2sphere - Mathbox for Alexander van der Vekens

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Theorem 2sphere 46825

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 9266	. . . 4 ⊢ {1, 2} ∈ Fin
3	1, 2	eqeltri 2834	. . 3 ⊢ 𝐼 ∈ Fin
4		simpl 483	. . 3 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) → 𝑀 ∈ 𝑃)
5		elrege0 13371	. . . . 5 ⊢ (𝑅 ∈ (0[,)+∞) ↔ (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅))
6	5	simplbi 498	. . . 4 ⊢ (𝑅 ∈ (0[,)+∞) → 𝑅 ∈ ℝ)
7	6	adantl 482	. . 3 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) → 𝑅 ∈ ℝ)
8		2sphere.e	. . . 4 ⊢ 𝐸 = (ℝ^‘𝐼)
9		2sphere.p	. . . 4 ⊢ 𝑃 = (ℝ ↑_m 𝐼)
10		eqid 2736	. . . 4 ⊢ (dist‘𝐸) = (dist‘𝐸)
11		2sphere.s	. . . 4 ⊢ 𝑆 = (Sphere‘𝐸)
12	8, 9, 10, 11	rrxsphere 46824	. . 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 215	. . . . . . . 8 ⊢ (𝑅 ∈ (0[,)+∞) → (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅))
16	15	ad2antlr 725	. . . . . . 7 ⊢ (((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) ∧ 𝑝 ∈ 𝑃) → (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅))
17		sqrtsq 15154	. . . . . . 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 2747	. . . . 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 46787	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘1) ∈ ℝ)
21	20	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → (𝑝‘1) ∈ ℝ)
22	1, 9	rrx2pxel 46787	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ 𝑃 → (𝑀‘1) ∈ ℝ)
23	22	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → (𝑀‘1) ∈ ℝ)
24	21, 23	resubcld 11583	. . . . . . . . . 10 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → ((𝑝‘1) − (𝑀‘1)) ∈ ℝ)
25	24	resqcld 14030	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → (((𝑝‘1) − (𝑀‘1))↑2) ∈ ℝ)
26	1, 9	rrx2pyel 46788	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘2) ∈ ℝ)
27	26	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → (𝑝‘2) ∈ ℝ)
28	1, 9	rrx2pyel 46788	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ 𝑃 → (𝑀‘2) ∈ ℝ)
29	28	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → (𝑀‘2) ∈ ℝ)
30	27, 29	resubcld 11583	. . . . . . . . . 10 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → ((𝑝‘2) − (𝑀‘2)) ∈ ℝ)
31	30	resqcld 14030	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → (((𝑝‘2) − (𝑀‘2))↑2) ∈ ℝ)
32	25, 31	readdcld 11184	. . . . . . . 8 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → ((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2)) ∈ ℝ)
33	24	sqge0d 14042	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → 0 ≤ (((𝑝‘1) − (𝑀‘1))↑2))
34	30	sqge0d 14042	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → 0 ≤ (((𝑝‘2) − (𝑀‘2))↑2))
35	25, 31, 33, 34	addge0d 11731	. . . . . . . 8 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → 0 ≤ ((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2)))
36	32, 35	jca 512	. . . . . . 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 713	. . . . . 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 14029	. . . . . . . . 9 ⊢ (𝑅 ∈ ℝ → (𝑅↑2) ∈ ℝ)
39		sqge0 14041	. . . . . . . . 9 ⊢ (𝑅 ∈ ℝ → 0 ≤ (𝑅↑2))
40	38, 39	jca 512	. . . . . . . 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 725	. . . . . 6 ⊢ (((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) ∧ 𝑝 ∈ 𝑃) → ((𝑅↑2) ∈ ℝ ∧ 0 ≤ (𝑅↑2)))
43		sqrt11 15147	. . . . . 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 12430	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ₀
47		eqid 2736	. . . . . . . . . . . . 13 ⊢ (𝔼_hil‘2) = (𝔼_hil‘2)
48	47	ehlval 24778	. . . . . . . . . . . 12 ⊢ (2 ∈ ℕ₀ → (𝔼_hil‘2) = (ℝ^‘(1...2)))
49	46, 48	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝔼_hil‘2) = (ℝ^‘(1...2))
50		fz12pr 13498	. . . . . . . . . . . . 13 ⊢ (1...2) = {1, 2}
51	50, 1	eqtr4i 2767	. . . . . . . . . . . 12 ⊢ (1...2) = 𝐼
52	51	fveq2i 6845	. . . . . . . . . . 11 ⊢ (ℝ^‘(1...2)) = (ℝ^‘𝐼)
53	49, 52	eqtri 2764	. . . . . . . . . 10 ⊢ (𝔼_hil‘2) = (ℝ^‘𝐼)
54	8, 53	eqtr4i 2767	. . . . . . . . 9 ⊢ 𝐸 = (𝔼_hil‘2)
55	1	oveq2i 7368	. . . . . . . . . 10 ⊢ (ℝ ↑_m 𝐼) = (ℝ ↑_m {1, 2})
56	9, 55	eqtri 2764	. . . . . . . . 9 ⊢ 𝑃 = (ℝ ↑_m {1, 2})
57	54, 56, 10	ehl2eudisval 24787	. . . . . . . 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 2742	. . . . . 6 ⊢ (((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) ∧ 𝑝 ∈ 𝑃) → (√‘((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2))) = (𝑝(dist‘𝐸)𝑀))
60	59	eqeq1d 2738	. . . . 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 308	. . . 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 3414	. . 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 2789	. 2 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) → {𝑝 ∈ 𝑃 ∣ (𝑝(dist‘𝐸)𝑀) = 𝑅} = 𝐶)
64	13, 63	eqtrd 2776	1 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) → (𝑀𝑆𝑅) = 𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {crab 3407 {cpr 4588 class class class wbr 5105 ‘cfv 6496 (class class class)co 7357 ↑_m cmap 8765 Fincfn 8883 ℝcr 11050 0cc0 11051 1c1 11052 + caddc 11054 +∞cpnf 11186 ≤ cle 11190 − cmin 11385 2c2 12208 ℕ₀cn0 12413 [,)cico 13266 ...cfz 13424 ↑cexp 13967 √csqrt 15118 distcds 17142 ℝ^crrx 24747 𝔼_hilcehl 24748 Spherecsph 46804

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-tpos 8157 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-sup 9378 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-sum 15571 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-0g 17323 df-gsum 17324 df-prds 17329 df-pws 17331 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-mhm 18601 df-grp 18751 df-minusg 18752 df-sbg 18753 df-subg 18925 df-ghm 19006 df-cntz 19097 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-ring 19966 df-cring 19967 df-oppr 20049 df-dvdsr 20070 df-unit 20071 df-invr 20101 df-dvr 20112 df-rnghom 20146 df-drng 20187 df-field 20188 df-subrg 20220 df-staf 20304 df-srng 20305 df-lmod 20324 df-lss 20393 df-sra 20633 df-rgmod 20634 df-xmet 20789 df-met 20790 df-cnfld 20797 df-refld 21009 df-dsmm 21138 df-frlm 21153 df-nm 23938 df-tng 23940 df-tcph 24533 df-rrx 24749 df-ehl 24750 df-sph 46806

This theorem is referenced by: 2sphere0 46826

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