2sphere - Mathbox for Alexander van der Vekens

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Theorem 2sphere 48483

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 9391	. . . 4 ⊢ {1, 2} ∈ Fin
3	1, 2	eqeltri 2840	. . 3 ⊢ 𝐼 ∈ Fin
4		simpl 482	. . 3 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) → 𝑀 ∈ 𝑃)
5		elrege0 13514	. . . . 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 2740	. . . 4 ⊢ (dist‘𝐸) = (dist‘𝐸)
11		2sphere.s	. . . 4 ⊢ 𝑆 = (Sphere‘𝐸)
12	8, 9, 10, 11	rrxsphere 48482	. . 3 ⊢ ((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) → (𝑀𝑆𝑅) = {𝑝 ∈ 𝑃 ∣ (𝑝(dist‘𝐸)𝑀) = 𝑅})
13	3, 4, 7, 12	mp3an2i 1466	. 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 726	. . . . . . 7 ⊢ (((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) ∧ 𝑝 ∈ 𝑃) → (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅))
17		sqrtsq 15318	. . . . . . 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 2751	. . . . 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 48445	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘1) ∈ ℝ)
21	20	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → (𝑝‘1) ∈ ℝ)
22	1, 9	rrx2pxel 48445	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ 𝑃 → (𝑀‘1) ∈ ℝ)
23	22	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → (𝑀‘1) ∈ ℝ)
24	21, 23	resubcld 11718	. . . . . . . . . 10 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → ((𝑝‘1) − (𝑀‘1)) ∈ ℝ)
25	24	resqcld 14175	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → (((𝑝‘1) − (𝑀‘1))↑2) ∈ ℝ)
26	1, 9	rrx2pyel 48446	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘2) ∈ ℝ)
27	26	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → (𝑝‘2) ∈ ℝ)
28	1, 9	rrx2pyel 48446	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ 𝑃 → (𝑀‘2) ∈ ℝ)
29	28	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → (𝑀‘2) ∈ ℝ)
30	27, 29	resubcld 11718	. . . . . . . . . 10 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → ((𝑝‘2) − (𝑀‘2)) ∈ ℝ)
31	30	resqcld 14175	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → (((𝑝‘2) − (𝑀‘2))↑2) ∈ ℝ)
32	25, 31	readdcld 11319	. . . . . . . 8 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → ((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2)) ∈ ℝ)
33	24	sqge0d 14187	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → 0 ≤ (((𝑝‘1) − (𝑀‘1))↑2))
34	30	sqge0d 14187	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → 0 ≤ (((𝑝‘2) − (𝑀‘2))↑2))
35	25, 31, 33, 34	addge0d 11866	. . . . . . . 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 714	. . . . . 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 14174	. . . . . . . . 9 ⊢ (𝑅 ∈ ℝ → (𝑅↑2) ∈ ℝ)
39		sqge0 14186	. . . . . . . . 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 726	. . . . . 6 ⊢ (((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) ∧ 𝑝 ∈ 𝑃) → ((𝑅↑2) ∈ ℝ ∧ 0 ≤ (𝑅↑2)))
43		sqrt11 15311	. . . . . 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 583	. . . . 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 615	. . . . . . . 8 ⊢ (((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) ∧ 𝑝 ∈ 𝑃) → (𝑝 ∈ 𝑃 ∧ 𝑀 ∈ 𝑃))
46		2nn0 12570	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ₀
47		eqid 2740	. . . . . . . . . . . . 13 ⊢ (𝔼_hil‘2) = (𝔼_hil‘2)
48	47	ehlval 25467	. . . . . . . . . . . 12 ⊢ (2 ∈ ℕ₀ → (𝔼_hil‘2) = (ℝ^‘(1...2)))
49	46, 48	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝔼_hil‘2) = (ℝ^‘(1...2))
50		fz12pr 13641	. . . . . . . . . . . . 13 ⊢ (1...2) = {1, 2}
51	50, 1	eqtr4i 2771	. . . . . . . . . . . 12 ⊢ (1...2) = 𝐼
52	51	fveq2i 6923	. . . . . . . . . . 11 ⊢ (ℝ^‘(1...2)) = (ℝ^‘𝐼)
53	49, 52	eqtri 2768	. . . . . . . . . 10 ⊢ (𝔼_hil‘2) = (ℝ^‘𝐼)
54	8, 53	eqtr4i 2771	. . . . . . . . 9 ⊢ 𝐸 = (𝔼_hil‘2)
55	1	oveq2i 7459	. . . . . . . . . 10 ⊢ (ℝ ↑_m 𝐼) = (ℝ ↑_m {1, 2})
56	9, 55	eqtri 2768	. . . . . . . . 9 ⊢ 𝑃 = (ℝ ↑_m {1, 2})
57	54, 56, 10	ehl2eudisval 25476	. . . . . . . 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 2746	. . . . . 6 ⊢ (((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) ∧ 𝑝 ∈ 𝑃) → (√‘((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2))) = (𝑝(dist‘𝐸)𝑀))
60	59	eqeq1d 2742	. . . . 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 3450	. . 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 2793	. 2 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) → {𝑝 ∈ 𝑃 ∣ (𝑝(dist‘𝐸)𝑀) = 𝑅} = 𝐶)
64	13, 63	eqtrd 2780	1 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) → (𝑀𝑆𝑅) = 𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 {crab 3443 {cpr 4650 class class class wbr 5166 ‘cfv 6573 (class class class)co 7448 ↑_m cmap 8884 Fincfn 9003 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 +∞cpnf 11321 ≤ cle 11325 − cmin 11520 2c2 12348 ℕ₀cn0 12553 [,)cico 13409 ...cfz 13567 ↑cexp 14112 √csqrt 15282 distcds 17320 ℝ^crrx 25436 𝔼_hilcehl 25437 Spherecsph 48462

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263 ax-mulf 11264

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-tpos 8267 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-sup 9511 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-clim 15534 df-sum 15735 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-0g 17501 df-gsum 17502 df-prds 17507 df-pws 17509 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-grp 18976 df-minusg 18977 df-sbg 18978 df-subg 19163 df-ghm 19253 df-cntz 19357 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-cring 20263 df-oppr 20360 df-dvdsr 20383 df-unit 20384 df-invr 20414 df-dvr 20427 df-rhm 20498 df-subrng 20572 df-subrg 20597 df-drng 20753 df-field 20754 df-staf 20862 df-srng 20863 df-lmod 20882 df-lss 20953 df-sra 21195 df-rgmod 21196 df-xmet 21380 df-met 21381 df-cnfld 21388 df-refld 21646 df-dsmm 21775 df-frlm 21790 df-nm 24616 df-tng 24618 df-tcph 25222 df-rrx 25438 df-ehl 25439 df-sph 48464

This theorem is referenced by: 2sphere0 48484

W3C validator