2sphere - Mathbox for Alexander van der Vekens

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Theorem 2sphere 48742

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 9281	. . . 4 ⊢ {1, 2} ∈ Fin
3	1, 2	eqeltri 2825	. . 3 ⊢ 𝐼 ∈ Fin
4		simpl 482	. . 3 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) → 𝑀 ∈ 𝑃)
5		elrege0 13422	. . . . 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 2730	. . . 4 ⊢ (dist‘𝐸) = (dist‘𝐸)
11		2sphere.s	. . . 4 ⊢ 𝑆 = (Sphere‘𝐸)
12	8, 9, 10, 11	rrxsphere 48741	. . 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 15242	. . . . . . 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 2741	. . . . 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 48704	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘1) ∈ ℝ)
21	20	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → (𝑝‘1) ∈ ℝ)
22	1, 9	rrx2pxel 48704	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ 𝑃 → (𝑀‘1) ∈ ℝ)
23	22	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → (𝑀‘1) ∈ ℝ)
24	21, 23	resubcld 11613	. . . . . . . . . 10 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → ((𝑝‘1) − (𝑀‘1)) ∈ ℝ)
25	24	resqcld 14097	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → (((𝑝‘1) − (𝑀‘1))↑2) ∈ ℝ)
26	1, 9	rrx2pyel 48705	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘2) ∈ ℝ)
27	26	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → (𝑝‘2) ∈ ℝ)
28	1, 9	rrx2pyel 48705	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ 𝑃 → (𝑀‘2) ∈ ℝ)
29	28	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → (𝑀‘2) ∈ ℝ)
30	27, 29	resubcld 11613	. . . . . . . . . 10 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → ((𝑝‘2) − (𝑀‘2)) ∈ ℝ)
31	30	resqcld 14097	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → (((𝑝‘2) − (𝑀‘2))↑2) ∈ ℝ)
32	25, 31	readdcld 11210	. . . . . . . 8 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → ((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2)) ∈ ℝ)
33	24	sqge0d 14109	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → 0 ≤ (((𝑝‘1) − (𝑀‘1))↑2))
34	30	sqge0d 14109	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → 0 ≤ (((𝑝‘2) − (𝑀‘2))↑2))
35	25, 31, 33, 34	addge0d 11761	. . . . . . . 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 14096	. . . . . . . . 9 ⊢ (𝑅 ∈ ℝ → (𝑅↑2) ∈ ℝ)
39		sqge0 14108	. . . . . . . . 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 15235	. . . . . 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 12466	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ₀
47		eqid 2730	. . . . . . . . . . . . 13 ⊢ (𝔼_hil‘2) = (𝔼_hil‘2)
48	47	ehlval 25321	. . . . . . . . . . . 12 ⊢ (2 ∈ ℕ₀ → (𝔼_hil‘2) = (ℝ^‘(1...2)))
49	46, 48	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝔼_hil‘2) = (ℝ^‘(1...2))
50		fz12pr 13549	. . . . . . . . . . . . 13 ⊢ (1...2) = {1, 2}
51	50, 1	eqtr4i 2756	. . . . . . . . . . . 12 ⊢ (1...2) = 𝐼
52	51	fveq2i 6864	. . . . . . . . . . 11 ⊢ (ℝ^‘(1...2)) = (ℝ^‘𝐼)
53	49, 52	eqtri 2753	. . . . . . . . . 10 ⊢ (𝔼_hil‘2) = (ℝ^‘𝐼)
54	8, 53	eqtr4i 2756	. . . . . . . . 9 ⊢ 𝐸 = (𝔼_hil‘2)
55	1	oveq2i 7401	. . . . . . . . . 10 ⊢ (ℝ ↑_m 𝐼) = (ℝ ↑_m {1, 2})
56	9, 55	eqtri 2753	. . . . . . . . 9 ⊢ 𝑃 = (ℝ ↑_m {1, 2})
57	54, 56, 10	ehl2eudisval 25330	. . . . . . . 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 2736	. . . . . 6 ⊢ (((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) ∧ 𝑝 ∈ 𝑃) → (√‘((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2))) = (𝑝(dist‘𝐸)𝑀))
60	59	eqeq1d 2732	. . . . 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 3415	. . 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 2778	. 2 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) → {𝑝 ∈ 𝑃 ∣ (𝑝(dist‘𝐸)𝑀) = 𝑅} = 𝐶)
64	13, 63	eqtrd 2765	1 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) → (𝑀𝑆𝑅) = 𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 {crab 3408 {cpr 4594 class class class wbr 5110 ‘cfv 6514 (class class class)co 7390 ↑_m cmap 8802 Fincfn 8921 ℝcr 11074 0cc0 11075 1c1 11076 + caddc 11078 +∞cpnf 11212 ≤ cle 11216 − cmin 11412 2c2 12248 ℕ₀cn0 12449 [,)cico 13315 ...cfz 13475 ↑cexp 14033 √csqrt 15206 distcds 17236 ℝ^crrx 25290 𝔼_hilcehl 25291 Spherecsph 48721

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154 ax-mulf 11155

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-tpos 8208 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-sup 9400 df-oi 9470 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ico 13319 df-icc 13320 df-fz 13476 df-fzo 13623 df-seq 13974 df-exp 14034 df-hash 14303 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15461 df-sum 15660 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-0g 17411 df-gsum 17412 df-prds 17417 df-pws 17419 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-mhm 18717 df-grp 18875 df-minusg 18876 df-sbg 18877 df-subg 19062 df-ghm 19152 df-cntz 19256 df-cmn 19719 df-abl 19720 df-mgp 20057 df-rng 20069 df-ur 20098 df-ring 20151 df-cring 20152 df-oppr 20253 df-dvdsr 20273 df-unit 20274 df-invr 20304 df-dvr 20317 df-rhm 20388 df-subrng 20462 df-subrg 20486 df-drng 20647 df-field 20648 df-staf 20755 df-srng 20756 df-lmod 20775 df-lss 20845 df-sra 21087 df-rgmod 21088 df-xmet 21264 df-met 21265 df-cnfld 21272 df-refld 21521 df-dsmm 21648 df-frlm 21663 df-nm 24477 df-tng 24479 df-tcph 25076 df-rrx 25292 df-ehl 25293 df-sph 48723

This theorem is referenced by: 2sphere0 48743

W3C validator