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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rrx2linesl | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
rrx2line.i | ⊢ 𝐼 = {1, 2} |
rrx2line.e | ⊢ 𝐸 = (ℝ^‘𝐼) |
rrx2line.b | ⊢ 𝑃 = (ℝ ↑𝑚 𝐼) |
rrx2line.l | ⊢ 𝐿 = (LineM‘𝐸) |
rrx2linesl.s | ⊢ 𝑆 = (((𝑌‘2) − (𝑋‘2)) / ((𝑌‘1) − (𝑋‘1))) |
Ref | Expression |
---|---|
rrx2linesl | ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) → (𝑋𝐿𝑌) = {𝑝 ∈ 𝑃 ∣ (𝑝‘2) = ((𝑆 · ((𝑝‘1) − (𝑋‘1))) + (𝑋‘2))}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq1 6447 | . . . 4 ⊢ (𝑋 = 𝑌 → (𝑋‘1) = (𝑌‘1)) | |
2 | 1 | necon3i 3001 | . . 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 = (LineM‘𝐸) | |
7 | 3, 4, 5, 6 | rrx2line 43490 | . . 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 1166 | . 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 10365 | . . . . . . . 8 ⊢ ℝ ∈ V | |
10 | prex 5143 | . . . . . . . . 9 ⊢ {1, 2} ∈ V | |
11 | 3, 10 | eqeltri 2855 | . . . . . . . 8 ⊢ 𝐼 ∈ V |
12 | 9, 11 | elmap 8171 | . . . . . . 7 ⊢ (𝑝 ∈ (ℝ ↑𝑚 𝐼) ↔ 𝑝:𝐼⟶ℝ) |
13 | id 22 | . . . . . . . 8 ⊢ (𝑝:𝐼⟶ℝ → 𝑝:𝐼⟶ℝ) | |
14 | 1ex 10374 | . . . . . . . . . . 11 ⊢ 1 ∈ V | |
15 | 14 | prid1 4529 | . . . . . . . . . 10 ⊢ 1 ∈ {1, 2} |
16 | 15, 3 | eleqtrri 2858 | . . . . . . . . 9 ⊢ 1 ∈ 𝐼 |
17 | 16 | a1i 11 | . . . . . . . 8 ⊢ (𝑝:𝐼⟶ℝ → 1 ∈ 𝐼) |
18 | 13, 17 | ffvelrnd 6626 | . . . . . . 7 ⊢ (𝑝:𝐼⟶ℝ → (𝑝‘1) ∈ ℝ) |
19 | 12, 18 | sylbi 209 | . . . . . 6 ⊢ (𝑝 ∈ (ℝ ↑𝑚 𝐼) → (𝑝‘1) ∈ ℝ) |
20 | 19, 5 | eleq2s 2877 | . . . . 5 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘1) ∈ ℝ) |
21 | 20 | adantl 475 | . . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) ∧ 𝑝 ∈ 𝑃) → (𝑝‘1) ∈ ℝ) |
22 | 9, 11 | elmap 8171 | . . . . . . . 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 | ffvelrnd 6626 | . . . . . . . 8 ⊢ (𝑋:𝐼⟶ℝ → (𝑋‘1) ∈ ℝ) |
26 | 22, 25 | sylbi 209 | . . . . . . 7 ⊢ (𝑋 ∈ (ℝ ↑𝑚 𝐼) → (𝑋‘1) ∈ ℝ) |
27 | 26, 5 | eleq2s 2877 | . . . . . 6 ⊢ (𝑋 ∈ 𝑃 → (𝑋‘1) ∈ ℝ) |
28 | 27 | 3ad2ant1 1124 | . . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) → (𝑋‘1) ∈ ℝ) |
29 | 28 | adantr 474 | . . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) ∧ 𝑝 ∈ 𝑃) → (𝑋‘1) ∈ ℝ) |
30 | 9, 11 | elmap 8171 | . . . . . . . 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 | ffvelrnd 6626 | . . . . . . . 8 ⊢ (𝑌:𝐼⟶ℝ → (𝑌‘1) ∈ ℝ) |
34 | 30, 33 | sylbi 209 | . . . . . . 7 ⊢ (𝑌 ∈ (ℝ ↑𝑚 𝐼) → (𝑌‘1) ∈ ℝ) |
35 | 34, 5 | eleq2s 2877 | . . . . . 6 ⊢ (𝑌 ∈ 𝑃 → (𝑌‘1) ∈ ℝ) |
36 | 35 | 3ad2ant2 1125 | . . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) → (𝑌‘1) ∈ ℝ) |
37 | 36 | adantr 474 | . . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) ∧ 𝑝 ∈ 𝑃) → (𝑌‘1) ∈ ℝ) |
38 | simpl3 1203 | . . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) ∧ 𝑝 ∈ 𝑃) → (𝑋‘1) ≠ (𝑌‘1)) | |
39 | 2ex 11457 | . . . . . . . . . . 11 ⊢ 2 ∈ V | |
40 | 39 | prid2 4530 | . . . . . . . . . 10 ⊢ 2 ∈ {1, 2} |
41 | 40, 3 | eleqtrri 2858 | . . . . . . . . 9 ⊢ 2 ∈ 𝐼 |
42 | 41 | a1i 11 | . . . . . . . 8 ⊢ (𝑝:𝐼⟶ℝ → 2 ∈ 𝐼) |
43 | 13, 42 | ffvelrnd 6626 | . . . . . . 7 ⊢ (𝑝:𝐼⟶ℝ → (𝑝‘2) ∈ ℝ) |
44 | 12, 43 | sylbi 209 | . . . . . 6 ⊢ (𝑝 ∈ (ℝ ↑𝑚 𝐼) → (𝑝‘2) ∈ ℝ) |
45 | 44, 5 | eleq2s 2877 | . . . . 5 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘2) ∈ ℝ) |
46 | 45 | adantl 475 | . . . 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 | ffvelrnd 6626 | . . . . . . . 8 ⊢ (𝑋:𝐼⟶ℝ → (𝑋‘2) ∈ ℝ) |
49 | 22, 48 | sylbi 209 | . . . . . . 7 ⊢ (𝑋 ∈ (ℝ ↑𝑚 𝐼) → (𝑋‘2) ∈ ℝ) |
50 | 49, 5 | eleq2s 2877 | . . . . . 6 ⊢ (𝑋 ∈ 𝑃 → (𝑋‘2) ∈ ℝ) |
51 | 50 | 3ad2ant1 1124 | . . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) → (𝑋‘2) ∈ ℝ) |
52 | 51 | adantr 474 | . . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) ∧ 𝑝 ∈ 𝑃) → (𝑋‘2) ∈ ℝ) |
53 | 5 | eleq2i 2851 | . . . . . . . 8 ⊢ (𝑌 ∈ 𝑃 ↔ 𝑌 ∈ (ℝ ↑𝑚 𝐼)) |
54 | 53, 30 | bitri 267 | . . . . . . 7 ⊢ (𝑌 ∈ 𝑃 ↔ 𝑌:𝐼⟶ℝ) |
55 | 41 | a1i 11 | . . . . . . . 8 ⊢ (𝑌:𝐼⟶ℝ → 2 ∈ 𝐼) |
56 | 31, 55 | ffvelrnd 6626 | . . . . . . 7 ⊢ (𝑌:𝐼⟶ℝ → (𝑌‘2) ∈ ℝ) |
57 | 54, 56 | sylbi 209 | . . . . . 6 ⊢ (𝑌 ∈ 𝑃 → (𝑌‘2) ∈ ℝ) |
58 | 57 | 3ad2ant2 1125 | . . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) → (𝑌‘2) ∈ ℝ) |
59 | 58 | adantr 474 | . . . 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 43452 | . . 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 3385 | . 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 2814 | 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 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ∃wrex 3091 {crab 3094 Vcvv 3398 {cpr 4400 ⟶wf 6133 ‘cfv 6137 (class class class)co 6924 ↑𝑚 cmap 8142 ℝcr 10273 1c1 10275 + caddc 10277 · cmul 10279 − cmin 10608 / cdiv 11035 2c2 11435 ℝ^crrx 23600 LineMcline 43477 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 ax-addf 10353 ax-mulf 10354 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-of 7176 df-om 7346 df-1st 7447 df-2nd 7448 df-supp 7579 df-tpos 7636 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-map 8144 df-ixp 8197 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fsupp 8566 df-sup 8638 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11036 df-nn 11380 df-2 11443 df-3 11444 df-4 11445 df-5 11446 df-6 11447 df-7 11448 df-8 11449 df-9 11450 df-n0 11648 df-z 11734 df-dec 11851 df-uz 11998 df-rp 12143 df-fz 12649 df-seq 13125 df-exp 13184 df-cj 14252 df-re 14253 df-im 14254 df-sqrt 14388 df-abs 14389 df-struct 16268 df-ndx 16269 df-slot 16270 df-base 16272 df-sets 16273 df-ress 16274 df-plusg 16362 df-mulr 16363 df-starv 16364 df-sca 16365 df-vsca 16366 df-ip 16367 df-tset 16368 df-ple 16369 df-ds 16371 df-unif 16372 df-hom 16373 df-cco 16374 df-0g 16499 df-prds 16505 df-pws 16507 df-mgm 17639 df-sgrp 17681 df-mnd 17692 df-mhm 17732 df-grp 17823 df-minusg 17824 df-sbg 17825 df-subg 17986 df-ghm 18053 df-cmn 18592 df-mgp 18888 df-ur 18900 df-ring 18947 df-cring 18948 df-oppr 19021 df-dvdsr 19039 df-unit 19040 df-invr 19070 df-dvr 19081 df-rnghom 19115 df-drng 19152 df-field 19153 df-subrg 19181 df-staf 19248 df-srng 19249 df-lmod 19268 df-lss 19336 df-sra 19580 df-rgmod 19581 df-cnfld 20154 df-refld 20359 df-dsmm 20486 df-frlm 20501 df-tng 22808 df-tcph 23387 df-rrx 23602 df-line 43479 |
This theorem is referenced by: line2 43502 |
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