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Theorem colperpex 27964

Description: In dimension 2 and above, on a line (𝐴𝐿𝐵) there is always a perpendicular 𝑃 from 𝐴 on a given plane (here given by 𝐶, in case 𝐶 does not lie on the line). Theorem 8.21 of [Schwabhauser] p. 63. (Contributed by Thierry Arnoux, 20-Nov-2019.)

**Hypotheses**
Ref	Expression
colperpex.p	⊢ 𝑃 = (Base‘𝐺)
colperpex.d	⊢ − = (dist‘𝐺)
colperpex.i	⊢ 𝐼 = (Itv‘𝐺)
colperpex.l	⊢ 𝐿 = (LineG‘𝐺)
colperpex.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
colperpex.1	⊢ (𝜑 → 𝐴 ∈ 𝑃)
colperpex.2	⊢ (𝜑 → 𝐵 ∈ 𝑃)
colperpex.3	⊢ (𝜑 → 𝐶 ∈ 𝑃)
colperpex.4	⊢ (𝜑 → 𝐴 ≠ 𝐵)
colperpex.5	⊢ (𝜑 → 𝐺Dim_TarskiG≥2)

**Assertion**
Ref	Expression
colperpex	⊢ (𝜑 → ∃𝑝 ∈ 𝑃 ((𝐴𝐿𝑝)(⟂G‘𝐺)(𝐴𝐿𝐵) ∧ ∃𝑡 ∈ 𝑃 ((𝑡 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵) ∧ 𝑡 ∈ (𝐶𝐼𝑝))))

Distinct variable groups: − ,𝑝,𝑡 𝐴,𝑝,𝑡 𝐵,𝑝,𝑡 𝐶,𝑝,𝑡 𝐺,𝑝,𝑡 𝐼,𝑝,𝑡 𝐿,𝑝,𝑡 𝑃,𝑝,𝑡 𝜑,𝑝,𝑡

Proof of Theorem colperpex Dummy variables 𝑠 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		colperpex.p	. . . . 5 ⊢ 𝑃 = (Base‘𝐺)
2		colperpex.d	. . . . 5 ⊢ − = (dist‘𝐺)
3		colperpex.i	. . . . 5 ⊢ 𝐼 = (Itv‘𝐺)
4		colperpex.l	. . . . 5 ⊢ 𝐿 = (LineG‘𝐺)
5		colperpex.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
6	5	ad3antrrr 729	. . . . 5 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐴𝐿𝐵)) ∧ 𝑑 ∈ 𝑃) ∧ ¬ 𝑑 ∈ (𝐴𝐿𝐵)) → 𝐺 ∈ TarskiG)
7		colperpex.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
8	7	ad3antrrr 729	. . . . 5 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐴𝐿𝐵)) ∧ 𝑑 ∈ 𝑃) ∧ ¬ 𝑑 ∈ (𝐴𝐿𝐵)) → 𝐴 ∈ 𝑃)
9		colperpex.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
10	9	ad3antrrr 729	. . . . 5 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐴𝐿𝐵)) ∧ 𝑑 ∈ 𝑃) ∧ ¬ 𝑑 ∈ (𝐴𝐿𝐵)) → 𝐵 ∈ 𝑃)
11		simplr 768	. . . . 5 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐴𝐿𝐵)) ∧ 𝑑 ∈ 𝑃) ∧ ¬ 𝑑 ∈ (𝐴𝐿𝐵)) → 𝑑 ∈ 𝑃)
12		colperpex.4	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
13	12	ad3antrrr 729	. . . . 5 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐴𝐿𝐵)) ∧ 𝑑 ∈ 𝑃) ∧ ¬ 𝑑 ∈ (𝐴𝐿𝐵)) → 𝐴 ≠ 𝐵)
14		simpr 486	. . . . 5 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐴𝐿𝐵)) ∧ 𝑑 ∈ 𝑃) ∧ ¬ 𝑑 ∈ (𝐴𝐿𝐵)) → ¬ 𝑑 ∈ (𝐴𝐿𝐵))
15	1, 2, 3, 4, 6, 8, 10, 11, 13, 14	colperpexlem3 27963	. . . 4 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐴𝐿𝐵)) ∧ 𝑑 ∈ 𝑃) ∧ ¬ 𝑑 ∈ (𝐴𝐿𝐵)) → ∃𝑝 ∈ 𝑃 ((𝐴𝐿𝑝)(⟂G‘𝐺)(𝐴𝐿𝐵) ∧ ∃𝑠 ∈ 𝑃 ((𝑠 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵) ∧ 𝑠 ∈ (𝑑𝐼𝑝))))
16		simprl 770	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝐶 ∈ (𝐴𝐿𝐵)) ∧ 𝑑 ∈ 𝑃) ∧ ¬ 𝑑 ∈ (𝐴𝐿𝐵)) ∧ 𝑝 ∈ 𝑃) ∧ ((𝐴𝐿𝑝)(⟂G‘𝐺)(𝐴𝐿𝐵) ∧ ∃𝑠 ∈ 𝑃 ((𝑠 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵) ∧ 𝑠 ∈ (𝑑𝐼𝑝)))) → (𝐴𝐿𝑝)(⟂G‘𝐺)(𝐴𝐿𝐵))
17		colperpex.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
18	17	ad5antr 733	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝐶 ∈ (𝐴𝐿𝐵)) ∧ 𝑑 ∈ 𝑃) ∧ ¬ 𝑑 ∈ (𝐴𝐿𝐵)) ∧ 𝑝 ∈ 𝑃) ∧ ((𝐴𝐿𝑝)(⟂G‘𝐺)(𝐴𝐿𝐵) ∧ ∃𝑠 ∈ 𝑃 ((𝑠 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵) ∧ 𝑠 ∈ (𝑑𝐼𝑝)))) → 𝐶 ∈ 𝑃)
19		simp-5r 785	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝐶 ∈ (𝐴𝐿𝐵)) ∧ 𝑑 ∈ 𝑃) ∧ ¬ 𝑑 ∈ (𝐴𝐿𝐵)) ∧ 𝑝 ∈ 𝑃) ∧ ((𝐴𝐿𝑝)(⟂G‘𝐺)(𝐴𝐿𝐵) ∧ ∃𝑠 ∈ 𝑃 ((𝑠 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵) ∧ 𝑠 ∈ (𝑑𝐼𝑝)))) → 𝐶 ∈ (𝐴𝐿𝐵))
20	19	orcd 872	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝐶 ∈ (𝐴𝐿𝐵)) ∧ 𝑑 ∈ 𝑃) ∧ ¬ 𝑑 ∈ (𝐴𝐿𝐵)) ∧ 𝑝 ∈ 𝑃) ∧ ((𝐴𝐿𝑝)(⟂G‘𝐺)(𝐴𝐿𝐵) ∧ ∃𝑠 ∈ 𝑃 ((𝑠 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵) ∧ 𝑠 ∈ (𝑑𝐼𝑝)))) → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
21	5	ad5antr 733	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝐶 ∈ (𝐴𝐿𝐵)) ∧ 𝑑 ∈ 𝑃) ∧ ¬ 𝑑 ∈ (𝐴𝐿𝐵)) ∧ 𝑝 ∈ 𝑃) ∧ ((𝐴𝐿𝑝)(⟂G‘𝐺)(𝐴𝐿𝐵) ∧ ∃𝑠 ∈ 𝑃 ((𝑠 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵) ∧ 𝑠 ∈ (𝑑𝐼𝑝)))) → 𝐺 ∈ TarskiG)
22		simplr 768	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝐶 ∈ (𝐴𝐿𝐵)) ∧ 𝑑 ∈ 𝑃) ∧ ¬ 𝑑 ∈ (𝐴𝐿𝐵)) ∧ 𝑝 ∈ 𝑃) ∧ ((𝐴𝐿𝑝)(⟂G‘𝐺)(𝐴𝐿𝐵) ∧ ∃𝑠 ∈ 𝑃 ((𝑠 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵) ∧ 𝑠 ∈ (𝑑𝐼𝑝)))) → 𝑝 ∈ 𝑃)
23	1, 2, 3, 21, 18, 22	tgbtwntriv1 27722	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝐶 ∈ (𝐴𝐿𝐵)) ∧ 𝑑 ∈ 𝑃) ∧ ¬ 𝑑 ∈ (𝐴𝐿𝐵)) ∧ 𝑝 ∈ 𝑃) ∧ ((𝐴𝐿𝑝)(⟂G‘𝐺)(𝐴𝐿𝐵) ∧ ∃𝑠 ∈ 𝑃 ((𝑠 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵) ∧ 𝑠 ∈ (𝑑𝐼𝑝)))) → 𝐶 ∈ (𝐶𝐼𝑝))
24		eleq1 2822	. . . . . . . . . . 11 ⊢ (𝑡 = 𝐶 → (𝑡 ∈ (𝐴𝐿𝐵) ↔ 𝐶 ∈ (𝐴𝐿𝐵)))
25	24	orbi1d 916	. . . . . . . . . 10 ⊢ (𝑡 = 𝐶 → ((𝑡 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵) ↔ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)))
26		eleq1 2822	. . . . . . . . . 10 ⊢ (𝑡 = 𝐶 → (𝑡 ∈ (𝐶𝐼𝑝) ↔ 𝐶 ∈ (𝐶𝐼𝑝)))
27	25, 26	anbi12d 632	. . . . . . . . 9 ⊢ (𝑡 = 𝐶 → (((𝑡 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵) ∧ 𝑡 ∈ (𝐶𝐼𝑝)) ↔ ((𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵) ∧ 𝐶 ∈ (𝐶𝐼𝑝))))
28	27	rspcev 3612	. . . . . . . 8 ⊢ ((𝐶 ∈ 𝑃 ∧ ((𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵) ∧ 𝐶 ∈ (𝐶𝐼𝑝))) → ∃𝑡 ∈ 𝑃 ((𝑡 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵) ∧ 𝑡 ∈ (𝐶𝐼𝑝)))
29	18, 20, 23, 28	syl12anc 836	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝐶 ∈ (𝐴𝐿𝐵)) ∧ 𝑑 ∈ 𝑃) ∧ ¬ 𝑑 ∈ (𝐴𝐿𝐵)) ∧ 𝑝 ∈ 𝑃) ∧ ((𝐴𝐿𝑝)(⟂G‘𝐺)(𝐴𝐿𝐵) ∧ ∃𝑠 ∈ 𝑃 ((𝑠 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵) ∧ 𝑠 ∈ (𝑑𝐼𝑝)))) → ∃𝑡 ∈ 𝑃 ((𝑡 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵) ∧ 𝑡 ∈ (𝐶𝐼𝑝)))
30	16, 29	jca 513	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝐶 ∈ (𝐴𝐿𝐵)) ∧ 𝑑 ∈ 𝑃) ∧ ¬ 𝑑 ∈ (𝐴𝐿𝐵)) ∧ 𝑝 ∈ 𝑃) ∧ ((𝐴𝐿𝑝)(⟂G‘𝐺)(𝐴𝐿𝐵) ∧ ∃𝑠 ∈ 𝑃 ((𝑠 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵) ∧ 𝑠 ∈ (𝑑𝐼𝑝)))) → ((𝐴𝐿𝑝)(⟂G‘𝐺)(𝐴𝐿𝐵) ∧ ∃𝑡 ∈ 𝑃 ((𝑡 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵) ∧ 𝑡 ∈ (𝐶𝐼𝑝))))
31	30	ex 414	. . . . 5 ⊢ (((((𝜑 ∧ 𝐶 ∈ (𝐴𝐿𝐵)) ∧ 𝑑 ∈ 𝑃) ∧ ¬ 𝑑 ∈ (𝐴𝐿𝐵)) ∧ 𝑝 ∈ 𝑃) → (((𝐴𝐿𝑝)(⟂G‘𝐺)(𝐴𝐿𝐵) ∧ ∃𝑠 ∈ 𝑃 ((𝑠 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵) ∧ 𝑠 ∈ (𝑑𝐼𝑝))) → ((𝐴𝐿𝑝)(⟂G‘𝐺)(𝐴𝐿𝐵) ∧ ∃𝑡 ∈ 𝑃 ((𝑡 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵) ∧ 𝑡 ∈ (𝐶𝐼𝑝)))))
32	31	reximdva 3169	. . . 4 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐴𝐿𝐵)) ∧ 𝑑 ∈ 𝑃) ∧ ¬ 𝑑 ∈ (𝐴𝐿𝐵)) → (∃𝑝 ∈ 𝑃 ((𝐴𝐿𝑝)(⟂G‘𝐺)(𝐴𝐿𝐵) ∧ ∃𝑠 ∈ 𝑃 ((𝑠 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵) ∧ 𝑠 ∈ (𝑑𝐼𝑝))) → ∃𝑝 ∈ 𝑃 ((𝐴𝐿𝑝)(⟂G‘𝐺)(𝐴𝐿𝐵) ∧ ∃𝑡 ∈ 𝑃 ((𝑡 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵) ∧ 𝑡 ∈ (𝐶𝐼𝑝)))))
33	15, 32	mpd 15	. . 3 ⊢ ((((𝜑 ∧ 𝐶 ∈ (𝐴𝐿𝐵)) ∧ 𝑑 ∈ 𝑃) ∧ ¬ 𝑑 ∈ (𝐴𝐿𝐵)) → ∃𝑝 ∈ 𝑃 ((𝐴𝐿𝑝)(⟂G‘𝐺)(𝐴𝐿𝐵) ∧ ∃𝑡 ∈ 𝑃 ((𝑡 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵) ∧ 𝑡 ∈ (𝐶𝐼𝑝))))
34	5	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐿𝐵)) → 𝐺 ∈ TarskiG)
35		colperpex.5	. . . . 5 ⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
36	35	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐿𝐵)) → 𝐺Dim_TarskiG≥2)
37	7	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐿𝐵)) → 𝐴 ∈ 𝑃)
38	9	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐿𝐵)) → 𝐵 ∈ 𝑃)
39	12	adantr 482	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐿𝐵)) → 𝐴 ≠ 𝐵)
40	1, 3, 4, 34, 36, 37, 38, 39	tglowdim2ln 27882	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐿𝐵)) → ∃𝑑 ∈ 𝑃 ¬ 𝑑 ∈ (𝐴𝐿𝐵))
41	33, 40	r19.29a 3163	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐿𝐵)) → ∃𝑝 ∈ 𝑃 ((𝐴𝐿𝑝)(⟂G‘𝐺)(𝐴𝐿𝐵) ∧ ∃𝑡 ∈ 𝑃 ((𝑡 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵) ∧ 𝑡 ∈ (𝐶𝐼𝑝))))
42	5	adantr 482	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ (𝐴𝐿𝐵)) → 𝐺 ∈ TarskiG)
43	7	adantr 482	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ (𝐴𝐿𝐵)) → 𝐴 ∈ 𝑃)
44	9	adantr 482	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ (𝐴𝐿𝐵)) → 𝐵 ∈ 𝑃)
45	17	adantr 482	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ (𝐴𝐿𝐵)) → 𝐶 ∈ 𝑃)
46	12	adantr 482	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ (𝐴𝐿𝐵)) → 𝐴 ≠ 𝐵)
47		simpr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ (𝐴𝐿𝐵)) → ¬ 𝐶 ∈ (𝐴𝐿𝐵))
48	1, 2, 3, 4, 42, 43, 44, 45, 46, 47	colperpexlem3 27963	. 2 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ (𝐴𝐿𝐵)) → ∃𝑝 ∈ 𝑃 ((𝐴𝐿𝑝)(⟂G‘𝐺)(𝐴𝐿𝐵) ∧ ∃𝑡 ∈ 𝑃 ((𝑡 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵) ∧ 𝑡 ∈ (𝐶𝐼𝑝))))
49	41, 48	pm2.61dan 812	1 ⊢ (𝜑 → ∃𝑝 ∈ 𝑃 ((𝐴𝐿𝑝)(⟂G‘𝐺)(𝐴𝐿𝐵) ∧ ∃𝑡 ∈ 𝑃 ((𝑡 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵) ∧ 𝑡 ∈ (𝐶𝐼𝑝))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∃wrex 3071 class class class wbr 5147 ‘cfv 6540 (class class class)co 7404 2c2 12263 Basecbs 17140 distcds 17202 TarskiGcstrkg 27658 Dim_TarskiG≥cstrkgld 27662 Itvcitv 27664 LineGclng 27665 ⟂Gcperpg 27926

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-rmo 3377 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-map 8818 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-trkgld 27683 df-trkg 27684 df-cgrg 27742 df-leg 27814 df-mir 27884 df-rag 27925 df-perpg 27927

This theorem is referenced by: midex 27968 oppperpex 27984

W3C validator