itsclinecirc0 - Mathbox for Alexander van der Vekens

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Theorem itsclinecirc0 48759

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 48730	. . . . 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 2814	. . 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 48698	. . . . . . 7 ⊢ (𝑌 ∈ 𝑃 → (𝑌‘2) ∈ ℝ)
13	12	3ad2ant1 1133	. . . . . 6 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → (𝑌‘2) ∈ ℝ)
14	1, 3	rrx2pyel 48698	. . . . . . 7 ⊢ (𝑍 ∈ 𝑃 → (𝑍‘2) ∈ ℝ)
15	14	3ad2ant2 1134	. . . . . 6 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → (𝑍‘2) ∈ ℝ)
16	13, 15	resubcld 11606	. . . . 5 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → ((𝑌‘2) − (𝑍‘2)) ∈ ℝ)
17	5, 16	eqeltrid 2832	. . . 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 48697	. . . . . . 7 ⊢ (𝑍 ∈ 𝑃 → (𝑍‘1) ∈ ℝ)
20	19	3ad2ant2 1134	. . . . . 6 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → (𝑍‘1) ∈ ℝ)
21	1, 3	rrx2pxel 48697	. . . . . . 7 ⊢ (𝑌 ∈ 𝑃 → (𝑌‘1) ∈ ℝ)
22	21	3ad2ant1 1133	. . . . . 6 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → (𝑌‘1) ∈ ℝ)
23	20, 22	resubcld 11606	. . . . 5 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → ((𝑍‘1) − (𝑌‘1)) ∈ ℝ)
24	6, 23	eqeltrid 2832	. . . 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 11204	. . . . . 6 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → ((𝑌‘2) · (𝑍‘1)) ∈ ℝ)
27	22, 15	remulcld 11204	. . . . . 6 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → ((𝑌‘1) · (𝑍‘2)) ∈ ℝ)
28	26, 27	resubcld 11606	. . . . 5 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → (((𝑌‘2) · (𝑍‘1)) − ((𝑌‘1) · (𝑍‘2))) ∈ ℝ)
29	7, 28	eqeltrid 2832	. . . 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 48703	. . . . 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 2729	. . . 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 48757	. . 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 1386	. 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 1540 ∈ wcel 2109 ≠ wne 2925 {crab 3405 {csn 4589 {cpr 4591 class class class wbr 5107 × cxp 5636 ‘cfv 6511 (class class class)co 7387 ↑_m cmap 8799 ℝcr 11067 0cc0 11068 1c1 11069 + caddc 11071 · cmul 11073 ≤ cle 11209 − cmin 11405 / cdiv 11835 2c2 12241 ℝ⁺crp 12951 ↑cexp 14026 √csqrt 15199 ℝ^crrx 25283 Line_Mcline 48713 Spherecsph 48714

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147 ax-mulf 11148

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 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 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-tpos 8205 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-sup 9393 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-seq 13967 df-exp 14027 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-clim 15454 df-sum 15653 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-0g 17404 df-gsum 17405 df-prds 17410 df-pws 17412 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18710 df-grp 18868 df-minusg 18869 df-sbg 18870 df-subg 19055 df-ghm 19145 df-cntz 19249 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-ring 20144 df-cring 20145 df-oppr 20246 df-dvdsr 20266 df-unit 20267 df-invr 20297 df-dvr 20310 df-rhm 20381 df-subrng 20455 df-subrg 20479 df-drng 20640 df-field 20641 df-staf 20748 df-srng 20749 df-lmod 20768 df-lss 20838 df-sra 21080 df-rgmod 21081 df-xmet 21257 df-met 21258 df-cnfld 21265 df-refld 21514 df-dsmm 21641 df-frlm 21656 df-nm 24470 df-tng 24472 df-tcph 25069 df-rrx 25285 df-ehl 25286 df-line 48715 df-sph 48716

This theorem is referenced by: (None)

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