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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 25280) with radius π in a two dimensional Euclidean space is a circle. (Contributed by AV, 5-Feb-2023.) |
Ref | Expression |
---|---|
2sphere.i | β’ πΌ = {1, 2} |
2sphere.e | β’ πΈ = (β^βπΌ) |
2sphere.p | β’ π = (β βm πΌ) |
2sphere.s | β’ π = (SphereβπΈ) |
2sphere0.0 | β’ 0 = (πΌ Γ {0}) |
2sphere0.c | β’ πΆ = {π β π β£ (((πβ1)β2) + ((πβ2)β2)) = (π β2)} |
Ref | Expression |
---|---|
2sphere0 | β’ (π β (0[,)+β) β ( 0 ππ ) = πΆ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2sphere.i | . . . . 5 β’ πΌ = {1, 2} | |
2 | prex 5425 | . . . . 5 β’ {1, 2} β V | |
3 | 1, 2 | eqeltri 2823 | . . . 4 β’ πΌ β V |
4 | 2sphere0.0 | . . . . 5 β’ 0 = (πΌ Γ {0}) | |
5 | 2sphere.p | . . . . 5 β’ π = (β βm πΌ) | |
6 | 4, 5 | rrx0el 25281 | . . . 4 β’ (πΌ β V β 0 β π) |
7 | 3, 6 | ax-mp 5 | . . 3 β’ 0 β π |
8 | 2sphere.e | . . . 4 β’ πΈ = (β^βπΌ) | |
9 | 2sphere.s | . . . 4 β’ π = (SphereβπΈ) | |
10 | eqid 2726 | . . . 4 β’ {π β π β£ ((((πβ1) β ( 0 β1))β2) + (((πβ2) β ( 0 β2))β2)) = (π β2)} = {π β π β£ ((((πβ1) β ( 0 β1))β2) + (((πβ2) β ( 0 β2))β2)) = (π β2)} | |
11 | 1, 8, 5, 9, 10 | 2sphere 47710 | . . 3 β’ (( 0 β π β§ π β (0[,)+β)) β ( 0 ππ ) = {π β π β£ ((((πβ1) β ( 0 β1))β2) + (((πβ2) β ( 0 β2))β2)) = (π β2)}) |
12 | 7, 11 | mpan 687 | . 2 β’ (π β (0[,)+β) β ( 0 ππ ) = {π β π β£ ((((πβ1) β ( 0 β1))β2) + (((πβ2) β ( 0 β2))β2)) = (π β2)}) |
13 | 4 | fveq1i 6886 | . . . . . . . . . . . 12 β’ ( 0 β1) = ((πΌ Γ {0})β1) |
14 | c0ex 11212 | . . . . . . . . . . . . 13 β’ 0 β V | |
15 | 1ex 11214 | . . . . . . . . . . . . . . 15 β’ 1 β V | |
16 | 15 | prid1 4761 | . . . . . . . . . . . . . 14 β’ 1 β {1, 2} |
17 | 16, 1 | eleqtrri 2826 | . . . . . . . . . . . . 13 β’ 1 β πΌ |
18 | fvconst2g 7199 | . . . . . . . . . . . . 13 β’ ((0 β V β§ 1 β πΌ) β ((πΌ Γ {0})β1) = 0) | |
19 | 14, 17, 18 | mp2an 689 | . . . . . . . . . . . 12 β’ ((πΌ Γ {0})β1) = 0 |
20 | 13, 19 | eqtri 2754 | . . . . . . . . . . 11 β’ ( 0 β1) = 0 |
21 | 20 | a1i 11 | . . . . . . . . . 10 β’ (π β π β ( 0 β1) = 0) |
22 | 21 | oveq2d 7421 | . . . . . . . . 9 β’ (π β π β ((πβ1) β ( 0 β1)) = ((πβ1) β 0)) |
23 | 1, 5 | rrx2pxel 47672 | . . . . . . . . . . 11 β’ (π β π β (πβ1) β β) |
24 | 23 | recnd 11246 | . . . . . . . . . 10 β’ (π β π β (πβ1) β β) |
25 | 24 | subid1d 11564 | . . . . . . . . 9 β’ (π β π β ((πβ1) β 0) = (πβ1)) |
26 | 22, 25 | eqtrd 2766 | . . . . . . . 8 β’ (π β π β ((πβ1) β ( 0 β1)) = (πβ1)) |
27 | 26 | oveq1d 7420 | . . . . . . 7 β’ (π β π β (((πβ1) β ( 0 β1))β2) = ((πβ1)β2)) |
28 | 4 | fveq1i 6886 | . . . . . . . . . . . 12 β’ ( 0 β2) = ((πΌ Γ {0})β2) |
29 | 2ex 12293 | . . . . . . . . . . . . . . 15 β’ 2 β V | |
30 | 29 | prid2 4762 | . . . . . . . . . . . . . 14 β’ 2 β {1, 2} |
31 | 30, 1 | eleqtrri 2826 | . . . . . . . . . . . . 13 β’ 2 β πΌ |
32 | fvconst2g 7199 | . . . . . . . . . . . . 13 β’ ((0 β V β§ 2 β πΌ) β ((πΌ Γ {0})β2) = 0) | |
33 | 14, 31, 32 | mp2an 689 | . . . . . . . . . . . 12 β’ ((πΌ Γ {0})β2) = 0 |
34 | 28, 33 | eqtri 2754 | . . . . . . . . . . 11 β’ ( 0 β2) = 0 |
35 | 34 | a1i 11 | . . . . . . . . . 10 β’ (π β π β ( 0 β2) = 0) |
36 | 35 | oveq2d 7421 | . . . . . . . . 9 β’ (π β π β ((πβ2) β ( 0 β2)) = ((πβ2) β 0)) |
37 | 1, 5 | rrx2pyel 47673 | . . . . . . . . . . 11 β’ (π β π β (πβ2) β β) |
38 | 37 | recnd 11246 | . . . . . . . . . 10 β’ (π β π β (πβ2) β β) |
39 | 38 | subid1d 11564 | . . . . . . . . 9 β’ (π β π β ((πβ2) β 0) = (πβ2)) |
40 | 36, 39 | eqtrd 2766 | . . . . . . . 8 β’ (π β π β ((πβ2) β ( 0 β2)) = (πβ2)) |
41 | 40 | oveq1d 7420 | . . . . . . 7 β’ (π β π β (((πβ2) β ( 0 β2))β2) = ((πβ2)β2)) |
42 | 27, 41 | oveq12d 7423 | . . . . . 6 β’ (π β π β ((((πβ1) β ( 0 β1))β2) + (((πβ2) β ( 0 β2))β2)) = (((πβ1)β2) + ((πβ2)β2))) |
43 | 42 | eqeq1d 2728 | . . . . 5 β’ (π β π β (((((πβ1) β ( 0 β1))β2) + (((πβ2) β ( 0 β2))β2)) = (π β2) β (((πβ1)β2) + ((πβ2)β2)) = (π β2))) |
44 | 43 | adantl 481 | . . . 4 β’ ((π β (0[,)+β) β§ π β π) β (((((πβ1) β ( 0 β1))β2) + (((πβ2) β ( 0 β2))β2)) = (π β2) β (((πβ1)β2) + ((πβ2)β2)) = (π β2))) |
45 | 44 | rabbidva 3433 | . . 3 β’ (π β (0[,)+β) β {π β π β£ ((((πβ1) β ( 0 β1))β2) + (((πβ2) β ( 0 β2))β2)) = (π β2)} = {π β π β£ (((πβ1)β2) + ((πβ2)β2)) = (π β2)}) |
46 | 2sphere0.c | . . 3 β’ πΆ = {π β π β£ (((πβ1)β2) + ((πβ2)β2)) = (π β2)} | |
47 | 45, 46 | eqtr4di 2784 | . 2 β’ (π β (0[,)+β) β {π β π β£ ((((πβ1) β ( 0 β1))β2) + (((πβ2) β ( 0 β2))β2)) = (π β2)} = πΆ) |
48 | 12, 47 | eqtrd 2766 | 1 β’ (π β (0[,)+β) β ( 0 ππ ) = πΆ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1533 β wcel 2098 {crab 3426 Vcvv 3468 {csn 4623 {cpr 4625 Γ cxp 5667 βcfv 6537 (class class class)co 7405 βm cmap 8822 βcr 11111 0cc0 11112 1c1 11113 + caddc 11115 +βcpnf 11249 β cmin 11448 2c2 12271 [,)cico 13332 βcexp 14032 β^crrx 25266 Spherecsph 47689 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-tpos 8212 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-ico 13336 df-icc 13337 df-fz 13491 df-fzo 13634 df-seq 13973 df-exp 14033 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438 df-sum 15639 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-hom 17230 df-cco 17231 df-0g 17396 df-gsum 17397 df-prds 17402 df-pws 17404 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18713 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19050 df-ghm 19139 df-cntz 19233 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-cring 20141 df-oppr 20236 df-dvdsr 20259 df-unit 20260 df-invr 20290 df-dvr 20303 df-rhm 20374 df-subrng 20446 df-subrg 20471 df-drng 20589 df-field 20590 df-staf 20688 df-srng 20689 df-lmod 20708 df-lss 20779 df-sra 21021 df-rgmod 21022 df-xmet 21233 df-met 21234 df-cnfld 21241 df-refld 21498 df-dsmm 21627 df-frlm 21642 df-nm 24446 df-tng 24448 df-tcph 25052 df-rrx 25268 df-ehl 25269 df-sph 47691 |
This theorem is referenced by: itsclc0 47732 itsclc0b 47733 itscnhlinecirc02p 47746 |
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