rrx2linesl - Mathbox for Alexander van der Vekens

	Mathbox for Alexander van der Vekens		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > rrx2linesl		Structured version Visualization version GIF version

Theorem rrx2linesl 48212

Description: The line passing through the two different points 𝑋 and 𝑌 in a real Euclidean space of dimension 2, expressed by the slope 𝑆 between the two points ("point-slope form"), sometimes also written as ((𝑝‘2) − (𝑋‘2)) = (𝑆 · ((𝑝‘1) − (𝑋‘1))). (Contributed by AV, 22-Jan-2023.)

**Hypotheses**
Ref	Expression
rrx2line.i	⊢ 𝐼 = {1, 2}
rrx2line.e	⊢ 𝐸 = (ℝ^‘𝐼)
rrx2line.b	⊢ 𝑃 = (ℝ ↑_m 𝐼)
rrx2line.l	⊢ 𝐿 = (Line_M‘𝐸)
rrx2linesl.s	⊢ 𝑆 = (((𝑌‘2) − (𝑋‘2)) / ((𝑌‘1) − (𝑋‘1)))

**Assertion**
Ref	Expression
rrx2linesl	⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) → (𝑋𝐿𝑌) = {𝑝 ∈ 𝑃 ∣ (𝑝‘2) = ((𝑆 · ((𝑝‘1) − (𝑋‘1))) + (𝑋‘2))})

Distinct variable groups: 𝐸,𝑝 𝐼,𝑝 𝑃,𝑝 𝑋,𝑝 𝑌,𝑝 Allowed substitution hints: 𝑆(𝑝) 𝐿(𝑝)

Proof of Theorem rrx2linesl Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 6904	. . . 4 ⊢ (𝑋 = 𝑌 → (𝑋‘1) = (𝑌‘1))
2	1	necon3i 2966	. . 3 ⊢ ((𝑋‘1) ≠ (𝑌‘1) → 𝑋 ≠ 𝑌)
3		rrx2line.i	. . . 4 ⊢ 𝐼 = {1, 2}
4		rrx2line.e	. . . 4 ⊢ 𝐸 = (ℝ^‘𝐼)
5		rrx2line.b	. . . 4 ⊢ 𝑃 = (ℝ ↑_m 𝐼)
6		rrx2line.l	. . . 4 ⊢ 𝐿 = (Line_M‘𝐸)
7	3, 4, 5, 6	rrx2line 48209	. . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (𝑋𝐿𝑌) = {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ ((𝑝‘1) = (((1 − 𝑡) · (𝑋‘1)) + (𝑡 · (𝑌‘1))) ∧ (𝑝‘2) = (((1 − 𝑡) · (𝑋‘2)) + (𝑡 · (𝑌‘2))))})
8	2, 7	syl3an3 1162	. 2 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) → (𝑋𝐿𝑌) = {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ ((𝑝‘1) = (((1 − 𝑡) · (𝑋‘1)) + (𝑡 · (𝑌‘1))) ∧ (𝑝‘2) = (((1 − 𝑡) · (𝑋‘2)) + (𝑡 · (𝑌‘2))))})
9		reex 11256	. . . . . . . 8 ⊢ ℝ ∈ V
10		prex 5442	. . . . . . . . 9 ⊢ {1, 2} ∈ V
11	3, 10	eqeltri 2825	. . . . . . . 8 ⊢ 𝐼 ∈ V
12	9, 11	elmap 8908	. . . . . . 7 ⊢ (𝑝 ∈ (ℝ ↑_m 𝐼) ↔ 𝑝:𝐼⟶ℝ)
13		id 22	. . . . . . . 8 ⊢ (𝑝:𝐼⟶ℝ → 𝑝:𝐼⟶ℝ)
14		1ex 11267	. . . . . . . . . . 11 ⊢ 1 ∈ V
15	14	prid1 4774	. . . . . . . . . 10 ⊢ 1 ∈ {1, 2}
16	15, 3	eleqtrri 2828	. . . . . . . . 9 ⊢ 1 ∈ 𝐼
17	16	a1i 11	. . . . . . . 8 ⊢ (𝑝:𝐼⟶ℝ → 1 ∈ 𝐼)
18	13, 17	ffvelcdmd 7104	. . . . . . 7 ⊢ (𝑝:𝐼⟶ℝ → (𝑝‘1) ∈ ℝ)
19	12, 18	sylbi 216	. . . . . 6 ⊢ (𝑝 ∈ (ℝ ↑_m 𝐼) → (𝑝‘1) ∈ ℝ)
20	19, 5	eleq2s 2847	. . . . 5 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘1) ∈ ℝ)
21	20	adantl 480	. . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) ∧ 𝑝 ∈ 𝑃) → (𝑝‘1) ∈ ℝ)
22	9, 11	elmap 8908	. . . . . . . 8 ⊢ (𝑋 ∈ (ℝ ↑_m 𝐼) ↔ 𝑋:𝐼⟶ℝ)
23		id 22	. . . . . . . . 9 ⊢ (𝑋:𝐼⟶ℝ → 𝑋:𝐼⟶ℝ)
24	16	a1i 11	. . . . . . . . 9 ⊢ (𝑋:𝐼⟶ℝ → 1 ∈ 𝐼)
25	23, 24	ffvelcdmd 7104	. . . . . . . 8 ⊢ (𝑋:𝐼⟶ℝ → (𝑋‘1) ∈ ℝ)
26	22, 25	sylbi 216	. . . . . . 7 ⊢ (𝑋 ∈ (ℝ ↑_m 𝐼) → (𝑋‘1) ∈ ℝ)
27	26, 5	eleq2s 2847	. . . . . 6 ⊢ (𝑋 ∈ 𝑃 → (𝑋‘1) ∈ ℝ)
28	27	3ad2ant1 1130	. . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) → (𝑋‘1) ∈ ℝ)
29	28	adantr 479	. . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) ∧ 𝑝 ∈ 𝑃) → (𝑋‘1) ∈ ℝ)
30	9, 11	elmap 8908	. . . . . . . 8 ⊢ (𝑌 ∈ (ℝ ↑_m 𝐼) ↔ 𝑌:𝐼⟶ℝ)
31		id 22	. . . . . . . . 9 ⊢ (𝑌:𝐼⟶ℝ → 𝑌:𝐼⟶ℝ)
32	16	a1i 11	. . . . . . . . 9 ⊢ (𝑌:𝐼⟶ℝ → 1 ∈ 𝐼)
33	31, 32	ffvelcdmd 7104	. . . . . . . 8 ⊢ (𝑌:𝐼⟶ℝ → (𝑌‘1) ∈ ℝ)
34	30, 33	sylbi 216	. . . . . . 7 ⊢ (𝑌 ∈ (ℝ ↑_m 𝐼) → (𝑌‘1) ∈ ℝ)
35	34, 5	eleq2s 2847	. . . . . 6 ⊢ (𝑌 ∈ 𝑃 → (𝑌‘1) ∈ ℝ)
36	35	3ad2ant2 1131	. . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) → (𝑌‘1) ∈ ℝ)
37	36	adantr 479	. . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) ∧ 𝑝 ∈ 𝑃) → (𝑌‘1) ∈ ℝ)
38		simpl3 1190	. . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) ∧ 𝑝 ∈ 𝑃) → (𝑋‘1) ≠ (𝑌‘1))
39		2ex 12351	. . . . . . . . . . 11 ⊢ 2 ∈ V
40	39	prid2 4775	. . . . . . . . . 10 ⊢ 2 ∈ {1, 2}
41	40, 3	eleqtrri 2828	. . . . . . . . 9 ⊢ 2 ∈ 𝐼
42	41	a1i 11	. . . . . . . 8 ⊢ (𝑝:𝐼⟶ℝ → 2 ∈ 𝐼)
43	13, 42	ffvelcdmd 7104	. . . . . . 7 ⊢ (𝑝:𝐼⟶ℝ → (𝑝‘2) ∈ ℝ)
44	12, 43	sylbi 216	. . . . . 6 ⊢ (𝑝 ∈ (ℝ ↑_m 𝐼) → (𝑝‘2) ∈ ℝ)
45	44, 5	eleq2s 2847	. . . . 5 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘2) ∈ ℝ)
46	45	adantl 480	. . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) ∧ 𝑝 ∈ 𝑃) → (𝑝‘2) ∈ ℝ)
47	41	a1i 11	. . . . . . . . 9 ⊢ (𝑋:𝐼⟶ℝ → 2 ∈ 𝐼)
48	23, 47	ffvelcdmd 7104	. . . . . . . 8 ⊢ (𝑋:𝐼⟶ℝ → (𝑋‘2) ∈ ℝ)
49	22, 48	sylbi 216	. . . . . . 7 ⊢ (𝑋 ∈ (ℝ ↑_m 𝐼) → (𝑋‘2) ∈ ℝ)
50	49, 5	eleq2s 2847	. . . . . 6 ⊢ (𝑋 ∈ 𝑃 → (𝑋‘2) ∈ ℝ)
51	50	3ad2ant1 1130	. . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) → (𝑋‘2) ∈ ℝ)
52	51	adantr 479	. . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) ∧ 𝑝 ∈ 𝑃) → (𝑋‘2) ∈ ℝ)
53	5	eleq2i 2821	. . . . . . . 8 ⊢ (𝑌 ∈ 𝑃 ↔ 𝑌 ∈ (ℝ ↑_m 𝐼))
54	53, 30	bitri 274	. . . . . . 7 ⊢ (𝑌 ∈ 𝑃 ↔ 𝑌:𝐼⟶ℝ)
55	41	a1i 11	. . . . . . . 8 ⊢ (𝑌:𝐼⟶ℝ → 2 ∈ 𝐼)
56	31, 55	ffvelcdmd 7104	. . . . . . 7 ⊢ (𝑌:𝐼⟶ℝ → (𝑌‘2) ∈ ℝ)
57	54, 56	sylbi 216	. . . . . 6 ⊢ (𝑌 ∈ 𝑃 → (𝑌‘2) ∈ ℝ)
58	57	3ad2ant2 1131	. . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) → (𝑌‘2) ∈ ℝ)
59	58	adantr 479	. . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) ∧ 𝑝 ∈ 𝑃) → (𝑌‘2) ∈ ℝ)
60		rrx2linesl.s	. . . 4 ⊢ 𝑆 = (((𝑌‘2) − (𝑋‘2)) / ((𝑌‘1) − (𝑋‘1)))
61	21, 29, 37, 38, 46, 52, 59, 60	affinecomb1 48171	. . 3 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) ∧ 𝑝 ∈ 𝑃) → (∃𝑡 ∈ ℝ ((𝑝‘1) = (((1 − 𝑡) · (𝑋‘1)) + (𝑡 · (𝑌‘1))) ∧ (𝑝‘2) = (((1 − 𝑡) · (𝑋‘2)) + (𝑡 · (𝑌‘2)))) ↔ (𝑝‘2) = ((𝑆 · ((𝑝‘1) − (𝑋‘1))) + (𝑋‘2))))
62	61	rabbidva 3435	. 2 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) → {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ ((𝑝‘1) = (((1 − 𝑡) · (𝑋‘1)) + (𝑡 · (𝑌‘1))) ∧ (𝑝‘2) = (((1 − 𝑡) · (𝑋‘2)) + (𝑡 · (𝑌‘2))))} = {𝑝 ∈ 𝑃 ∣ (𝑝‘2) = ((𝑆 · ((𝑝‘1) − (𝑋‘1))) + (𝑋‘2))})
63	8, 62	eqtrd 2769	1 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) → (𝑋𝐿𝑌) = {𝑝 ∈ 𝑃 ∣ (𝑝‘2) = ((𝑆 · ((𝑝‘1) − (𝑋‘1))) + (𝑋‘2))})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2100 ≠ wne 2933 ∃wrex 3063 {crab 3428 Vcvv 3472 {cpr 4638 ⟶wf 6554 ‘cfv 6558 (class class class)co 7430 ↑_m cmap 8863 ℝcr 11164 1c1 11166 + caddc 11168 · cmul 11170 − cmin 11501 / cdiv 11928 2c2 12329 ℝ^crrx 25425 Line_Mcline 48196

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2102 ax-9 2110 ax-10 2133 ax-11 2150 ax-12 2170 ax-ext 2700 ax-rep 5293 ax-sep 5307 ax-nul 5314 ax-pow 5373 ax-pr 5437 ax-un 7751 ax-cnex 11221 ax-resscn 11222 ax-1cn 11223 ax-icn 11224 ax-addcl 11225 ax-addrcl 11226 ax-mulcl 11227 ax-mulrcl 11228 ax-mulcom 11229 ax-addass 11230 ax-mulass 11231 ax-distr 11232 ax-i2m1 11233 ax-1ne0 11234 ax-1rid 11235 ax-rnegex 11236 ax-rrecex 11237 ax-cnre 11238 ax-pre-lttri 11239 ax-pre-lttrn 11240 ax-pre-ltadd 11241 ax-pre-mulgt0 11242 ax-pre-sup 11243 ax-addf 11244 ax-mulf 11245

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2062 df-mo 2532 df-eu 2561 df-clab 2707 df-cleq 2721 df-clel 2806 df-nfc 2881 df-ne 2934 df-nel 3040 df-ral 3055 df-rex 3064 df-rmo 3373 df-reu 3374 df-rab 3429 df-v 3474 df-sbc 3789 df-csb 3905 df-dif 3962 df-un 3964 df-in 3966 df-ss 3976 df-pss 3979 df-nul 4336 df-if 4537 df-pw 4612 df-sn 4637 df-pr 4639 df-tp 4641 df-op 4643 df-uni 4919 df-iun 5008 df-br 5157 df-opab 5219 df-mpt 5240 df-tr 5274 df-id 5584 df-eprel 5590 df-po 5598 df-so 5599 df-fr 5641 df-we 5643 df-xp 5692 df-rel 5693 df-cnv 5694 df-co 5695 df-dm 5696 df-rn 5697 df-res 5698 df-ima 5699 df-pred 6317 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6510 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7386 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7695 df-om 7885 df-1st 8011 df-2nd 8012 df-supp 8183 df-tpos 8249 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8742 df-map 8865 df-ixp 8935 df-en 8983 df-dom 8984 df-sdom 8985 df-fin 8986 df-fsupp 9413 df-sup 9492 df-pnf 11307 df-mnf 11308 df-xr 11309 df-ltxr 11310 df-le 11311 df-sub 11503 df-neg 11504 df-div 11929 df-nn 12275 df-2 12337 df-3 12338 df-4 12339 df-5 12340 df-6 12341 df-7 12342 df-8 12343 df-9 12344 df-n0 12535 df-z 12621 df-dec 12740 df-uz 12885 df-rp 13039 df-fz 13549 df-seq 14033 df-exp 14093 df-cj 15125 df-re 15126 df-im 15127 df-sqrt 15261 df-abs 15262 df-struct 17170 df-sets 17187 df-slot 17205 df-ndx 17217 df-base 17235 df-ress 17264 df-plusg 17300 df-mulr 17301 df-starv 17302 df-sca 17303 df-vsca 17304 df-ip 17305 df-tset 17306 df-ple 17307 df-ds 17309 df-unif 17310 df-hom 17311 df-cco 17312 df-0g 17477 df-prds 17483 df-pws 17485 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mhm 18794 df-grp 18952 df-minusg 18953 df-sbg 18954 df-subg 19139 df-ghm 19229 df-cmn 19802 df-abl 19803 df-mgp 20140 df-rng 20158 df-ur 20187 df-ring 20240 df-cring 20241 df-oppr 20338 df-dvdsr 20361 df-unit 20362 df-invr 20392 df-dvr 20405 df-rhm 20476 df-subrng 20550 df-subrg 20575 df-drng 20731 df-field 20732 df-staf 20840 df-srng 20841 df-lmod 20860 df-lss 20931 df-sra 21173 df-rgmod 21174 df-cnfld 21366 df-refld 21623 df-dsmm 21752 df-frlm 21767 df-tng 24607 df-tcph 25211 df-rrx 25427 df-line 48198

This theorem is referenced by: line2 48221

W3C validator