![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
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 | ⊢ 𝐼 = {1, 2} |
inlinecirc02p.e | ⊢ 𝐸 = (ℝ^‘𝐼) |
inlinecirc02p.p | ⊢ 𝑃 = (ℝ ↑m 𝐼) |
inlinecirc02p.s | ⊢ 𝑆 = (Sphere‘𝐸) |
inlinecirc02p.0 | ⊢ 0 = (𝐼 × {0}) |
inlinecirc02p.l | ⊢ 𝐿 = (LineM‘𝐸) |
inlinecirc02p.d | ⊢ 𝐷 = (dist‘𝐸) |
Ref | Expression |
---|---|
inlinecirc02preu | ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) ∧ (𝑅 ∈ ℝ+ ∧ (𝑋𝐷 0 ) < 𝑅)) → ∃!𝑝 ∈ 𝒫 𝑃((♯‘𝑝) = 2 ∧ 𝑝 = (( 0 𝑆𝑅) ∩ (𝑋𝐿𝑌)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inlinecirc02p.i | . . . 4 ⊢ 𝐼 = {1, 2} | |
2 | inlinecirc02p.e | . . . 4 ⊢ 𝐸 = (ℝ^‘𝐼) | |
3 | inlinecirc02p.p | . . . 4 ⊢ 𝑃 = (ℝ ↑m 𝐼) | |
4 | inlinecirc02p.s | . . . 4 ⊢ 𝑆 = (Sphere‘𝐸) | |
5 | inlinecirc02p.0 | . . . 4 ⊢ 0 = (𝐼 × {0}) | |
6 | inlinecirc02p.l | . . . 4 ⊢ 𝐿 = (LineM‘𝐸) | |
7 | inlinecirc02p.d | . . . 4 ⊢ 𝐷 = (dist‘𝐸) | |
8 | 1, 2, 3, 4, 5, 6, 7 | inlinecirc02p 47561 | . . 3 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) ∧ (𝑅 ∈ ℝ+ ∧ (𝑋𝐷 0 ) < 𝑅)) → (( 0 𝑆𝑅) ∩ (𝑋𝐿𝑌)) ∈ (Pairsproper‘𝑃)) |
9 | reueq 3733 | . . 3 ⊢ ((( 0 𝑆𝑅) ∩ (𝑋𝐿𝑌)) ∈ (Pairsproper‘𝑃) ↔ ∃!𝑝 ∈ (Pairsproper‘𝑃)𝑝 = (( 0 𝑆𝑅) ∩ (𝑋𝐿𝑌))) | |
10 | 8, 9 | sylib 217 | . 2 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) ∧ (𝑅 ∈ ℝ+ ∧ (𝑋𝐷 0 ) < 𝑅)) → ∃!𝑝 ∈ (Pairsproper‘𝑃)𝑝 = (( 0 𝑆𝑅) ∩ (𝑋𝐿𝑌))) |
11 | 3 | ovexi 7446 | . . 3 ⊢ 𝑃 ∈ V |
12 | prprreueq 46487 | . . 3 ⊢ (𝑃 ∈ V → (∃!𝑝 ∈ (Pairsproper‘𝑃)𝑝 = (( 0 𝑆𝑅) ∩ (𝑋𝐿𝑌)) ↔ ∃!𝑝 ∈ 𝒫 𝑃((♯‘𝑝) = 2 ∧ 𝑝 = (( 0 𝑆𝑅) ∩ (𝑋𝐿𝑌))))) | |
13 | 11, 12 | mp1i 13 | . 2 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) ∧ (𝑅 ∈ ℝ+ ∧ (𝑋𝐷 0 ) < 𝑅)) → (∃!𝑝 ∈ (Pairsproper‘𝑃)𝑝 = (( 0 𝑆𝑅) ∩ (𝑋𝐿𝑌)) ↔ ∃!𝑝 ∈ 𝒫 𝑃((♯‘𝑝) = 2 ∧ 𝑝 = (( 0 𝑆𝑅) ∩ (𝑋𝐿𝑌))))) |
14 | 10, 13 | mpbid 231 | 1 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) ∧ (𝑅 ∈ ℝ+ ∧ (𝑋𝐷 0 ) < 𝑅)) → ∃!𝑝 ∈ 𝒫 𝑃((♯‘𝑝) = 2 ∧ 𝑝 = (( 0 𝑆𝑅) ∩ (𝑋𝐿𝑌)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∃!wreu 3373 Vcvv 3473 ∩ cin 3947 𝒫 cpw 4602 {csn 4628 {cpr 4630 class class class wbr 5148 × cxp 5674 ‘cfv 6543 (class class class)co 7412 ↑m cmap 8824 ℝcr 11113 0cc0 11114 1c1 11115 < clt 11253 2c2 12272 ℝ+crp 12979 ♯chash 14295 distcds 17211 ℝ^crrx 25132 Pairspropercprpr 46479 LineMcline 47501 Spherecsph 47502 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9640 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 ax-addf 11193 ax-mulf 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8151 df-tpos 8215 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-2o 8471 df-oadd 8474 df-er 8707 df-map 8826 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9366 df-sup 9441 df-oi 9509 df-dju 9900 df-card 9938 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-rp 12980 df-xneg 13097 df-xadd 13098 df-xmul 13099 df-ico 13335 df-icc 13336 df-fz 13490 df-fzo 13633 df-seq 13972 df-exp 14033 df-hash 14296 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-clim 15437 df-sum 15638 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-0g 17392 df-gsum 17393 df-prds 17398 df-pws 17400 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-mhm 18706 df-grp 18859 df-minusg 18860 df-sbg 18861 df-subg 19040 df-ghm 19129 df-cntz 19223 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-cring 20131 df-oppr 20226 df-dvdsr 20249 df-unit 20250 df-invr 20280 df-dvr 20293 df-rhm 20364 df-subrng 20435 df-subrg 20460 df-drng 20503 df-field 20504 df-staf 20597 df-srng 20598 df-lmod 20617 df-lss 20688 df-sra 20931 df-rgmod 20932 df-xmet 21138 df-met 21139 df-cnfld 21146 df-refld 21378 df-dsmm 21507 df-frlm 21522 df-nm 24312 df-tng 24314 df-tcph 24918 df-rrx 25134 df-ehl 25135 df-prpr 46480 df-line 47503 df-sph 47504 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |