2sphere - Mathbox for Alexander van der Vekens

	Mathbox for Alexander van der Vekens		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > 2sphere		Structured version Visualization version GIF version

Theorem 2sphere 49557

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 2858	. . 3 ⊢ 𝐼 ∈ Fin
4		simpl 487	. . 3 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) → 𝑀 ∈ 𝑃)
5		elrege0 13487	. . . . 5 ⊢ (𝑅 ∈ (0[,)+∞) ↔ (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅))
6	5	simplbi 501	. . . 4 ⊢ (𝑅 ∈ (0[,)+∞) → 𝑅 ∈ ℝ)
7	6	adantl 486	. . 3 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) → 𝑅 ∈ ℝ)
8		2sphere.e	. . . 4 ⊢ 𝐸 = (ℝ^‘𝐼)
9		2sphere.p	. . . 4 ⊢ 𝑃 = (ℝ ↑_m 𝐼)
10		eqid 2762	. . . 4 ⊢ (dist‘𝐸) = (dist‘𝐸)
11		2sphere.s	. . . 4 ⊢ 𝑆 = (Sphere‘𝐸)
12	8, 9, 10, 11	rrxsphere 49556	. . 3 ⊢ ((𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ) → (𝑀𝑆𝑅) = {𝑝 ∈ 𝑃 ∣ (𝑝(dist‘𝐸)𝑀) = 𝑅})
13	3, 4, 7, 12	mp3an2i 1494	. 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 219	. . . . . . . 8 ⊢ (𝑅 ∈ (0[,)+∞) → (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅))
16	15	ad2antlr 739	. . . . . . 7 ⊢ (((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) ∧ 𝑝 ∈ 𝑃) → (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅))
17		sqrtsq 15327	. . . . . . 7 ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (√‘(𝑅↑2)) = 𝑅)
18	16, 17	syl 18	. . . . . 6 ⊢ (((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) ∧ 𝑝 ∈ 𝑃) → (√‘(𝑅↑2)) = 𝑅)
19	18	eqeq2d 2773	. . . . 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 49519	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘1) ∈ ℝ)
21	20	adantl 486	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → (𝑝‘1) ∈ ℝ)
22	1, 9	rrx2pxel 49519	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ 𝑃 → (𝑀‘1) ∈ ℝ)
23	22	adantr 485	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → (𝑀‘1) ∈ ℝ)
24	21, 23	resubcld 11648	. . . . . . . . . 10 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → ((𝑝‘1) − (𝑀‘1)) ∈ ℝ)
25	24	resqcld 14168	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → (((𝑝‘1) − (𝑀‘1))↑2) ∈ ℝ)
26	1, 9	rrx2pyel 49520	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘2) ∈ ℝ)
27	26	adantl 486	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → (𝑝‘2) ∈ ℝ)
28	1, 9	rrx2pyel 49520	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ 𝑃 → (𝑀‘2) ∈ ℝ)
29	28	adantr 485	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → (𝑀‘2) ∈ ℝ)
30	27, 29	resubcld 11648	. . . . . . . . . 10 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → ((𝑝‘2) − (𝑀‘2)) ∈ ℝ)
31	30	resqcld 14168	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → (((𝑝‘2) − (𝑀‘2))↑2) ∈ ℝ)
32	25, 31	readdcld 11244	. . . . . . . 8 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → ((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2)) ∈ ℝ)
33	24	sqge0d 14180	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → 0 ≤ (((𝑝‘1) − (𝑀‘1))↑2))
34	30	sqge0d 14180	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → 0 ≤ (((𝑝‘2) − (𝑀‘2))↑2))
35	25, 31, 33, 34	addge0d 11796	. . . . . . . 8 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃) → 0 ≤ ((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2)))
36	32, 35	jca 520	. . . . . . 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 727	. . . . . 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 14167	. . . . . . . . 9 ⊢ (𝑅 ∈ ℝ → (𝑅↑2) ∈ ℝ)
39		sqge0 14179	. . . . . . . . 9 ⊢ (𝑅 ∈ ℝ → 0 ≤ (𝑅↑2))
40	38, 39	jca 520	. . . . . . . 8 ⊢ (𝑅 ∈ ℝ → ((𝑅↑2) ∈ ℝ ∧ 0 ≤ (𝑅↑2)))
41	6, 40	syl 18	. . . . . . 7 ⊢ (𝑅 ∈ (0[,)+∞) → ((𝑅↑2) ∈ ℝ ∧ 0 ≤ (𝑅↑2)))
42	41	ad2antlr 739	. . . . . 6 ⊢ (((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) ∧ 𝑝 ∈ 𝑃) → ((𝑅↑2) ∈ ℝ ∧ 0 ≤ (𝑅↑2)))
43		sqrt11 15320	. . . . . 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 595	. . . . 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 627	. . . . . . . 8 ⊢ (((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) ∧ 𝑝 ∈ 𝑃) → (𝑝 ∈ 𝑃 ∧ 𝑀 ∈ 𝑃))
46		2nn0 12527	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ₀
47		eqid 2762	. . . . . . . . . . . . 13 ⊢ (𝔼_hil‘2) = (𝔼_hil‘2)
48	47	ehlval 25584	. . . . . . . . . . . 12 ⊢ (2 ∈ ℕ₀ → (𝔼_hil‘2) = (ℝ^‘(1...2)))
49	46, 48	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝔼_hil‘2) = (ℝ^‘(1...2))
50		fz12pr 13616	. . . . . . . . . . . . 13 ⊢ (1...2) = {1, 2}
51	50, 1	eqtr4i 2788	. . . . . . . . . . . 12 ⊢ (1...2) = 𝐼
52	51	fveq2i 6884	. . . . . . . . . . 11 ⊢ (ℝ^‘(1...2)) = (ℝ^‘𝐼)
53	49, 52	eqtri 2785	. . . . . . . . . 10 ⊢ (𝔼_hil‘2) = (ℝ^‘𝐼)
54	8, 53	eqtr4i 2788	. . . . . . . . 9 ⊢ 𝐸 = (𝔼_hil‘2)
55	1	oveq2i 7423	. . . . . . . . . 10 ⊢ (ℝ ↑_m 𝐼) = (ℝ ↑_m {1, 2})
56	9, 55	eqtri 2785	. . . . . . . . 9 ⊢ 𝑃 = (ℝ ↑_m {1, 2})
57	54, 56, 10	ehl2eudisval 25593	. . . . . . . 8 ⊢ ((𝑝 ∈ 𝑃 ∧ 𝑀 ∈ 𝑃) → (𝑝(dist‘𝐸)𝑀) = (√‘((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2))))
58	45, 57	syl 18	. . . . . . 7 ⊢ (((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) ∧ 𝑝 ∈ 𝑃) → (𝑝(dist‘𝐸)𝑀) = (√‘((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2))))
59	58	eqcomd 2768	. . . . . 6 ⊢ (((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) ∧ 𝑝 ∈ 𝑃) → (√‘((((𝑝‘1) − (𝑀‘1))↑2) + (((𝑝‘2) − (𝑀‘2))↑2))) = (𝑝(dist‘𝐸)𝑀))
60	59	eqeq1d 2764	. . . . 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 312	. . . 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 3421	. . 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 2810	. 2 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) → {𝑝 ∈ 𝑃 ∣ (𝑝(dist‘𝐸)𝑀) = 𝑅} = 𝐶)
64	13, 63	eqtrd 2797	1 ⊢ ((𝑀 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) → (𝑀𝑆𝑅) = 𝐶)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1569 ∈ wcel 2142 {crab 3415 {cpr 4590 class class class wbr 5108 ‘cfv 6536 (class class class)co 7412 ↑_m cmap 8822 Fincfn 8941 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 +∞cpnf 11246 ≤ cle 11250 − cmin 11447 2c2 12301 ℕ₀cn0 12510 [,)cico 13380 ...cfz 13541 ↑cexp 14104 √csqrt 15291 distcds 17325 ℝ^crrx 25553 𝔼_hilcehl 25554 Spherecsph 49536

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-inf2 9608 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 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-se 5614 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-om 7861 df-1st 7984 df-2nd 7985 df-supp 8155 df-tpos 8220 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-2o 8452 df-er 8692 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9320 df-sup 9400 df-oi 9470 df-card 9932 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-div 11878 df-nn 12240 df-2 12309 df-3 12310 df-4 12311 df-5 12312 df-6 12313 df-7 12314 df-8 12315 df-9 12316 df-n0 12511 df-z 12598 df-dec 12718 df-uz 12869 df-rp 13023 df-xneg 13143 df-xadd 13144 df-xmul 13145 df-ico 13384 df-icc 13385 df-fz 13542 df-fzo 13690 df-seq 14045 df-exp 14105 df-hash 14374 df-cj 15157 df-re 15158 df-im 15159 df-sqrt 15293 df-abs 15294 df-clim 15546 df-sum 15745 df-struct 17213 df-sets 17230 df-slot 17248 df-ndx 17260 df-base 17276 df-ress 17297 df-plusg 17329 df-mulr 17330 df-starv 17331 df-sca 17332 df-vsca 17333 df-ip 17334 df-tset 17335 df-ple 17336 df-ds 17338 df-unif 17339 df-hom 17340 df-cco 17341 df-0g 17500 df-gsum 17501 df-prds 17506 df-pws 17508 df-mgm 18704 df-sgrp 18783 df-mnd 18799 df-mhm 18847 df-grp 19009 df-minusg 19010 df-sbg 19011 df-subg 19195 df-ghm 19290 df-cntz 19393 df-cmn 19858 df-abl 19859 df-mgp 20223 df-rng 20237 df-ur 20270 df-ring 20323 df-cring 20324 df-oppr 20426 df-dvdsr 20446 df-unit 20447 df-invr 20477 df-dvr 20490 df-rhm 20561 df-subrng 20656 df-subrg 20680 df-drng 20840 df-field 20841 df-staf 20953 df-srng 20954 df-lmod 20994 df-lss 21064 df-sra 21305 df-rgmod 21306 df-xmet 21526 df-met 21527 df-cnfld 21534 df-refld 21766 df-dsmm 21893 df-frlm 21908 df-nm 24750 df-tng 24752 df-tcph 25339 df-rrx 25555 df-ehl 25556 df-sph 49538

This theorem is used by: 2sphere0 49558

W3C validator