validator