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Mirrors > Home > MPE Home > Th. List > Mathboxes > itsclinecirc0 | Structured version Visualization version GIF version |
Description: The intersection points of a line through two different points 𝑌 and 𝑍 and a circle around the origin, using the definition of a line in a two dimensional Euclidean space. (Contributed by AV, 25-Feb-2023.) (Proof shortened by AV, 16-May-2023.) |
Ref | Expression |
---|---|
itsclc0.i | ⊢ 𝐼 = {1, 2} |
itsclc0.e | ⊢ 𝐸 = (ℝ^‘𝐼) |
itsclc0.p | ⊢ 𝑃 = (ℝ ↑m 𝐼) |
itsclc0.s | ⊢ 𝑆 = (Sphere‘𝐸) |
itsclc0.0 | ⊢ 0 = (𝐼 × {0}) |
itsclc0.q | ⊢ 𝑄 = ((𝐴↑2) + (𝐵↑2)) |
itsclc0.d | ⊢ 𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2)) |
itsclinecirc0.l | ⊢ 𝐿 = (LineM‘𝐸) |
itsclinecirc0.a | ⊢ 𝐴 = ((𝑌‘2) − (𝑍‘2)) |
itsclinecirc0.b | ⊢ 𝐵 = ((𝑍‘1) − (𝑌‘1)) |
itsclinecirc0.c | ⊢ 𝐶 = (((𝑌‘2) · (𝑍‘1)) − ((𝑌‘1) · (𝑍‘2))) |
Ref | Expression |
---|---|
itsclinecirc0 | ⊢ (((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷)) → ((𝑋 ∈ ( 0 𝑆𝑅) ∧ 𝑋 ∈ (𝑌𝐿𝑍)) → (((𝑋‘1) = (((𝐴 · 𝐶) + (𝐵 · (√‘𝐷))) / 𝑄) ∧ (𝑋‘2) = (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄)) ∨ ((𝑋‘1) = (((𝐴 · 𝐶) − (𝐵 · (√‘𝐷))) / 𝑄) ∧ (𝑋‘2) = (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | itsclc0.i | . . . . . 6 ⊢ 𝐼 = {1, 2} | |
2 | itsclc0.e | . . . . . 6 ⊢ 𝐸 = (ℝ^‘𝐼) | |
3 | itsclc0.p | . . . . . 6 ⊢ 𝑃 = (ℝ ↑m 𝐼) | |
4 | itsclinecirc0.l | . . . . . 6 ⊢ 𝐿 = (LineM‘𝐸) | |
5 | itsclinecirc0.a | . . . . . 6 ⊢ 𝐴 = ((𝑌‘2) − (𝑍‘2)) | |
6 | itsclinecirc0.b | . . . . . 6 ⊢ 𝐵 = ((𝑍‘1) − (𝑌‘1)) | |
7 | itsclinecirc0.c | . . . . . 6 ⊢ 𝐶 = (((𝑌‘2) · (𝑍‘1)) − ((𝑌‘1) · (𝑍‘2))) | |
8 | 1, 2, 3, 4, 5, 6, 7 | rrx2linest2 48594 | . . . . 5 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → (𝑌𝐿𝑍) = {𝑝 ∈ 𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶}) |
9 | 8 | adantr 480 | . . . 4 ⊢ (((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷)) → (𝑌𝐿𝑍) = {𝑝 ∈ 𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶}) |
10 | 9 | eleq2d 2825 | . . 3 ⊢ (((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷)) → (𝑋 ∈ (𝑌𝐿𝑍) ↔ 𝑋 ∈ {𝑝 ∈ 𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶})) |
11 | 10 | anbi2d 630 | . 2 ⊢ (((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷)) → ((𝑋 ∈ ( 0 𝑆𝑅) ∧ 𝑋 ∈ (𝑌𝐿𝑍)) ↔ (𝑋 ∈ ( 0 𝑆𝑅) ∧ 𝑋 ∈ {𝑝 ∈ 𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶}))) |
12 | 1, 3 | rrx2pyel 48562 | . . . . . . 7 ⊢ (𝑌 ∈ 𝑃 → (𝑌‘2) ∈ ℝ) |
13 | 12 | 3ad2ant1 1132 | . . . . . 6 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → (𝑌‘2) ∈ ℝ) |
14 | 1, 3 | rrx2pyel 48562 | . . . . . . 7 ⊢ (𝑍 ∈ 𝑃 → (𝑍‘2) ∈ ℝ) |
15 | 14 | 3ad2ant2 1133 | . . . . . 6 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → (𝑍‘2) ∈ ℝ) |
16 | 13, 15 | resubcld 11689 | . . . . 5 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → ((𝑌‘2) − (𝑍‘2)) ∈ ℝ) |
17 | 5, 16 | eqeltrid 2843 | . . . 4 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → 𝐴 ∈ ℝ) |
18 | 17 | adantr 480 | . . 3 ⊢ (((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷)) → 𝐴 ∈ ℝ) |
19 | 1, 3 | rrx2pxel 48561 | . . . . . . 7 ⊢ (𝑍 ∈ 𝑃 → (𝑍‘1) ∈ ℝ) |
20 | 19 | 3ad2ant2 1133 | . . . . . 6 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → (𝑍‘1) ∈ ℝ) |
21 | 1, 3 | rrx2pxel 48561 | . . . . . . 7 ⊢ (𝑌 ∈ 𝑃 → (𝑌‘1) ∈ ℝ) |
22 | 21 | 3ad2ant1 1132 | . . . . . 6 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → (𝑌‘1) ∈ ℝ) |
23 | 20, 22 | resubcld 11689 | . . . . 5 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → ((𝑍‘1) − (𝑌‘1)) ∈ ℝ) |
24 | 6, 23 | eqeltrid 2843 | . . . 4 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → 𝐵 ∈ ℝ) |
25 | 24 | adantr 480 | . . 3 ⊢ (((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷)) → 𝐵 ∈ ℝ) |
26 | 13, 20 | remulcld 11289 | . . . . . 6 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → ((𝑌‘2) · (𝑍‘1)) ∈ ℝ) |
27 | 22, 15 | remulcld 11289 | . . . . . 6 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → ((𝑌‘1) · (𝑍‘2)) ∈ ℝ) |
28 | 26, 27 | resubcld 11689 | . . . . 5 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → (((𝑌‘2) · (𝑍‘1)) − ((𝑌‘1) · (𝑍‘2))) ∈ ℝ) |
29 | 7, 28 | eqeltrid 2843 | . . . 4 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → 𝐶 ∈ ℝ) |
30 | 29 | adantr 480 | . . 3 ⊢ (((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷)) → 𝐶 ∈ ℝ) |
31 | 1, 3, 6, 5 | rrx2pnedifcoorneorr 48567 | . . . . 5 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → (𝐵 ≠ 0 ∨ 𝐴 ≠ 0)) |
32 | 31 | orcomd 871 | . . . 4 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → (𝐴 ≠ 0 ∨ 𝐵 ≠ 0)) |
33 | 32 | adantr 480 | . . 3 ⊢ (((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷)) → (𝐴 ≠ 0 ∨ 𝐵 ≠ 0)) |
34 | simpr 484 | . . 3 ⊢ (((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷)) → (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷)) | |
35 | itsclc0.s | . . . 4 ⊢ 𝑆 = (Sphere‘𝐸) | |
36 | itsclc0.0 | . . . 4 ⊢ 0 = (𝐼 × {0}) | |
37 | itsclc0.q | . . . 4 ⊢ 𝑄 = ((𝐴↑2) + (𝐵↑2)) | |
38 | itsclc0.d | . . . 4 ⊢ 𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2)) | |
39 | eqid 2735 | . . . 4 ⊢ {𝑝 ∈ 𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶} = {𝑝 ∈ 𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶} | |
40 | 1, 2, 3, 35, 36, 37, 38, 39 | itsclc0 48621 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 ≠ 0 ∨ 𝐵 ≠ 0) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷)) → ((𝑋 ∈ ( 0 𝑆𝑅) ∧ 𝑋 ∈ {𝑝 ∈ 𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶}) → (((𝑋‘1) = (((𝐴 · 𝐶) + (𝐵 · (√‘𝐷))) / 𝑄) ∧ (𝑋‘2) = (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄)) ∨ ((𝑋‘1) = (((𝐴 · 𝐶) − (𝐵 · (√‘𝐷))) / 𝑄) ∧ (𝑋‘2) = (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄))))) |
41 | 18, 25, 30, 33, 34, 40 | syl311anc 1383 | . 2 ⊢ (((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷)) → ((𝑋 ∈ ( 0 𝑆𝑅) ∧ 𝑋 ∈ {𝑝 ∈ 𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶}) → (((𝑋‘1) = (((𝐴 · 𝐶) + (𝐵 · (√‘𝐷))) / 𝑄) ∧ (𝑋‘2) = (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄)) ∨ ((𝑋‘1) = (((𝐴 · 𝐶) − (𝐵 · (√‘𝐷))) / 𝑄) ∧ (𝑋‘2) = (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄))))) |
42 | 11, 41 | sylbid 240 | 1 ⊢ (((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) ∧ (𝑅 ∈ ℝ+ ∧ 0 ≤ 𝐷)) → ((𝑋 ∈ ( 0 𝑆𝑅) ∧ 𝑋 ∈ (𝑌𝐿𝑍)) → (((𝑋‘1) = (((𝐴 · 𝐶) + (𝐵 · (√‘𝐷))) / 𝑄) ∧ (𝑋‘2) = (((𝐵 · 𝐶) − (𝐴 · (√‘𝐷))) / 𝑄)) ∨ ((𝑋‘1) = (((𝐴 · 𝐶) − (𝐵 · (√‘𝐷))) / 𝑄) ∧ (𝑋‘2) = (((𝐵 · 𝐶) + (𝐴 · (√‘𝐷))) / 𝑄))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 {crab 3433 {csn 4631 {cpr 4633 class class class wbr 5148 × cxp 5687 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 ≤ cle 11294 − cmin 11490 / cdiv 11918 2c2 12319 ℝ+crp 13032 ↑cexp 14099 √csqrt 15269 ℝ^crrx 25431 LineMcline 48577 Spherecsph 48578 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-0g 17488 df-gsum 17489 df-prds 17494 df-pws 17496 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-ghm 19244 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-rhm 20489 df-subrng 20563 df-subrg 20587 df-drng 20748 df-field 20749 df-staf 20857 df-srng 20858 df-lmod 20877 df-lss 20948 df-sra 21190 df-rgmod 21191 df-xmet 21375 df-met 21376 df-cnfld 21383 df-refld 21641 df-dsmm 21770 df-frlm 21785 df-nm 24611 df-tng 24613 df-tcph 25217 df-rrx 25433 df-ehl 25434 df-line 48579 df-sph 48580 |
This theorem is referenced by: (None) |
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