rrx2linesl - Mathbox for Alexander van der Vekens

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Theorem rrx2linesl 48477

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 6919	. . . 4 ⊢ (𝑋 = 𝑌 → (𝑋‘1) = (𝑌‘1))
2	1	necon3i 2979	. . 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 48474	. . 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 1165	. 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 11275	. . . . . . . 8 ⊢ ℝ ∈ V
10		prex 5452	. . . . . . . . 9 ⊢ {1, 2} ∈ V
11	3, 10	eqeltri 2840	. . . . . . . 8 ⊢ 𝐼 ∈ V
12	9, 11	elmap 8929	. . . . . . 7 ⊢ (𝑝 ∈ (ℝ ↑_m 𝐼) ↔ 𝑝:𝐼⟶ℝ)
13		id 22	. . . . . . . 8 ⊢ (𝑝:𝐼⟶ℝ → 𝑝:𝐼⟶ℝ)
14		1ex 11286	. . . . . . . . . . 11 ⊢ 1 ∈ V
15	14	prid1 4787	. . . . . . . . . 10 ⊢ 1 ∈ {1, 2}
16	15, 3	eleqtrri 2843	. . . . . . . . 9 ⊢ 1 ∈ 𝐼
17	16	a1i 11	. . . . . . . 8 ⊢ (𝑝:𝐼⟶ℝ → 1 ∈ 𝐼)
18	13, 17	ffvelcdmd 7119	. . . . . . 7 ⊢ (𝑝:𝐼⟶ℝ → (𝑝‘1) ∈ ℝ)
19	12, 18	sylbi 217	. . . . . 6 ⊢ (𝑝 ∈ (ℝ ↑_m 𝐼) → (𝑝‘1) ∈ ℝ)
20	19, 5	eleq2s 2862	. . . . 5 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘1) ∈ ℝ)
21	20	adantl 481	. . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) ∧ 𝑝 ∈ 𝑃) → (𝑝‘1) ∈ ℝ)
22	9, 11	elmap 8929	. . . . . . . 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 7119	. . . . . . . 8 ⊢ (𝑋:𝐼⟶ℝ → (𝑋‘1) ∈ ℝ)
26	22, 25	sylbi 217	. . . . . . 7 ⊢ (𝑋 ∈ (ℝ ↑_m 𝐼) → (𝑋‘1) ∈ ℝ)
27	26, 5	eleq2s 2862	. . . . . 6 ⊢ (𝑋 ∈ 𝑃 → (𝑋‘1) ∈ ℝ)
28	27	3ad2ant1 1133	. . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) → (𝑋‘1) ∈ ℝ)
29	28	adantr 480	. . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) ∧ 𝑝 ∈ 𝑃) → (𝑋‘1) ∈ ℝ)
30	9, 11	elmap 8929	. . . . . . . 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 7119	. . . . . . . 8 ⊢ (𝑌:𝐼⟶ℝ → (𝑌‘1) ∈ ℝ)
34	30, 33	sylbi 217	. . . . . . 7 ⊢ (𝑌 ∈ (ℝ ↑_m 𝐼) → (𝑌‘1) ∈ ℝ)
35	34, 5	eleq2s 2862	. . . . . 6 ⊢ (𝑌 ∈ 𝑃 → (𝑌‘1) ∈ ℝ)
36	35	3ad2ant2 1134	. . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) → (𝑌‘1) ∈ ℝ)
37	36	adantr 480	. . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) ∧ 𝑝 ∈ 𝑃) → (𝑌‘1) ∈ ℝ)
38		simpl3 1193	. . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) ∧ 𝑝 ∈ 𝑃) → (𝑋‘1) ≠ (𝑌‘1))
39		2ex 12370	. . . . . . . . . . 11 ⊢ 2 ∈ V
40	39	prid2 4788	. . . . . . . . . 10 ⊢ 2 ∈ {1, 2}
41	40, 3	eleqtrri 2843	. . . . . . . . 9 ⊢ 2 ∈ 𝐼
42	41	a1i 11	. . . . . . . 8 ⊢ (𝑝:𝐼⟶ℝ → 2 ∈ 𝐼)
43	13, 42	ffvelcdmd 7119	. . . . . . 7 ⊢ (𝑝:𝐼⟶ℝ → (𝑝‘2) ∈ ℝ)
44	12, 43	sylbi 217	. . . . . 6 ⊢ (𝑝 ∈ (ℝ ↑_m 𝐼) → (𝑝‘2) ∈ ℝ)
45	44, 5	eleq2s 2862	. . . . 5 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘2) ∈ ℝ)
46	45	adantl 481	. . . 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 7119	. . . . . . . 8 ⊢ (𝑋:𝐼⟶ℝ → (𝑋‘2) ∈ ℝ)
49	22, 48	sylbi 217	. . . . . . 7 ⊢ (𝑋 ∈ (ℝ ↑_m 𝐼) → (𝑋‘2) ∈ ℝ)
50	49, 5	eleq2s 2862	. . . . . 6 ⊢ (𝑋 ∈ 𝑃 → (𝑋‘2) ∈ ℝ)
51	50	3ad2ant1 1133	. . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) → (𝑋‘2) ∈ ℝ)
52	51	adantr 480	. . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) ∧ 𝑝 ∈ 𝑃) → (𝑋‘2) ∈ ℝ)
53	5	eleq2i 2836	. . . . . . . 8 ⊢ (𝑌 ∈ 𝑃 ↔ 𝑌 ∈ (ℝ ↑_m 𝐼))
54	53, 30	bitri 275	. . . . . . 7 ⊢ (𝑌 ∈ 𝑃 ↔ 𝑌:𝐼⟶ℝ)
55	41	a1i 11	. . . . . . . 8 ⊢ (𝑌:𝐼⟶ℝ → 2 ∈ 𝐼)
56	31, 55	ffvelcdmd 7119	. . . . . . 7 ⊢ (𝑌:𝐼⟶ℝ → (𝑌‘2) ∈ ℝ)
57	54, 56	sylbi 217	. . . . . 6 ⊢ (𝑌 ∈ 𝑃 → (𝑌‘2) ∈ ℝ)
58	57	3ad2ant2 1134	. . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) → (𝑌‘2) ∈ ℝ)
59	58	adantr 480	. . . 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 48436	. . 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 3450	. 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 2780	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 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∃wrex 3076 {crab 3443 Vcvv 3488 {cpr 4650 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ↑_m cmap 8884 ℝcr 11183 1c1 11185 + caddc 11187 · cmul 11189 − cmin 11520 / cdiv 11947 2c2 12348 ℝ^crrx 25436 Line_Mcline 48461

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263 ax-mulf 11264

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-tpos 8267 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-sup 9511 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-rp 13058 df-fz 13568 df-seq 14053 df-exp 14113 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-0g 17501 df-prds 17507 df-pws 17509 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-grp 18976 df-minusg 18977 df-sbg 18978 df-subg 19163 df-ghm 19253 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-cring 20263 df-oppr 20360 df-dvdsr 20383 df-unit 20384 df-invr 20414 df-dvr 20427 df-rhm 20498 df-subrng 20572 df-subrg 20597 df-drng 20753 df-field 20754 df-staf 20862 df-srng 20863 df-lmod 20882 df-lss 20953 df-sra 21195 df-rgmod 21196 df-cnfld 21388 df-refld 21646 df-dsmm 21775 df-frlm 21790 df-tng 24618 df-tcph 25222 df-rrx 25438 df-line 48463

This theorem is referenced by: line2 48486

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