itsclinecirc0 - Mathbox for Alexander van der Vekens

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Theorem itsclinecirc0 48768

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 48739	. . . . 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 48707	. . . . . . 7 ⊢ (𝑌 ∈ 𝑃 → (𝑌‘2) ∈ ℝ)
13	12	3ad2ant1 1133	. . . . . 6 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → (𝑌‘2) ∈ ℝ)
14	1, 3	rrx2pyel 48707	. . . . . . 7 ⊢ (𝑍 ∈ 𝑃 → (𝑍‘2) ∈ ℝ)
15	14	3ad2ant2 1134	. . . . . 6 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → (𝑍‘2) ∈ ℝ)
16	13, 15	resubcld 11548	. . . . 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 48706	. . . . . . 7 ⊢ (𝑍 ∈ 𝑃 → (𝑍‘1) ∈ ℝ)
20	19	3ad2ant2 1134	. . . . . 6 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → (𝑍‘1) ∈ ℝ)
21	1, 3	rrx2pxel 48706	. . . . . . 7 ⊢ (𝑌 ∈ 𝑃 → (𝑌‘1) ∈ ℝ)
22	21	3ad2ant1 1133	. . . . . 6 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → (𝑌‘1) ∈ ℝ)
23	20, 22	resubcld 11548	. . . . 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 11145	. . . . . 6 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → ((𝑌‘2) · (𝑍‘1)) ∈ ℝ)
27	22, 15	remulcld 11145	. . . . . 6 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → ((𝑌‘1) · (𝑍‘2)) ∈ ℝ)
28	26, 27	resubcld 11548	. . . . 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 48712	. . . . 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 48766	. . 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 3394 {csn 4577 {cpr 4579 class class class wbr 5092 × cxp 5617 ‘cfv 6482 (class class class)co 7349 ↑_m cmap 8753 ℝcr 11008 0cc0 11009 1c1 11010 + caddc 11012 · cmul 11014 ≤ cle 11150 − cmin 11347 / cdiv 11777 2c2 12183 ℝ⁺crp 12893 ↑cexp 13968 √csqrt 15140 ℝ^crrx 25281 Line_Mcline 48722 Spherecsph 48723

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 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088 ax-mulf 11089

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 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-tpos 8159 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-er 8625 df-map 8755 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-sup 9332 df-oi 9402 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-rp 12894 df-xneg 13014 df-xadd 13015 df-xmul 13016 df-ico 13254 df-icc 13255 df-fz 13411 df-fzo 13558 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-sum 15594 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-0g 17345 df-gsum 17346 df-prds 17351 df-pws 17353 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-mhm 18657 df-grp 18815 df-minusg 18816 df-sbg 18817 df-subg 19002 df-ghm 19092 df-cntz 19196 df-cmn 19661 df-abl 19662 df-mgp 20026 df-rng 20038 df-ur 20067 df-ring 20120 df-cring 20121 df-oppr 20222 df-dvdsr 20242 df-unit 20243 df-invr 20273 df-dvr 20286 df-rhm 20357 df-subrng 20431 df-subrg 20455 df-drng 20616 df-field 20617 df-staf 20724 df-srng 20725 df-lmod 20765 df-lss 20835 df-sra 21077 df-rgmod 21078 df-xmet 21254 df-met 21255 df-cnfld 21262 df-refld 21512 df-dsmm 21639 df-frlm 21654 df-nm 24468 df-tng 24470 df-tcph 25067 df-rrx 25283 df-ehl 25284 df-line 48724 df-sph 48725

This theorem is referenced by: (None)

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