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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 24000) 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 5333 | . . . . 5 ⊢ {1, 2} ∈ V | |
3 | 1, 2 | eqeltri 2909 | . . . 4 ⊢ 𝐼 ∈ V |
4 | 2sphere0.0 | . . . . 5 ⊢ 0 = (𝐼 × {0}) | |
5 | 2sphere.p | . . . . 5 ⊢ 𝑃 = (ℝ ↑m 𝐼) | |
6 | 4, 5 | rrx0el 24001 | . . . 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 2821 | . . . 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 44785 | . . 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 688 | . 2 ⊢ (𝑅 ∈ (0[,)+∞) → ( 0 𝑆𝑅) = {𝑝 ∈ 𝑃 ∣ ((((𝑝‘1) − ( 0 ‘1))↑2) + (((𝑝‘2) − ( 0 ‘2))↑2)) = (𝑅↑2)}) |
13 | 4 | fveq1i 6671 | . . . . . . . . . . . 12 ⊢ ( 0 ‘1) = ((𝐼 × {0})‘1) |
14 | c0ex 10635 | . . . . . . . . . . . . 13 ⊢ 0 ∈ V | |
15 | 1ex 10637 | . . . . . . . . . . . . . . 15 ⊢ 1 ∈ V | |
16 | 15 | prid1 4698 | . . . . . . . . . . . . . 14 ⊢ 1 ∈ {1, 2} |
17 | 16, 1 | eleqtrri 2912 | . . . . . . . . . . . . 13 ⊢ 1 ∈ 𝐼 |
18 | fvconst2g 6964 | . . . . . . . . . . . . 13 ⊢ ((0 ∈ V ∧ 1 ∈ 𝐼) → ((𝐼 × {0})‘1) = 0) | |
19 | 14, 17, 18 | mp2an 690 | . . . . . . . . . . . 12 ⊢ ((𝐼 × {0})‘1) = 0 |
20 | 13, 19 | eqtri 2844 | . . . . . . . . . . 11 ⊢ ( 0 ‘1) = 0 |
21 | 20 | a1i 11 | . . . . . . . . . 10 ⊢ (𝑝 ∈ 𝑃 → ( 0 ‘1) = 0) |
22 | 21 | oveq2d 7172 | . . . . . . . . 9 ⊢ (𝑝 ∈ 𝑃 → ((𝑝‘1) − ( 0 ‘1)) = ((𝑝‘1) − 0)) |
23 | 1, 5 | rrx2pxel 44747 | . . . . . . . . . . 11 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘1) ∈ ℝ) |
24 | 23 | recnd 10669 | . . . . . . . . . 10 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘1) ∈ ℂ) |
25 | 24 | subid1d 10986 | . . . . . . . . 9 ⊢ (𝑝 ∈ 𝑃 → ((𝑝‘1) − 0) = (𝑝‘1)) |
26 | 22, 25 | eqtrd 2856 | . . . . . . . 8 ⊢ (𝑝 ∈ 𝑃 → ((𝑝‘1) − ( 0 ‘1)) = (𝑝‘1)) |
27 | 26 | oveq1d 7171 | . . . . . . 7 ⊢ (𝑝 ∈ 𝑃 → (((𝑝‘1) − ( 0 ‘1))↑2) = ((𝑝‘1)↑2)) |
28 | 4 | fveq1i 6671 | . . . . . . . . . . . 12 ⊢ ( 0 ‘2) = ((𝐼 × {0})‘2) |
29 | 2ex 11715 | . . . . . . . . . . . . . . 15 ⊢ 2 ∈ V | |
30 | 29 | prid2 4699 | . . . . . . . . . . . . . 14 ⊢ 2 ∈ {1, 2} |
31 | 30, 1 | eleqtrri 2912 | . . . . . . . . . . . . 13 ⊢ 2 ∈ 𝐼 |
32 | fvconst2g 6964 | . . . . . . . . . . . . 13 ⊢ ((0 ∈ V ∧ 2 ∈ 𝐼) → ((𝐼 × {0})‘2) = 0) | |
33 | 14, 31, 32 | mp2an 690 | . . . . . . . . . . . 12 ⊢ ((𝐼 × {0})‘2) = 0 |
34 | 28, 33 | eqtri 2844 | . . . . . . . . . . 11 ⊢ ( 0 ‘2) = 0 |
35 | 34 | a1i 11 | . . . . . . . . . 10 ⊢ (𝑝 ∈ 𝑃 → ( 0 ‘2) = 0) |
36 | 35 | oveq2d 7172 | . . . . . . . . 9 ⊢ (𝑝 ∈ 𝑃 → ((𝑝‘2) − ( 0 ‘2)) = ((𝑝‘2) − 0)) |
37 | 1, 5 | rrx2pyel 44748 | . . . . . . . . . . 11 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘2) ∈ ℝ) |
38 | 37 | recnd 10669 | . . . . . . . . . 10 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘2) ∈ ℂ) |
39 | 38 | subid1d 10986 | . . . . . . . . 9 ⊢ (𝑝 ∈ 𝑃 → ((𝑝‘2) − 0) = (𝑝‘2)) |
40 | 36, 39 | eqtrd 2856 | . . . . . . . 8 ⊢ (𝑝 ∈ 𝑃 → ((𝑝‘2) − ( 0 ‘2)) = (𝑝‘2)) |
41 | 40 | oveq1d 7171 | . . . . . . 7 ⊢ (𝑝 ∈ 𝑃 → (((𝑝‘2) − ( 0 ‘2))↑2) = ((𝑝‘2)↑2)) |
42 | 27, 41 | oveq12d 7174 | . . . . . 6 ⊢ (𝑝 ∈ 𝑃 → ((((𝑝‘1) − ( 0 ‘1))↑2) + (((𝑝‘2) − ( 0 ‘2))↑2)) = (((𝑝‘1)↑2) + ((𝑝‘2)↑2))) |
43 | 42 | eqeq1d 2823 | . . . . 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 484 | . . . 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 3478 | . . 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 2874 | . 2 ⊢ (𝑅 ∈ (0[,)+∞) → {𝑝 ∈ 𝑃 ∣ ((((𝑝‘1) − ( 0 ‘1))↑2) + (((𝑝‘2) − ( 0 ‘2))↑2)) = (𝑅↑2)} = 𝐶) |
48 | 12, 47 | eqtrd 2856 | 1 ⊢ (𝑅 ∈ (0[,)+∞) → ( 0 𝑆𝑅) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 {crab 3142 Vcvv 3494 {csn 4567 {cpr 4569 × cxp 5553 ‘cfv 6355 (class class class)co 7156 ↑m cmap 8406 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 +∞cpnf 10672 − cmin 10870 2c2 11693 [,)cico 12741 ↑cexp 13430 ℝ^crrx 23986 Spherecsph 44764 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-sup 8906 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-sum 15043 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-0g 16715 df-gsum 16716 df-prds 16721 df-pws 16723 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-ghm 18356 df-cntz 18447 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-dvr 19433 df-rnghom 19467 df-drng 19504 df-field 19505 df-subrg 19533 df-staf 19616 df-srng 19617 df-lmod 19636 df-lss 19704 df-sra 19944 df-rgmod 19945 df-xmet 20538 df-met 20539 df-cnfld 20546 df-refld 20749 df-dsmm 20876 df-frlm 20891 df-nm 23192 df-tng 23194 df-tcph 23773 df-rrx 23988 df-ehl 23989 df-sph 44766 |
This theorem is referenced by: itsclc0 44807 itsclc0b 44808 itscnhlinecirc02p 44821 |
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