rrx2linesl - Mathbox for Alexander van der Vekens

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Theorem rrx2linesl 48723

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 6875	. . . 4 ⊢ (𝑋 = 𝑌 → (𝑋‘1) = (𝑌‘1))
2	1	necon3i 2964	. . 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 48720	. . 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 11220	. . . . . . . 8 ⊢ ℝ ∈ V
10		prex 5407	. . . . . . . . 9 ⊢ {1, 2} ∈ V
11	3, 10	eqeltri 2830	. . . . . . . 8 ⊢ 𝐼 ∈ V
12	9, 11	elmap 8885	. . . . . . 7 ⊢ (𝑝 ∈ (ℝ ↑_m 𝐼) ↔ 𝑝:𝐼⟶ℝ)
13		id 22	. . . . . . . 8 ⊢ (𝑝:𝐼⟶ℝ → 𝑝:𝐼⟶ℝ)
14		1ex 11231	. . . . . . . . . . 11 ⊢ 1 ∈ V
15	14	prid1 4738	. . . . . . . . . 10 ⊢ 1 ∈ {1, 2}
16	15, 3	eleqtrri 2833	. . . . . . . . 9 ⊢ 1 ∈ 𝐼
17	16	a1i 11	. . . . . . . 8 ⊢ (𝑝:𝐼⟶ℝ → 1 ∈ 𝐼)
18	13, 17	ffvelcdmd 7075	. . . . . . 7 ⊢ (𝑝:𝐼⟶ℝ → (𝑝‘1) ∈ ℝ)
19	12, 18	sylbi 217	. . . . . 6 ⊢ (𝑝 ∈ (ℝ ↑_m 𝐼) → (𝑝‘1) ∈ ℝ)
20	19, 5	eleq2s 2852	. . . . 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 8885	. . . . . . . 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 7075	. . . . . . . 8 ⊢ (𝑋:𝐼⟶ℝ → (𝑋‘1) ∈ ℝ)
26	22, 25	sylbi 217	. . . . . . 7 ⊢ (𝑋 ∈ (ℝ ↑_m 𝐼) → (𝑋‘1) ∈ ℝ)
27	26, 5	eleq2s 2852	. . . . . 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 8885	. . . . . . . 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 7075	. . . . . . . 8 ⊢ (𝑌:𝐼⟶ℝ → (𝑌‘1) ∈ ℝ)
34	30, 33	sylbi 217	. . . . . . 7 ⊢ (𝑌 ∈ (ℝ ↑_m 𝐼) → (𝑌‘1) ∈ ℝ)
35	34, 5	eleq2s 2852	. . . . . 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 1194	. . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) ∧ 𝑝 ∈ 𝑃) → (𝑋‘1) ≠ (𝑌‘1))
39		2ex 12317	. . . . . . . . . . 11 ⊢ 2 ∈ V
40	39	prid2 4739	. . . . . . . . . 10 ⊢ 2 ∈ {1, 2}
41	40, 3	eleqtrri 2833	. . . . . . . . 9 ⊢ 2 ∈ 𝐼
42	41	a1i 11	. . . . . . . 8 ⊢ (𝑝:𝐼⟶ℝ → 2 ∈ 𝐼)
43	13, 42	ffvelcdmd 7075	. . . . . . 7 ⊢ (𝑝:𝐼⟶ℝ → (𝑝‘2) ∈ ℝ)
44	12, 43	sylbi 217	. . . . . 6 ⊢ (𝑝 ∈ (ℝ ↑_m 𝐼) → (𝑝‘2) ∈ ℝ)
45	44, 5	eleq2s 2852	. . . . 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 7075	. . . . . . . 8 ⊢ (𝑋:𝐼⟶ℝ → (𝑋‘2) ∈ ℝ)
49	22, 48	sylbi 217	. . . . . . 7 ⊢ (𝑋 ∈ (ℝ ↑_m 𝐼) → (𝑋‘2) ∈ ℝ)
50	49, 5	eleq2s 2852	. . . . . 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 2826	. . . . . . . 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 7075	. . . . . . 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 48682	. . 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 3422	. 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 2770	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 1086 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 ∃wrex 3060 {crab 3415 Vcvv 3459 {cpr 4603 ⟶wf 6527 ‘cfv 6531 (class class class)co 7405 ↑_m cmap 8840 ℝcr 11128 1c1 11130 + caddc 11132 · cmul 11134 − cmin 11466 / cdiv 11894 2c2 12295 ℝ^crrx 25335 Line_Mcline 48707

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207 ax-addf 11208 ax-mulf 11209

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7671 df-om 7862 df-1st 7988 df-2nd 7989 df-supp 8160 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8719 df-map 8842 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9374 df-sup 9454 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-z 12589 df-dec 12709 df-uz 12853 df-rp 13009 df-fz 13525 df-seq 14020 df-exp 14080 df-cj 15118 df-re 15119 df-im 15120 df-sqrt 15254 df-abs 15255 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-starv 17286 df-sca 17287 df-vsca 17288 df-ip 17289 df-tset 17290 df-ple 17291 df-ds 17293 df-unif 17294 df-hom 17295 df-cco 17296 df-0g 17455 df-prds 17461 df-pws 17463 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-mhm 18761 df-grp 18919 df-minusg 18920 df-sbg 18921 df-subg 19106 df-ghm 19196 df-cmn 19763 df-abl 19764 df-mgp 20101 df-rng 20113 df-ur 20142 df-ring 20195 df-cring 20196 df-oppr 20297 df-dvdsr 20317 df-unit 20318 df-invr 20348 df-dvr 20361 df-rhm 20432 df-subrng 20506 df-subrg 20530 df-drng 20691 df-field 20692 df-staf 20799 df-srng 20800 df-lmod 20819 df-lss 20889 df-sra 21131 df-rgmod 21132 df-cnfld 21316 df-refld 21565 df-dsmm 21692 df-frlm 21707 df-tng 24523 df-tcph 25121 df-rrx 25337 df-line 48709

This theorem is referenced by: line2 48732

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