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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 24001) 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 5298 | . . . . 5 ⊢ {1, 2} ∈ V | |
3 | 1, 2 | eqeltri 2886 | . . . 4 ⊢ 𝐼 ∈ V |
4 | 2sphere0.0 | . . . . 5 ⊢ 0 = (𝐼 × {0}) | |
5 | 2sphere.p | . . . . 5 ⊢ 𝑃 = (ℝ ↑m 𝐼) | |
6 | 4, 5 | rrx0el 24002 | . . . 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 2798 | . . . 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 45163 | . . 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 689 | . 2 ⊢ (𝑅 ∈ (0[,)+∞) → ( 0 𝑆𝑅) = {𝑝 ∈ 𝑃 ∣ ((((𝑝‘1) − ( 0 ‘1))↑2) + (((𝑝‘2) − ( 0 ‘2))↑2)) = (𝑅↑2)}) |
13 | 4 | fveq1i 6646 | . . . . . . . . . . . 12 ⊢ ( 0 ‘1) = ((𝐼 × {0})‘1) |
14 | c0ex 10624 | . . . . . . . . . . . . 13 ⊢ 0 ∈ V | |
15 | 1ex 10626 | . . . . . . . . . . . . . . 15 ⊢ 1 ∈ V | |
16 | 15 | prid1 4658 | . . . . . . . . . . . . . 14 ⊢ 1 ∈ {1, 2} |
17 | 16, 1 | eleqtrri 2889 | . . . . . . . . . . . . 13 ⊢ 1 ∈ 𝐼 |
18 | fvconst2g 6941 | . . . . . . . . . . . . 13 ⊢ ((0 ∈ V ∧ 1 ∈ 𝐼) → ((𝐼 × {0})‘1) = 0) | |
19 | 14, 17, 18 | mp2an 691 | . . . . . . . . . . . 12 ⊢ ((𝐼 × {0})‘1) = 0 |
20 | 13, 19 | eqtri 2821 | . . . . . . . . . . 11 ⊢ ( 0 ‘1) = 0 |
21 | 20 | a1i 11 | . . . . . . . . . 10 ⊢ (𝑝 ∈ 𝑃 → ( 0 ‘1) = 0) |
22 | 21 | oveq2d 7151 | . . . . . . . . 9 ⊢ (𝑝 ∈ 𝑃 → ((𝑝‘1) − ( 0 ‘1)) = ((𝑝‘1) − 0)) |
23 | 1, 5 | rrx2pxel 45125 | . . . . . . . . . . 11 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘1) ∈ ℝ) |
24 | 23 | recnd 10658 | . . . . . . . . . 10 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘1) ∈ ℂ) |
25 | 24 | subid1d 10975 | . . . . . . . . 9 ⊢ (𝑝 ∈ 𝑃 → ((𝑝‘1) − 0) = (𝑝‘1)) |
26 | 22, 25 | eqtrd 2833 | . . . . . . . 8 ⊢ (𝑝 ∈ 𝑃 → ((𝑝‘1) − ( 0 ‘1)) = (𝑝‘1)) |
27 | 26 | oveq1d 7150 | . . . . . . 7 ⊢ (𝑝 ∈ 𝑃 → (((𝑝‘1) − ( 0 ‘1))↑2) = ((𝑝‘1)↑2)) |
28 | 4 | fveq1i 6646 | . . . . . . . . . . . 12 ⊢ ( 0 ‘2) = ((𝐼 × {0})‘2) |
29 | 2ex 11702 | . . . . . . . . . . . . . . 15 ⊢ 2 ∈ V | |
30 | 29 | prid2 4659 | . . . . . . . . . . . . . 14 ⊢ 2 ∈ {1, 2} |
31 | 30, 1 | eleqtrri 2889 | . . . . . . . . . . . . 13 ⊢ 2 ∈ 𝐼 |
32 | fvconst2g 6941 | . . . . . . . . . . . . 13 ⊢ ((0 ∈ V ∧ 2 ∈ 𝐼) → ((𝐼 × {0})‘2) = 0) | |
33 | 14, 31, 32 | mp2an 691 | . . . . . . . . . . . 12 ⊢ ((𝐼 × {0})‘2) = 0 |
34 | 28, 33 | eqtri 2821 | . . . . . . . . . . 11 ⊢ ( 0 ‘2) = 0 |
35 | 34 | a1i 11 | . . . . . . . . . 10 ⊢ (𝑝 ∈ 𝑃 → ( 0 ‘2) = 0) |
36 | 35 | oveq2d 7151 | . . . . . . . . 9 ⊢ (𝑝 ∈ 𝑃 → ((𝑝‘2) − ( 0 ‘2)) = ((𝑝‘2) − 0)) |
37 | 1, 5 | rrx2pyel 45126 | . . . . . . . . . . 11 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘2) ∈ ℝ) |
38 | 37 | recnd 10658 | . . . . . . . . . 10 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘2) ∈ ℂ) |
39 | 38 | subid1d 10975 | . . . . . . . . 9 ⊢ (𝑝 ∈ 𝑃 → ((𝑝‘2) − 0) = (𝑝‘2)) |
40 | 36, 39 | eqtrd 2833 | . . . . . . . 8 ⊢ (𝑝 ∈ 𝑃 → ((𝑝‘2) − ( 0 ‘2)) = (𝑝‘2)) |
41 | 40 | oveq1d 7150 | . . . . . . 7 ⊢ (𝑝 ∈ 𝑃 → (((𝑝‘2) − ( 0 ‘2))↑2) = ((𝑝‘2)↑2)) |
42 | 27, 41 | oveq12d 7153 | . . . . . 6 ⊢ (𝑝 ∈ 𝑃 → ((((𝑝‘1) − ( 0 ‘1))↑2) + (((𝑝‘2) − ( 0 ‘2))↑2)) = (((𝑝‘1)↑2) + ((𝑝‘2)↑2))) |
43 | 42 | eqeq1d 2800 | . . . . 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 485 | . . . 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 3425 | . . 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 | eqtr4di 2851 | . 2 ⊢ (𝑅 ∈ (0[,)+∞) → {𝑝 ∈ 𝑃 ∣ ((((𝑝‘1) − ( 0 ‘1))↑2) + (((𝑝‘2) − ( 0 ‘2))↑2)) = (𝑅↑2)} = 𝐶) |
48 | 12, 47 | eqtrd 2833 | 1 ⊢ (𝑅 ∈ (0[,)+∞) → ( 0 𝑆𝑅) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 {crab 3110 Vcvv 3441 {csn 4525 {cpr 4527 × cxp 5517 ‘cfv 6324 (class class class)co 7135 ↑m cmap 8389 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 +∞cpnf 10661 − cmin 10859 2c2 11680 [,)cico 12728 ↑cexp 13425 ℝ^crrx 23987 Spherecsph 45142 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-sup 8890 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-0g 16707 df-gsum 16708 df-prds 16713 df-pws 16715 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-ghm 18348 df-cntz 18439 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-rnghom 19463 df-drng 19497 df-field 19498 df-subrg 19526 df-staf 19609 df-srng 19610 df-lmod 19629 df-lss 19697 df-sra 19937 df-rgmod 19938 df-xmet 20084 df-met 20085 df-cnfld 20092 df-refld 20294 df-dsmm 20421 df-frlm 20436 df-nm 23189 df-tng 23191 df-tcph 23774 df-rrx 23989 df-ehl 23990 df-sph 45144 |
This theorem is referenced by: itsclc0 45185 itsclc0b 45186 itscnhlinecirc02p 45199 |
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