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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 24777) 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 5390 | . . . . 5 β’ {1, 2} β V | |
3 | 1, 2 | eqeltri 2830 | . . . 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 24778 | . . . 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 2733 | . . . 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 46921 | . . 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 689 | . 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 6844 | . . . . . . . . . . . 12 β’ ( 0 β1) = ((πΌ Γ {0})β1) |
14 | c0ex 11154 | . . . . . . . . . . . . 13 β’ 0 β V | |
15 | 1ex 11156 | . . . . . . . . . . . . . . 15 β’ 1 β V | |
16 | 15 | prid1 4724 | . . . . . . . . . . . . . 14 β’ 1 β {1, 2} |
17 | 16, 1 | eleqtrri 2833 | . . . . . . . . . . . . 13 β’ 1 β πΌ |
18 | fvconst2g 7152 | . . . . . . . . . . . . 13 β’ ((0 β V β§ 1 β πΌ) β ((πΌ Γ {0})β1) = 0) | |
19 | 14, 17, 18 | mp2an 691 | . . . . . . . . . . . 12 β’ ((πΌ Γ {0})β1) = 0 |
20 | 13, 19 | eqtri 2761 | . . . . . . . . . . 11 β’ ( 0 β1) = 0 |
21 | 20 | a1i 11 | . . . . . . . . . 10 β’ (π β π β ( 0 β1) = 0) |
22 | 21 | oveq2d 7374 | . . . . . . . . 9 β’ (π β π β ((πβ1) β ( 0 β1)) = ((πβ1) β 0)) |
23 | 1, 5 | rrx2pxel 46883 | . . . . . . . . . . 11 β’ (π β π β (πβ1) β β) |
24 | 23 | recnd 11188 | . . . . . . . . . 10 β’ (π β π β (πβ1) β β) |
25 | 24 | subid1d 11506 | . . . . . . . . 9 β’ (π β π β ((πβ1) β 0) = (πβ1)) |
26 | 22, 25 | eqtrd 2773 | . . . . . . . 8 β’ (π β π β ((πβ1) β ( 0 β1)) = (πβ1)) |
27 | 26 | oveq1d 7373 | . . . . . . 7 β’ (π β π β (((πβ1) β ( 0 β1))β2) = ((πβ1)β2)) |
28 | 4 | fveq1i 6844 | . . . . . . . . . . . 12 β’ ( 0 β2) = ((πΌ Γ {0})β2) |
29 | 2ex 12235 | . . . . . . . . . . . . . . 15 β’ 2 β V | |
30 | 29 | prid2 4725 | . . . . . . . . . . . . . 14 β’ 2 β {1, 2} |
31 | 30, 1 | eleqtrri 2833 | . . . . . . . . . . . . 13 β’ 2 β πΌ |
32 | fvconst2g 7152 | . . . . . . . . . . . . 13 β’ ((0 β V β§ 2 β πΌ) β ((πΌ Γ {0})β2) = 0) | |
33 | 14, 31, 32 | mp2an 691 | . . . . . . . . . . . 12 β’ ((πΌ Γ {0})β2) = 0 |
34 | 28, 33 | eqtri 2761 | . . . . . . . . . . 11 β’ ( 0 β2) = 0 |
35 | 34 | a1i 11 | . . . . . . . . . 10 β’ (π β π β ( 0 β2) = 0) |
36 | 35 | oveq2d 7374 | . . . . . . . . 9 β’ (π β π β ((πβ2) β ( 0 β2)) = ((πβ2) β 0)) |
37 | 1, 5 | rrx2pyel 46884 | . . . . . . . . . . 11 β’ (π β π β (πβ2) β β) |
38 | 37 | recnd 11188 | . . . . . . . . . 10 β’ (π β π β (πβ2) β β) |
39 | 38 | subid1d 11506 | . . . . . . . . 9 β’ (π β π β ((πβ2) β 0) = (πβ2)) |
40 | 36, 39 | eqtrd 2773 | . . . . . . . 8 β’ (π β π β ((πβ2) β ( 0 β2)) = (πβ2)) |
41 | 40 | oveq1d 7373 | . . . . . . 7 β’ (π β π β (((πβ2) β ( 0 β2))β2) = ((πβ2)β2)) |
42 | 27, 41 | oveq12d 7376 | . . . . . 6 β’ (π β π β ((((πβ1) β ( 0 β1))β2) + (((πβ2) β ( 0 β2))β2)) = (((πβ1)β2) + ((πβ2)β2))) |
43 | 42 | eqeq1d 2735 | . . . . 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 483 | . . . 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 3413 | . . 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 2791 | . 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 2773 | 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 1542 β wcel 2107 {crab 3406 Vcvv 3444 {csn 4587 {cpr 4589 Γ cxp 5632 βcfv 6497 (class class class)co 7358 βm cmap 8768 βcr 11055 0cc0 11056 1c1 11057 + caddc 11059 +βcpnf 11191 β cmin 11390 2c2 12213 [,)cico 13272 βcexp 13973 β^crrx 24763 Spherecsph 46900 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9582 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 ax-addf 11135 ax-mulf 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7618 df-om 7804 df-1st 7922 df-2nd 7923 df-supp 8094 df-tpos 8158 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-map 8770 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9309 df-sup 9383 df-oi 9451 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-rp 12921 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ico 13276 df-icc 13277 df-fz 13431 df-fzo 13574 df-seq 13913 df-exp 13974 df-hash 14237 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-clim 15376 df-sum 15577 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-starv 17153 df-sca 17154 df-vsca 17155 df-ip 17156 df-tset 17157 df-ple 17158 df-ds 17160 df-unif 17161 df-hom 17162 df-cco 17163 df-0g 17328 df-gsum 17329 df-prds 17334 df-pws 17336 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-mhm 18606 df-grp 18756 df-minusg 18757 df-sbg 18758 df-subg 18930 df-ghm 19011 df-cntz 19102 df-cmn 19569 df-abl 19570 df-mgp 19902 df-ur 19919 df-ring 19971 df-cring 19972 df-oppr 20054 df-dvdsr 20075 df-unit 20076 df-invr 20106 df-dvr 20117 df-rnghom 20153 df-drng 20199 df-field 20200 df-subrg 20234 df-staf 20318 df-srng 20319 df-lmod 20338 df-lss 20408 df-sra 20649 df-rgmod 20650 df-xmet 20805 df-met 20806 df-cnfld 20813 df-refld 21025 df-dsmm 21154 df-frlm 21169 df-nm 23954 df-tng 23956 df-tcph 24549 df-rrx 24765 df-ehl 24766 df-sph 46902 |
This theorem is referenced by: itsclc0 46943 itsclc0b 46944 itscnhlinecirc02p 46957 |
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