itsclinecirc0 - Mathbox for Alexander van der Vekens

	Mathbox for Alexander van der Vekens		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > itsclinecirc0		Structured version Visualization version GIF version

Theorem itsclinecirc0 48098

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 48069	. . . . 5 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → (𝑌𝐿𝑍) = {𝑝 ∈ 𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶})
9	8	adantr 479	. . . 4 ⊢ (((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) ∧ (𝑅 ∈ ℝ⁺ ∧ 0 ≤ 𝐷)) → (𝑌𝐿𝑍) = {𝑝 ∈ 𝑃 ∣ ((𝐴 · (𝑝‘1)) + (𝐵 · (𝑝‘2))) = 𝐶})
10	9	eleq2d 2811	. . 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 628	. 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 48037	. . . . . . 7 ⊢ (𝑌 ∈ 𝑃 → (𝑌‘2) ∈ ℝ)
13	12	3ad2ant1 1130	. . . . . 6 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → (𝑌‘2) ∈ ℝ)
14	1, 3	rrx2pyel 48037	. . . . . . 7 ⊢ (𝑍 ∈ 𝑃 → (𝑍‘2) ∈ ℝ)
15	14	3ad2ant2 1131	. . . . . 6 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → (𝑍‘2) ∈ ℝ)
16	13, 15	resubcld 11688	. . . . 5 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → ((𝑌‘2) − (𝑍‘2)) ∈ ℝ)
17	5, 16	eqeltrid 2829	. . . 4 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → 𝐴 ∈ ℝ)
18	17	adantr 479	. . 3 ⊢ (((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) ∧ (𝑅 ∈ ℝ⁺ ∧ 0 ≤ 𝐷)) → 𝐴 ∈ ℝ)
19	1, 3	rrx2pxel 48036	. . . . . . 7 ⊢ (𝑍 ∈ 𝑃 → (𝑍‘1) ∈ ℝ)
20	19	3ad2ant2 1131	. . . . . 6 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → (𝑍‘1) ∈ ℝ)
21	1, 3	rrx2pxel 48036	. . . . . . 7 ⊢ (𝑌 ∈ 𝑃 → (𝑌‘1) ∈ ℝ)
22	21	3ad2ant1 1130	. . . . . 6 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → (𝑌‘1) ∈ ℝ)
23	20, 22	resubcld 11688	. . . . 5 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → ((𝑍‘1) − (𝑌‘1)) ∈ ℝ)
24	6, 23	eqeltrid 2829	. . . 4 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → 𝐵 ∈ ℝ)
25	24	adantr 479	. . 3 ⊢ (((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) ∧ (𝑅 ∈ ℝ⁺ ∧ 0 ≤ 𝐷)) → 𝐵 ∈ ℝ)
26	13, 20	remulcld 11290	. . . . . 6 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → ((𝑌‘2) · (𝑍‘1)) ∈ ℝ)
27	22, 15	remulcld 11290	. . . . . 6 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → ((𝑌‘1) · (𝑍‘2)) ∈ ℝ)
28	26, 27	resubcld 11688	. . . . 5 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → (((𝑌‘2) · (𝑍‘1)) − ((𝑌‘1) · (𝑍‘2))) ∈ ℝ)
29	7, 28	eqeltrid 2829	. . . 4 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → 𝐶 ∈ ℝ)
30	29	adantr 479	. . 3 ⊢ (((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) ∧ (𝑅 ∈ ℝ⁺ ∧ 0 ≤ 𝐷)) → 𝐶 ∈ ℝ)
31	1, 3, 6, 5	rrx2pnedifcoorneorr 48042	. . . . 5 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → (𝐵 ≠ 0 ∨ 𝐴 ≠ 0))
32	31	orcomd 869	. . . 4 ⊢ ((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) → (𝐴 ≠ 0 ∨ 𝐵 ≠ 0))
33	32	adantr 479	. . 3 ⊢ (((𝑌 ∈ 𝑃 ∧ 𝑍 ∈ 𝑃 ∧ 𝑌 ≠ 𝑍) ∧ (𝑅 ∈ ℝ⁺ ∧ 0 ≤ 𝐷)) → (𝐴 ≠ 0 ∨ 𝐵 ≠ 0))
34		simpr 483	. . 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 2725	. . . 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 48096	. . 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 1381	. 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 239	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 394 ∨ wo 845 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 {crab 3418 {csn 4632 {cpr 4634 class class class wbr 5152 × cxp 5679 ‘cfv 6553 (class class class)co 7423 ↑_m cmap 8854 ℝcr 11153 0cc0 11154 1c1 11155 + caddc 11157 · cmul 11159 ≤ cle 11295 − cmin 11490 / cdiv 11917 2c2 12314 ℝ⁺crp 13023 ↑cexp 14076 √csqrt 15233 ℝ^crrx 25394 Line_Mcline 48052 Spherecsph 48053

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-inf2 9680 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 ax-pre-sup 11232 ax-addf 11233 ax-mulf 11234

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-se 5637 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-of 7689 df-om 7876 df-1st 8002 df-2nd 8003 df-supp 8174 df-tpos 8240 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-2o 8496 df-er 8733 df-map 8856 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9402 df-sup 9481 df-oi 9549 df-card 9978 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-div 11918 df-nn 12260 df-2 12322 df-3 12323 df-4 12324 df-5 12325 df-6 12326 df-7 12327 df-8 12328 df-9 12329 df-n0 12520 df-z 12606 df-dec 12725 df-uz 12870 df-rp 13024 df-xneg 13141 df-xadd 13142 df-xmul 13143 df-ico 13379 df-icc 13380 df-fz 13534 df-fzo 13677 df-seq 14017 df-exp 14077 df-hash 14343 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-clim 15485 df-sum 15686 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-starv 17276 df-sca 17277 df-vsca 17278 df-ip 17279 df-tset 17280 df-ple 17281 df-ds 17283 df-unif 17284 df-hom 17285 df-cco 17286 df-0g 17451 df-gsum 17452 df-prds 17457 df-pws 17459 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-mhm 18768 df-grp 18926 df-minusg 18927 df-sbg 18928 df-subg 19112 df-ghm 19202 df-cntz 19306 df-cmn 19775 df-abl 19776 df-mgp 20113 df-rng 20131 df-ur 20160 df-ring 20213 df-cring 20214 df-oppr 20311 df-dvdsr 20334 df-unit 20335 df-invr 20365 df-dvr 20378 df-rhm 20449 df-subrng 20523 df-subrg 20548 df-drng 20666 df-field 20667 df-staf 20765 df-srng 20766 df-lmod 20785 df-lss 20856 df-sra 21098 df-rgmod 21099 df-xmet 21328 df-met 21329 df-cnfld 21336 df-refld 21593 df-dsmm 21722 df-frlm 21737 df-nm 24574 df-tng 24576 df-tcph 25180 df-rrx 25396 df-ehl 25397 df-line 48054 df-sph 48055

This theorem is referenced by: (None)

W3C validator