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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 25353) 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 5438 | . . . . 5 β’ {1, 2} β V | |
3 | 1, 2 | eqeltri 2825 | . . . 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 25354 | . . . 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 2728 | . . . 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 47918 | . . 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 688 | . 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 6903 | . . . . . . . . . . . 12 β’ ( 0 β1) = ((πΌ Γ {0})β1) |
14 | c0ex 11248 | . . . . . . . . . . . . 13 β’ 0 β V | |
15 | 1ex 11250 | . . . . . . . . . . . . . . 15 β’ 1 β V | |
16 | 15 | prid1 4771 | . . . . . . . . . . . . . 14 β’ 1 β {1, 2} |
17 | 16, 1 | eleqtrri 2828 | . . . . . . . . . . . . 13 β’ 1 β πΌ |
18 | fvconst2g 7220 | . . . . . . . . . . . . 13 β’ ((0 β V β§ 1 β πΌ) β ((πΌ Γ {0})β1) = 0) | |
19 | 14, 17, 18 | mp2an 690 | . . . . . . . . . . . 12 β’ ((πΌ Γ {0})β1) = 0 |
20 | 13, 19 | eqtri 2756 | . . . . . . . . . . 11 β’ ( 0 β1) = 0 |
21 | 20 | a1i 11 | . . . . . . . . . 10 β’ (π β π β ( 0 β1) = 0) |
22 | 21 | oveq2d 7442 | . . . . . . . . 9 β’ (π β π β ((πβ1) β ( 0 β1)) = ((πβ1) β 0)) |
23 | 1, 5 | rrx2pxel 47880 | . . . . . . . . . . 11 β’ (π β π β (πβ1) β β) |
24 | 23 | recnd 11282 | . . . . . . . . . 10 β’ (π β π β (πβ1) β β) |
25 | 24 | subid1d 11600 | . . . . . . . . 9 β’ (π β π β ((πβ1) β 0) = (πβ1)) |
26 | 22, 25 | eqtrd 2768 | . . . . . . . 8 β’ (π β π β ((πβ1) β ( 0 β1)) = (πβ1)) |
27 | 26 | oveq1d 7441 | . . . . . . 7 β’ (π β π β (((πβ1) β ( 0 β1))β2) = ((πβ1)β2)) |
28 | 4 | fveq1i 6903 | . . . . . . . . . . . 12 β’ ( 0 β2) = ((πΌ Γ {0})β2) |
29 | 2ex 12329 | . . . . . . . . . . . . . . 15 β’ 2 β V | |
30 | 29 | prid2 4772 | . . . . . . . . . . . . . 14 β’ 2 β {1, 2} |
31 | 30, 1 | eleqtrri 2828 | . . . . . . . . . . . . 13 β’ 2 β πΌ |
32 | fvconst2g 7220 | . . . . . . . . . . . . 13 β’ ((0 β V β§ 2 β πΌ) β ((πΌ Γ {0})β2) = 0) | |
33 | 14, 31, 32 | mp2an 690 | . . . . . . . . . . . 12 β’ ((πΌ Γ {0})β2) = 0 |
34 | 28, 33 | eqtri 2756 | . . . . . . . . . . 11 β’ ( 0 β2) = 0 |
35 | 34 | a1i 11 | . . . . . . . . . 10 β’ (π β π β ( 0 β2) = 0) |
36 | 35 | oveq2d 7442 | . . . . . . . . 9 β’ (π β π β ((πβ2) β ( 0 β2)) = ((πβ2) β 0)) |
37 | 1, 5 | rrx2pyel 47881 | . . . . . . . . . . 11 β’ (π β π β (πβ2) β β) |
38 | 37 | recnd 11282 | . . . . . . . . . 10 β’ (π β π β (πβ2) β β) |
39 | 38 | subid1d 11600 | . . . . . . . . 9 β’ (π β π β ((πβ2) β 0) = (πβ2)) |
40 | 36, 39 | eqtrd 2768 | . . . . . . . 8 β’ (π β π β ((πβ2) β ( 0 β2)) = (πβ2)) |
41 | 40 | oveq1d 7441 | . . . . . . 7 β’ (π β π β (((πβ2) β ( 0 β2))β2) = ((πβ2)β2)) |
42 | 27, 41 | oveq12d 7444 | . . . . . 6 β’ (π β π β ((((πβ1) β ( 0 β1))β2) + (((πβ2) β ( 0 β2))β2)) = (((πβ1)β2) + ((πβ2)β2))) |
43 | 42 | eqeq1d 2730 | . . . . 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 480 | . . . 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 3437 | . . 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 2786 | . 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 2768 | 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 3430 Vcvv 3473 {csn 4632 {cpr 4634 Γ cxp 5680 βcfv 6553 (class class class)co 7426 βm cmap 8853 βcr 11147 0cc0 11148 1c1 11149 + caddc 11151 +βcpnf 11285 β cmin 11484 2c2 12307 [,)cico 13368 βcexp 14068 β^crrx 25339 Spherecsph 47897 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-inf2 9674 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-pre-sup 11226 ax-addf 11227 ax-mulf 11228 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7692 df-om 7879 df-1st 8001 df-2nd 8002 df-supp 8174 df-tpos 8240 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-map 8855 df-ixp 8925 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-fsupp 9396 df-sup 9475 df-oi 9543 df-card 9972 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-7 12320 df-8 12321 df-9 12322 df-n0 12513 df-z 12599 df-dec 12718 df-uz 12863 df-rp 13017 df-xneg 13134 df-xadd 13135 df-xmul 13136 df-ico 13372 df-icc 13373 df-fz 13527 df-fzo 13670 df-seq 14009 df-exp 14069 df-hash 14332 df-cj 15088 df-re 15089 df-im 15090 df-sqrt 15224 df-abs 15225 df-clim 15474 df-sum 15675 df-struct 17125 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-mulr 17256 df-starv 17257 df-sca 17258 df-vsca 17259 df-ip 17260 df-tset 17261 df-ple 17262 df-ds 17264 df-unif 17265 df-hom 17266 df-cco 17267 df-0g 17432 df-gsum 17433 df-prds 17438 df-pws 17440 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-mhm 18749 df-grp 18907 df-minusg 18908 df-sbg 18909 df-subg 19092 df-ghm 19182 df-cntz 19282 df-cmn 19751 df-abl 19752 df-mgp 20089 df-rng 20107 df-ur 20136 df-ring 20189 df-cring 20190 df-oppr 20287 df-dvdsr 20310 df-unit 20311 df-invr 20341 df-dvr 20354 df-rhm 20425 df-subrng 20497 df-subrg 20522 df-drng 20640 df-field 20641 df-staf 20739 df-srng 20740 df-lmod 20759 df-lss 20830 df-sra 21072 df-rgmod 21073 df-xmet 21286 df-met 21287 df-cnfld 21294 df-refld 21551 df-dsmm 21680 df-frlm 21695 df-nm 24519 df-tng 24521 df-tcph 25125 df-rrx 25341 df-ehl 25342 df-sph 47899 |
This theorem is referenced by: itsclc0 47940 itsclc0b 47941 itscnhlinecirc02p 47954 |
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