itsclinecirc0 - Mathbox for Alexander van der Vekens

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Theorem itsclinecirc0 48699

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.)

**Hypotheses**
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 = (Line_M‘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)))

**Assertion**
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)))))

Proof of Theorem itsclinecirc0 Dummy variable 𝑝 is distinct from all other variables.

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 = (Line_M‘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 48670	. . . . 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 2826	. . 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 48638	. . . . . . 7 ⊢ (𝑌 ∈ 𝑃 → (𝑌‘2) ∈ ℝ)
13	12	3ad2ant1 1133	. . . . . 6 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → (𝑌‘2) ∈ ℝ)
14	1, 3	rrx2pyel 48638	. . . . . . 7 ⊢ (𝑍 ∈ 𝑃 → (𝑍‘2) ∈ ℝ)
15	14	3ad2ant2 1134	. . . . . 6 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → (𝑍‘2) ∈ ℝ)
16	13, 15	resubcld 11692	. . . . 5 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → ((𝑌‘2) − (𝑍‘2)) ∈ ℝ)
17	5, 16	eqeltrid 2844	. . . 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 48637	. . . . . . 7 ⊢ (𝑍 ∈ 𝑃 → (𝑍‘1) ∈ ℝ)
20	19	3ad2ant2 1134	. . . . . 6 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → (𝑍‘1) ∈ ℝ)
21	1, 3	rrx2pxel 48637	. . . . . . 7 ⊢ (𝑌 ∈ 𝑃 → (𝑌‘1) ∈ ℝ)
22	21	3ad2ant1 1133	. . . . . 6 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → (𝑌‘1) ∈ ℝ)
23	20, 22	resubcld 11692	. . . . 5 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → ((𝑍‘1) − (𝑌‘1)) ∈ ℝ)
24	6, 23	eqeltrid 2844	. . . 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 11292	. . . . . 6 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → ((𝑌‘2) · (𝑍‘1)) ∈ ℝ)
27	22, 15	remulcld 11292	. . . . . 6 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → ((𝑌‘1) · (𝑍‘2)) ∈ ℝ)
28	26, 27	resubcld 11692	. . . . 5 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → (((𝑌‘2) · (𝑍‘1)) − ((𝑌‘1) · (𝑍‘2))) ∈ ℝ)
29	7, 28	eqeltrid 2844	. . . 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 48643	. . . . 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 2736	. . . 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 48697	. . 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 1385	. 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 1539 ∈ wcel 2107 ≠ wne 2939 {crab 3435 {csn 4625 {cpr 4627 class class class wbr 5142 × cxp 5682 ‘cfv 6560 (class class class)co 7432 ↑_m cmap 8867 ℝcr 11155 0cc0 11156 1c1 11157 + caddc 11159 · cmul 11161 ≤ cle 11297 − cmin 11493 / cdiv 11921 2c2 12322 ℝ⁺crp 13035 ↑cexp 14103 √csqrt 15273 ℝ^crrx 25418 Line_Mcline 48653 Spherecsph 48654

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235 ax-mulf 11236

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-tpos 8252 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-map 8869 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-sup 9483 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ico 13394 df-icc 13395 df-fz 13549 df-fzo 13696 df-seq 14044 df-exp 14104 df-hash 14371 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-clim 15525 df-sum 15724 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 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 17487 df-gsum 17488 df-prds 17493 df-pws 17495 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mhm 18797 df-grp 18955 df-minusg 18956 df-sbg 18957 df-subg 19142 df-ghm 19232 df-cntz 19336 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-ring 20233 df-cring 20234 df-oppr 20335 df-dvdsr 20358 df-unit 20359 df-invr 20389 df-dvr 20402 df-rhm 20473 df-subrng 20547 df-subrg 20571 df-drng 20732 df-field 20733 df-staf 20841 df-srng 20842 df-lmod 20861 df-lss 20931 df-sra 21173 df-rgmod 21174 df-xmet 21358 df-met 21359 df-cnfld 21366 df-refld 21624 df-dsmm 21753 df-frlm 21768 df-nm 24596 df-tng 24598 df-tcph 25204 df-rrx 25420 df-ehl 25421 df-line 48655 df-sph 48656

This theorem is referenced by: (None)

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