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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > inlinecirc02preu | Structured version Visualization version GIF version |
Description: Intersection of a line with a circle: A line passing through a point within a circle around the origin intersects the circle at exactly two different points, expressed with restricted uniqueness (and without the definition of proper pairs). (Contributed by AV, 16-May-2023.) |
Ref | Expression |
---|---|
inlinecirc02p.i | β’ πΌ = {1, 2} |
inlinecirc02p.e | β’ πΈ = (β^βπΌ) |
inlinecirc02p.p | β’ π = (β βm πΌ) |
inlinecirc02p.s | β’ π = (SphereβπΈ) |
inlinecirc02p.0 | β’ 0 = (πΌ Γ {0}) |
inlinecirc02p.l | β’ πΏ = (LineMβπΈ) |
inlinecirc02p.d | β’ π· = (distβπΈ) |
Ref | Expression |
---|---|
inlinecirc02preu | β’ (((π β π β§ π β π β§ π β π) β§ (π β β+ β§ (ππ· 0 ) < π )) β β!π β π« π((β―βπ) = 2 β§ π = (( 0 ππ ) β© (ππΏπ)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inlinecirc02p.i | . . . 4 β’ πΌ = {1, 2} | |
2 | inlinecirc02p.e | . . . 4 β’ πΈ = (β^βπΌ) | |
3 | inlinecirc02p.p | . . . 4 β’ π = (β βm πΌ) | |
4 | inlinecirc02p.s | . . . 4 β’ π = (SphereβπΈ) | |
5 | inlinecirc02p.0 | . . . 4 β’ 0 = (πΌ Γ {0}) | |
6 | inlinecirc02p.l | . . . 4 β’ πΏ = (LineMβπΈ) | |
7 | inlinecirc02p.d | . . . 4 β’ π· = (distβπΈ) | |
8 | 1, 2, 3, 4, 5, 6, 7 | inlinecirc02p 46947 | . . 3 β’ (((π β π β§ π β π β§ π β π) β§ (π β β+ β§ (ππ· 0 ) < π )) β (( 0 ππ ) β© (ππΏπ)) β (Pairsproperβπ)) |
9 | reueq 3700 | . . 3 β’ ((( 0 ππ ) β© (ππΏπ)) β (Pairsproperβπ) β β!π β (Pairsproperβπ)π = (( 0 ππ ) β© (ππΏπ))) | |
10 | 8, 9 | sylib 217 | . 2 β’ (((π β π β§ π β π β§ π β π) β§ (π β β+ β§ (ππ· 0 ) < π )) β β!π β (Pairsproperβπ)π = (( 0 ππ ) β© (ππΏπ))) |
11 | 3 | ovexi 7396 | . . 3 β’ π β V |
12 | prprreueq 45786 | . . 3 β’ (π β V β (β!π β (Pairsproperβπ)π = (( 0 ππ ) β© (ππΏπ)) β β!π β π« π((β―βπ) = 2 β§ π = (( 0 ππ ) β© (ππΏπ))))) | |
13 | 11, 12 | mp1i 13 | . 2 β’ (((π β π β§ π β π β§ π β π) β§ (π β β+ β§ (ππ· 0 ) < π )) β (β!π β (Pairsproperβπ)π = (( 0 ππ ) β© (ππΏπ)) β β!π β π« π((β―βπ) = 2 β§ π = (( 0 ππ ) β© (ππΏπ))))) |
14 | 10, 13 | mpbid 231 | 1 β’ (((π β π β§ π β π β§ π β π) β§ (π β β+ β§ (ππ· 0 ) < π )) β β!π β π« π((β―βπ) = 2 β§ π = (( 0 ππ ) β© (ππΏπ)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2944 β!wreu 3354 Vcvv 3448 β© cin 3914 π« cpw 4565 {csn 4591 {cpr 4593 class class class wbr 5110 Γ cxp 5636 βcfv 6501 (class class class)co 7362 βm cmap 8772 βcr 11057 0cc0 11058 1c1 11059 < clt 11196 2c2 12215 β+crp 12922 β―chash 14237 distcds 17149 β^crrx 24763 Pairspropercprpr 45778 LineMcline 46887 Spherecsph 46888 |
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 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 ax-addf 11137 ax-mulf 11138 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-om 7808 df-1st 7926 df-2nd 7927 df-supp 8098 df-tpos 8162 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-oadd 8421 df-er 8655 df-map 8774 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9313 df-sup 9385 df-oi 9453 df-dju 9844 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-rp 12923 df-xneg 13040 df-xadd 13041 df-xmul 13042 df-ico 13277 df-icc 13278 df-fz 13432 df-fzo 13575 df-seq 13914 df-exp 13975 df-hash 14238 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-clim 15377 df-sum 15578 df-struct 17026 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-plusg 17153 df-mulr 17154 df-starv 17155 df-sca 17156 df-vsca 17157 df-ip 17158 df-tset 17159 df-ple 17160 df-ds 17162 df-unif 17163 df-hom 17164 df-cco 17165 df-0g 17330 df-gsum 17331 df-prds 17336 df-pws 17338 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-mhm 18608 df-grp 18758 df-minusg 18759 df-sbg 18760 df-subg 18932 df-ghm 19013 df-cntz 19104 df-cmn 19571 df-abl 19572 df-mgp 19904 df-ur 19921 df-ring 19973 df-cring 19974 df-oppr 20056 df-dvdsr 20077 df-unit 20078 df-invr 20108 df-dvr 20119 df-rnghom 20155 df-drng 20201 df-field 20202 df-subrg 20236 df-staf 20320 df-srng 20321 df-lmod 20340 df-lss 20409 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-prpr 45779 df-line 46889 df-sph 46890 |
This theorem is referenced by: (None) |
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