rrx2linesl - Mathbox for Alexander van der Vekens

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Theorem rrx2linesl 48728

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 6821	. . . 4 ⊢ (𝑋 = 𝑌 → (𝑋‘1) = (𝑌‘1))
2	1	necon3i 2957	. . 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 48725	. . 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 11100	. . . . . . . 8 ⊢ ℝ ∈ V
10		prex 5376	. . . . . . . . 9 ⊢ {1, 2} ∈ V
11	3, 10	eqeltri 2824	. . . . . . . 8 ⊢ 𝐼 ∈ V
12	9, 11	elmap 8798	. . . . . . 7 ⊢ (𝑝 ∈ (ℝ ↑_m 𝐼) ↔ 𝑝:𝐼⟶ℝ)
13		id 22	. . . . . . . 8 ⊢ (𝑝:𝐼⟶ℝ → 𝑝:𝐼⟶ℝ)
14		1ex 11111	. . . . . . . . . . 11 ⊢ 1 ∈ V
15	14	prid1 4714	. . . . . . . . . 10 ⊢ 1 ∈ {1, 2}
16	15, 3	eleqtrri 2827	. . . . . . . . 9 ⊢ 1 ∈ 𝐼
17	16	a1i 11	. . . . . . . 8 ⊢ (𝑝:𝐼⟶ℝ → 1 ∈ 𝐼)
18	13, 17	ffvelcdmd 7019	. . . . . . 7 ⊢ (𝑝:𝐼⟶ℝ → (𝑝‘1) ∈ ℝ)
19	12, 18	sylbi 217	. . . . . 6 ⊢ (𝑝 ∈ (ℝ ↑_m 𝐼) → (𝑝‘1) ∈ ℝ)
20	19, 5	eleq2s 2846	. . . . 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 8798	. . . . . . . 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 7019	. . . . . . . 8 ⊢ (𝑋:𝐼⟶ℝ → (𝑋‘1) ∈ ℝ)
26	22, 25	sylbi 217	. . . . . . 7 ⊢ (𝑋 ∈ (ℝ ↑_m 𝐼) → (𝑋‘1) ∈ ℝ)
27	26, 5	eleq2s 2846	. . . . . 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 8798	. . . . . . . 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 7019	. . . . . . . 8 ⊢ (𝑌:𝐼⟶ℝ → (𝑌‘1) ∈ ℝ)
34	30, 33	sylbi 217	. . . . . . 7 ⊢ (𝑌 ∈ (ℝ ↑_m 𝐼) → (𝑌‘1) ∈ ℝ)
35	34, 5	eleq2s 2846	. . . . . 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 12205	. . . . . . . . . . 11 ⊢ 2 ∈ V
40	39	prid2 4715	. . . . . . . . . 10 ⊢ 2 ∈ {1, 2}
41	40, 3	eleqtrri 2827	. . . . . . . . 9 ⊢ 2 ∈ 𝐼
42	41	a1i 11	. . . . . . . 8 ⊢ (𝑝:𝐼⟶ℝ → 2 ∈ 𝐼)
43	13, 42	ffvelcdmd 7019	. . . . . . 7 ⊢ (𝑝:𝐼⟶ℝ → (𝑝‘2) ∈ ℝ)
44	12, 43	sylbi 217	. . . . . 6 ⊢ (𝑝 ∈ (ℝ ↑_m 𝐼) → (𝑝‘2) ∈ ℝ)
45	44, 5	eleq2s 2846	. . . . 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 7019	. . . . . . . 8 ⊢ (𝑋:𝐼⟶ℝ → (𝑋‘2) ∈ ℝ)
49	22, 48	sylbi 217	. . . . . . 7 ⊢ (𝑋 ∈ (ℝ ↑_m 𝐼) → (𝑋‘2) ∈ ℝ)
50	49, 5	eleq2s 2846	. . . . . 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 2820	. . . . . . . 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 7019	. . . . . . 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 48687	. . 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 3401	. 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 2764	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 2109 ≠ wne 2925 ∃wrex 3053 {crab 3394 Vcvv 3436 {cpr 4579 ⟶wf 6478 ‘cfv 6482 (class class class)co 7349 ↑_m cmap 8753 ℝcr 11008 1c1 11010 + caddc 11012 · cmul 11014 − cmin 11347 / cdiv 11777 2c2 12183 ℝ^crrx 25281 Line_Mcline 48712

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-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-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-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-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-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-fz 13411 df-seq 13909 df-exp 13969 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 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-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-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-cnfld 21262 df-refld 21512 df-dsmm 21639 df-frlm 21654 df-tng 24470 df-tcph 25067 df-rrx 25283 df-line 48714

This theorem is referenced by: line2 48737

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