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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2sphere0 | Structured version Visualization version GIF version |
Description: The sphere around the origin 0 (see rrx0 23572) with radius 𝑅 in a two dimensional Euclidean space is a circle. (Contributed by AV, 5-Feb-2023.) |
Ref | Expression |
---|---|
2sphere.i | ⊢ 𝐼 = {1, 2} |
2sphere.e | ⊢ 𝐸 = (ℝ^‘𝐼) |
2sphere.p | ⊢ 𝑃 = (ℝ ↑𝑚 𝐼) |
2sphere.s | ⊢ 𝑆 = (Sphere‘𝐸) |
2sphere0.0 | ⊢ 0 = (𝐼 × {0}) |
2sphere0.c | ⊢ 𝐶 = {𝑝 ∈ 𝑃 ∣ (((𝑝‘1)↑2) + ((𝑝‘2)↑2)) = (𝑅↑2)} |
Ref | Expression |
---|---|
2sphere0 | ⊢ (𝑅 ∈ (0[,)+∞) → ( 0 𝑆𝑅) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2sphere.i | . . . . 5 ⊢ 𝐼 = {1, 2} | |
2 | prex 5132 | . . . . 5 ⊢ {1, 2} ∈ V | |
3 | 1, 2 | eqeltri 2902 | . . . 4 ⊢ 𝐼 ∈ V |
4 | 2sphere0.0 | . . . . 5 ⊢ 0 = (𝐼 × {0}) | |
5 | 2sphere.p | . . . . 5 ⊢ 𝑃 = (ℝ ↑𝑚 𝐼) | |
6 | 4, 5 | rrx0el 23573 | . . . 4 ⊢ (𝐼 ∈ V → 0 ∈ 𝑃) |
7 | 3, 6 | ax-mp 5 | . . 3 ⊢ 0 ∈ 𝑃 |
8 | 2sphere.e | . . . 4 ⊢ 𝐸 = (ℝ^‘𝐼) | |
9 | 2sphere.s | . . . 4 ⊢ 𝑆 = (Sphere‘𝐸) | |
10 | eqid 2825 | . . . 4 ⊢ {𝑝 ∈ 𝑃 ∣ ((((𝑝‘1) − ( 0 ‘1))↑2) + (((𝑝‘2) − ( 0 ‘2))↑2)) = (𝑅↑2)} = {𝑝 ∈ 𝑃 ∣ ((((𝑝‘1) − ( 0 ‘1))↑2) + (((𝑝‘2) − ( 0 ‘2))↑2)) = (𝑅↑2)} | |
11 | 1, 8, 5, 9, 10 | 2sphere 43311 | . . 3 ⊢ (( 0 ∈ 𝑃 ∧ 𝑅 ∈ (0[,)+∞)) → ( 0 𝑆𝑅) = {𝑝 ∈ 𝑃 ∣ ((((𝑝‘1) − ( 0 ‘1))↑2) + (((𝑝‘2) − ( 0 ‘2))↑2)) = (𝑅↑2)}) |
12 | 7, 11 | mpan 681 | . 2 ⊢ (𝑅 ∈ (0[,)+∞) → ( 0 𝑆𝑅) = {𝑝 ∈ 𝑃 ∣ ((((𝑝‘1) − ( 0 ‘1))↑2) + (((𝑝‘2) − ( 0 ‘2))↑2)) = (𝑅↑2)}) |
13 | 4 | fveq1i 6438 | . . . . . . . . . . . 12 ⊢ ( 0 ‘1) = ((𝐼 × {0})‘1) |
14 | c0ex 10357 | . . . . . . . . . . . . 13 ⊢ 0 ∈ V | |
15 | 1ex 10359 | . . . . . . . . . . . . . . 15 ⊢ 1 ∈ V | |
16 | 15 | prid1 4517 | . . . . . . . . . . . . . 14 ⊢ 1 ∈ {1, 2} |
17 | 16, 1 | eleqtrri 2905 | . . . . . . . . . . . . 13 ⊢ 1 ∈ 𝐼 |
18 | fvconst2g 6728 | . . . . . . . . . . . . 13 ⊢ ((0 ∈ V ∧ 1 ∈ 𝐼) → ((𝐼 × {0})‘1) = 0) | |
19 | 14, 17, 18 | mp2an 683 | . . . . . . . . . . . 12 ⊢ ((𝐼 × {0})‘1) = 0 |
20 | 13, 19 | eqtri 2849 | . . . . . . . . . . 11 ⊢ ( 0 ‘1) = 0 |
21 | 20 | a1i 11 | . . . . . . . . . 10 ⊢ (𝑝 ∈ 𝑃 → ( 0 ‘1) = 0) |
22 | 21 | oveq2d 6926 | . . . . . . . . 9 ⊢ (𝑝 ∈ 𝑃 → ((𝑝‘1) − ( 0 ‘1)) = ((𝑝‘1) − 0)) |
23 | 1, 5 | rrx2pxel 42271 | . . . . . . . . . . 11 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘1) ∈ ℝ) |
24 | 23 | recnd 10392 | . . . . . . . . . 10 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘1) ∈ ℂ) |
25 | 24 | subid1d 10709 | . . . . . . . . 9 ⊢ (𝑝 ∈ 𝑃 → ((𝑝‘1) − 0) = (𝑝‘1)) |
26 | 22, 25 | eqtrd 2861 | . . . . . . . 8 ⊢ (𝑝 ∈ 𝑃 → ((𝑝‘1) − ( 0 ‘1)) = (𝑝‘1)) |
27 | 26 | oveq1d 6925 | . . . . . . 7 ⊢ (𝑝 ∈ 𝑃 → (((𝑝‘1) − ( 0 ‘1))↑2) = ((𝑝‘1)↑2)) |
28 | 4 | fveq1i 6438 | . . . . . . . . . . . 12 ⊢ ( 0 ‘2) = ((𝐼 × {0})‘2) |
29 | 2ex 11435 | . . . . . . . . . . . . . . 15 ⊢ 2 ∈ V | |
30 | 29 | prid2 4518 | . . . . . . . . . . . . . 14 ⊢ 2 ∈ {1, 2} |
31 | 30, 1 | eleqtrri 2905 | . . . . . . . . . . . . 13 ⊢ 2 ∈ 𝐼 |
32 | fvconst2g 6728 | . . . . . . . . . . . . 13 ⊢ ((0 ∈ V ∧ 2 ∈ 𝐼) → ((𝐼 × {0})‘2) = 0) | |
33 | 14, 31, 32 | mp2an 683 | . . . . . . . . . . . 12 ⊢ ((𝐼 × {0})‘2) = 0 |
34 | 28, 33 | eqtri 2849 | . . . . . . . . . . 11 ⊢ ( 0 ‘2) = 0 |
35 | 34 | a1i 11 | . . . . . . . . . 10 ⊢ (𝑝 ∈ 𝑃 → ( 0 ‘2) = 0) |
36 | 35 | oveq2d 6926 | . . . . . . . . 9 ⊢ (𝑝 ∈ 𝑃 → ((𝑝‘2) − ( 0 ‘2)) = ((𝑝‘2) − 0)) |
37 | 1, 5 | rrx2pyel 42272 | . . . . . . . . . . 11 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘2) ∈ ℝ) |
38 | 37 | recnd 10392 | . . . . . . . . . 10 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘2) ∈ ℂ) |
39 | 38 | subid1d 10709 | . . . . . . . . 9 ⊢ (𝑝 ∈ 𝑃 → ((𝑝‘2) − 0) = (𝑝‘2)) |
40 | 36, 39 | eqtrd 2861 | . . . . . . . 8 ⊢ (𝑝 ∈ 𝑃 → ((𝑝‘2) − ( 0 ‘2)) = (𝑝‘2)) |
41 | 40 | oveq1d 6925 | . . . . . . 7 ⊢ (𝑝 ∈ 𝑃 → (((𝑝‘2) − ( 0 ‘2))↑2) = ((𝑝‘2)↑2)) |
42 | 27, 41 | oveq12d 6928 | . . . . . 6 ⊢ (𝑝 ∈ 𝑃 → ((((𝑝‘1) − ( 0 ‘1))↑2) + (((𝑝‘2) − ( 0 ‘2))↑2)) = (((𝑝‘1)↑2) + ((𝑝‘2)↑2))) |
43 | 42 | eqeq1d 2827 | . . . . 5 ⊢ (𝑝 ∈ 𝑃 → (((((𝑝‘1) − ( 0 ‘1))↑2) + (((𝑝‘2) − ( 0 ‘2))↑2)) = (𝑅↑2) ↔ (((𝑝‘1)↑2) + ((𝑝‘2)↑2)) = (𝑅↑2))) |
44 | 43 | adantl 475 | . . . 4 ⊢ ((𝑅 ∈ (0[,)+∞) ∧ 𝑝 ∈ 𝑃) → (((((𝑝‘1) − ( 0 ‘1))↑2) + (((𝑝‘2) − ( 0 ‘2))↑2)) = (𝑅↑2) ↔ (((𝑝‘1)↑2) + ((𝑝‘2)↑2)) = (𝑅↑2))) |
45 | 44 | rabbidva 3401 | . . 3 ⊢ (𝑅 ∈ (0[,)+∞) → {𝑝 ∈ 𝑃 ∣ ((((𝑝‘1) − ( 0 ‘1))↑2) + (((𝑝‘2) − ( 0 ‘2))↑2)) = (𝑅↑2)} = {𝑝 ∈ 𝑃 ∣ (((𝑝‘1)↑2) + ((𝑝‘2)↑2)) = (𝑅↑2)}) |
46 | 2sphere0.c | . . 3 ⊢ 𝐶 = {𝑝 ∈ 𝑃 ∣ (((𝑝‘1)↑2) + ((𝑝‘2)↑2)) = (𝑅↑2)} | |
47 | 45, 46 | syl6eqr 2879 | . 2 ⊢ (𝑅 ∈ (0[,)+∞) → {𝑝 ∈ 𝑃 ∣ ((((𝑝‘1) − ( 0 ‘1))↑2) + (((𝑝‘2) − ( 0 ‘2))↑2)) = (𝑅↑2)} = 𝐶) |
48 | 12, 47 | eqtrd 2861 | 1 ⊢ (𝑅 ∈ (0[,)+∞) → ( 0 𝑆𝑅) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1656 ∈ wcel 2164 {crab 3121 Vcvv 3414 {csn 4399 {cpr 4401 × cxp 5344 ‘cfv 6127 (class class class)co 6910 ↑𝑚 cmap 8127 ℝcr 10258 0cc0 10259 1c1 10260 + caddc 10262 +∞cpnf 10395 − cmin 10592 2c2 11413 [,)cico 12472 ↑cexp 13161 ℝ^crrx 23558 Spherecsph 43292 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-pre-sup 10337 ax-addf 10338 ax-mulf 10339 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-se 5306 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-isom 6136 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-of 7162 df-om 7332 df-1st 7433 df-2nd 7434 df-supp 7565 df-tpos 7622 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-map 8129 df-ixp 8182 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-fsupp 8551 df-sup 8623 df-oi 8691 df-card 9085 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-dec 11829 df-uz 11976 df-rp 12120 df-xneg 12239 df-xadd 12240 df-xmul 12241 df-ico 12476 df-icc 12477 df-fz 12627 df-fzo 12768 df-seq 13103 df-exp 13162 df-hash 13418 df-cj 14223 df-re 14224 df-im 14225 df-sqrt 14359 df-abs 14360 df-clim 14603 df-sum 14801 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-starv 16327 df-sca 16328 df-vsca 16329 df-ip 16330 df-tset 16331 df-ple 16332 df-ds 16334 df-unif 16335 df-hom 16336 df-cco 16337 df-0g 16462 df-gsum 16463 df-prds 16468 df-pws 16470 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-mhm 17695 df-grp 17786 df-minusg 17787 df-sbg 17788 df-subg 17949 df-ghm 18016 df-cntz 18107 df-cmn 18555 df-abl 18556 df-mgp 18851 df-ur 18863 df-ring 18910 df-cring 18911 df-oppr 18984 df-dvdsr 19002 df-unit 19003 df-invr 19033 df-dvr 19044 df-rnghom 19078 df-drng 19112 df-field 19113 df-subrg 19141 df-staf 19208 df-srng 19209 df-lmod 19228 df-lss 19296 df-sra 19540 df-rgmod 19541 df-xmet 20106 df-met 20107 df-cnfld 20114 df-refld 20319 df-dsmm 20446 df-frlm 20461 df-nm 22764 df-tng 22766 df-tcph 23345 df-rrx 23560 df-ehl 23561 df-sph 43294 |
This theorem is referenced by: itsclc0 43323 |
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