2sphere - Mathbox for Alexander van der Vekens

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Theorem 2sphere 48670

Description: The sphere with center 𝑀 and radius 𝑅 in a two dimensional Euclidean space is a circle. (Contributed by AV, 5-Feb-2023.)

**Hypotheses**
Ref	Expression
2sphere.i	⊢ 𝐼 = {1, 2}
2sphere.e	⊢ 𝐸 = (ℝ^‘𝐼)
2sphere.p	⊢ 𝑃 = (ℝ ↑_m 𝐼)
2sphere.s	⊢ 𝑆 = (Sphere‘𝐸)
2sphere.c	⊢ 𝐶 = {𝑝 ∈ 𝑃 ∣ ((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2)) = (𝑅↑2)}

**Assertion**
Ref	Expression
2sphere	⊢ ((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) → (𝑀𝑆𝑅) = 𝐶)

Distinct variable groups: 𝐸,𝑝 𝐼,𝑝 𝑀,𝑝 𝑃,𝑝 𝑅,𝑝 Allowed substitution hints: 𝐶(𝑝) 𝑆(𝑝)

**Proof of Theorem 2sphere**
Step	Hyp	Ref	Expression
1		2sphere.i	. . . 4 ⊢ 𝐼 = {1, 2}
2		prfi 9363	. . . 4 ⊢ {1, 2} ∈ Fin
3	1, 2	eqeltri 2837	. . 3 ⊢ 𝐼 ∈ Fin
4		simpl 482	. . 3 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) → 𝑀 ∈ 𝑃)
5		elrege0 13494	. . . . 5 ⊢ (𝑅 ∈ (0[,)+∞) ↔ (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅))
6	5	simplbi 497	. . . 4 ⊢ (𝑅 ∈ (0[,)+∞) → 𝑅 ∈ ℝ)
7	6	adantl 481	. . 3 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) → 𝑅 ∈ ℝ)
8		2sphere.e	. . . 4 ⊢ 𝐸 = (ℝ^‘𝐼)
9		2sphere.p	. . . 4 ⊢ 𝑃 = (ℝ ↑_m 𝐼)
10		eqid 2737	. . . 4 ⊢ (dist‘𝐸) = (dist‘𝐸)
11		2sphere.s	. . . 4 ⊢ 𝑆 = (Sphere‘𝐸)
12	8, 9, 10, 11	rrxsphere 48669	. . 3 ⊢ ((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) → (𝑀𝑆𝑅) = {𝑝 ∈ 𝑃 ∣ (𝑝(dist‘𝐸)𝑀) = 𝑅})
13	3, 4, 7, 12	mp3an2i 1468	. 2 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) → (𝑀𝑆𝑅) = {𝑝 ∈ 𝑃 ∣ (𝑝(dist‘𝐸)𝑀) = 𝑅})
14		2sphere.c	. . 3 ⊢ 𝐶 = {𝑝 ∈ 𝑃 ∣ ((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2)) = (𝑅↑2)}
15	5	biimpi 216	. . . . . . . 8 ⊢ (𝑅 ∈ (0[,)+∞) → (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅))
16	15	ad2antlr 727	. . . . . . 7 ⊢ (((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) ∧ 𝑝 ∈ 𝑃) → (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅))
17		sqrtsq 15308	. . . . . . 7 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (√‘(𝑅↑2)) = 𝑅)
18	16, 17	syl 17	. . . . . 6 ⊢ (((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) ∧ 𝑝 ∈ 𝑃) → (√‘(𝑅↑2)) = 𝑅)
19	18	eqeq2d 2748	. . . . 5 ⊢ (((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) ∧ 𝑝 ∈ 𝑃) → ((√‘((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2))) = (√‘(𝑅↑2)) ↔ (√‘((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2))) = 𝑅))
20	1, 9	rrx2pxel 48632	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘1) ∈ ℝ)
21	20	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → (𝑝‘1) ∈ ℝ)
22	1, 9	rrx2pxel 48632	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ 𝑃 → (𝑀‘1) ∈ ℝ)
23	22	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → (𝑀‘1) ∈ ℝ)
24	21, 23	resubcld 11691	. . . . . . . . . 10 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → ((𝑝‘1) − (𝑀‘1)) ∈ ℝ)
25	24	resqcld 14165	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → (((𝑝‘1) − (𝑀‘1))↑2) ∈ ℝ)
26	1, 9	rrx2pyel 48633	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘2) ∈ ℝ)
27	26	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → (𝑝‘2) ∈ ℝ)
28	1, 9	rrx2pyel 48633	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ 𝑃 → (𝑀‘2) ∈ ℝ)
29	28	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → (𝑀‘2) ∈ ℝ)
30	27, 29	resubcld 11691	. . . . . . . . . 10 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → ((𝑝‘2) − (𝑀‘2)) ∈ ℝ)
31	30	resqcld 14165	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → (((𝑝‘2) − (𝑀‘2))↑2) ∈ ℝ)
32	25, 31	readdcld 11290	. . . . . . . 8 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → ((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2)) ∈ ℝ)
33	24	sqge0d 14177	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → 0 ≤ (((𝑝‘1) − (𝑀‘1))↑2))
34	30	sqge0d 14177	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → 0 ≤ (((𝑝‘2) − (𝑀‘2))↑2))
35	25, 31, 33, 34	addge0d 11839	. . . . . . . 8 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → 0 ≤ ((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2)))
36	32, 35	jca 511	. . . . . . 7 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → (((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2)) ∈ ℝ ∧ 0 ≤ ((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2))))
37	36	adantlr 715	. . . . . 6 ⊢ (((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) ∧ 𝑝 ∈ 𝑃) → (((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2)) ∈ ℝ ∧ 0 ≤ ((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2))))
38		resqcl 14164	. . . . . . . . 9 ⊢ (𝑅 ∈ ℝ → (𝑅↑2) ∈ ℝ)
39		sqge0 14176	. . . . . . . . 9 ⊢ (𝑅 ∈ ℝ → 0 ≤ (𝑅↑2))
40	38, 39	jca 511	. . . . . . . 8 ⊢ (𝑅 ∈ ℝ → ((𝑅↑2) ∈ ℝ ∧ 0 ≤ (𝑅↑2)))
41	6, 40	syl 17	. . . . . . 7 ⊢ (𝑅 ∈ (0[,)+∞) → ((𝑅↑2) ∈ ℝ ∧ 0 ≤ (𝑅↑2)))
42	41	ad2antlr 727	. . . . . 6 ⊢ (((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) ∧ 𝑝 ∈ 𝑃) → ((𝑅↑2) ∈ ℝ ∧ 0 ≤ (𝑅↑2)))
43		sqrt11 15301	. . . . . 6 ⊢ (((((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2)) ∈ ℝ ∧ 0 ≤ ((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2))) ∧ ((𝑅↑2) ∈ ℝ ∧ 0 ≤ (𝑅↑2))) → ((√‘((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2))) = (√‘(𝑅↑2)) ↔ ((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2)) = (𝑅↑2)))
44	37, 42, 43	syl2anc 584	. . . . 5 ⊢ (((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) ∧ 𝑝 ∈ 𝑃) → ((√‘((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2))) = (√‘(𝑅↑2)) ↔ ((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2)) = (𝑅↑2)))
45	4	anim1ci 616	. . . . . . . 8 ⊢ (((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) ∧ 𝑝 ∈ 𝑃) → (𝑝 ∈ 𝑃 ∧ 𝑀 ∈ 𝑃))
46		2nn0 12543	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ₀
47		eqid 2737	. . . . . . . . . . . . 13 ⊢ (𝔼_hil‘2) = (𝔼_hil‘2)
48	47	ehlval 25448	. . . . . . . . . . . 12 ⊢ (2 ∈ ℕ₀ → (𝔼_hil‘2) = (ℝ^‘(1...2)))
49	46, 48	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝔼_hil‘2) = (ℝ^‘(1...2))
50		fz12pr 13621	. . . . . . . . . . . . 13 ⊢ (1...2) = {1, 2}
51	50, 1	eqtr4i 2768	. . . . . . . . . . . 12 ⊢ (1...2) = 𝐼
52	51	fveq2i 6909	. . . . . . . . . . 11 ⊢ (ℝ^‘(1...2)) = (ℝ^‘𝐼)
53	49, 52	eqtri 2765	. . . . . . . . . 10 ⊢ (𝔼_hil‘2) = (ℝ^‘𝐼)
54	8, 53	eqtr4i 2768	. . . . . . . . 9 ⊢ 𝐸 = (𝔼_hil‘2)
55	1	oveq2i 7442	. . . . . . . . . 10 ⊢ (ℝ ↑_m 𝐼) = (ℝ ↑_m {1, 2})
56	9, 55	eqtri 2765	. . . . . . . . 9 ⊢ 𝑃 = (ℝ ↑_m {1, 2})
57	54, 56, 10	ehl2eudisval 25457	. . . . . . . 8 ⊢ ((𝑝 ∈ 𝑃 ∧ 𝑀 ∈ 𝑃) → (𝑝(dist‘𝐸)𝑀) = (√‘((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2))))
58	45, 57	syl 17	. . . . . . 7 ⊢ (((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) ∧ 𝑝 ∈ 𝑃) → (𝑝(dist‘𝐸)𝑀) = (√‘((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2))))
59	58	eqcomd 2743	. . . . . 6 ⊢ (((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) ∧ 𝑝 ∈ 𝑃) → (√‘((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2))) = (𝑝(dist‘𝐸)𝑀))
60	59	eqeq1d 2739	. . . . 5 ⊢ (((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) ∧ 𝑝 ∈ 𝑃) → ((√‘((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2))) = 𝑅 ↔ (𝑝(dist‘𝐸)𝑀) = 𝑅))
61	19, 44, 60	3bitr3d 309	. . . 4 ⊢ (((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) ∧ 𝑝 ∈ 𝑃) → (((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2)) = (𝑅↑2) ↔ (𝑝(dist‘𝐸)𝑀) = 𝑅))
62	61	rabbidva 3443	. . 3 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) → {𝑝 ∈ 𝑃 ∣ ((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2)) = (𝑅↑2)} = {𝑝 ∈ 𝑃 ∣ (𝑝(dist‘𝐸)𝑀) = 𝑅})
63	14, 62	eqtr2id 2790	. 2 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) → {𝑝 ∈ 𝑃 ∣ (𝑝(dist‘𝐸)𝑀) = 𝑅} = 𝐶)
64	13, 63	eqtrd 2777	1 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) → (𝑀𝑆𝑅) = 𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 {cpr 4628 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ↑_m cmap 8866 Fincfn 8985 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 +∞cpnf 11292 ≤ cle 11296 − cmin 11492 2c2 12321 ℕ₀cn0 12526 [,)cico 13389 ...cfz 13547 ↑cexp 14102 √csqrt 15272 distcds 17306 ℝ^crrx 25417 𝔼_hilcehl 25418 Spherecsph 48649

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-sum 15723 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-ghm 19231 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-rhm 20472 df-subrng 20546 df-subrg 20570 df-drng 20731 df-field 20732 df-staf 20840 df-srng 20841 df-lmod 20860 df-lss 20930 df-sra 21172 df-rgmod 21173 df-xmet 21357 df-met 21358 df-cnfld 21365 df-refld 21623 df-dsmm 21752 df-frlm 21767 df-nm 24595 df-tng 24597 df-tcph 25203 df-rrx 25419 df-ehl 25420 df-sph 48651

This theorem is referenced by: 2sphere0 48671

W3C validator