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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 6644 | . . . 4 ⊢ (𝑋 = 𝑌 → (𝑋‘1) = (𝑌‘1)) | |
2 | 1 | necon3i 3019 | . . 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 45154 | . . 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 10617 | . . . . . . . 8 ⊢ ℝ ∈ V | |
10 | prex 5298 | . . . . . . . . 9 ⊢ {1, 2} ∈ V | |
11 | 3, 10 | eqeltri 2886 | . . . . . . . 8 ⊢ 𝐼 ∈ V |
12 | 9, 11 | elmap 8418 | . . . . . . 7 ⊢ (𝑝 ∈ (ℝ ↑m 𝐼) ↔ 𝑝:𝐼⟶ℝ) |
13 | id 22 | . . . . . . . 8 ⊢ (𝑝:𝐼⟶ℝ → 𝑝:𝐼⟶ℝ) | |
14 | 1ex 10626 | . . . . . . . . . . 11 ⊢ 1 ∈ V | |
15 | 14 | prid1 4658 | . . . . . . . . . 10 ⊢ 1 ∈ {1, 2} |
16 | 15, 3 | eleqtrri 2889 | . . . . . . . . 9 ⊢ 1 ∈ 𝐼 |
17 | 16 | a1i 11 | . . . . . . . 8 ⊢ (𝑝:𝐼⟶ℝ → 1 ∈ 𝐼) |
18 | 13, 17 | ffvelrnd 6829 | . . . . . . 7 ⊢ (𝑝:𝐼⟶ℝ → (𝑝‘1) ∈ ℝ) |
19 | 12, 18 | sylbi 220 | . . . . . 6 ⊢ (𝑝 ∈ (ℝ ↑m 𝐼) → (𝑝‘1) ∈ ℝ) |
20 | 19, 5 | eleq2s 2908 | . . . . 5 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘1) ∈ ℝ) |
21 | 20 | adantl 485 | . . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) ∧ 𝑝 ∈ 𝑃) → (𝑝‘1) ∈ ℝ) |
22 | 9, 11 | elmap 8418 | . . . . . . . 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 6829 | . . . . . . . 8 ⊢ (𝑋:𝐼⟶ℝ → (𝑋‘1) ∈ ℝ) |
26 | 22, 25 | sylbi 220 | . . . . . . 7 ⊢ (𝑋 ∈ (ℝ ↑m 𝐼) → (𝑋‘1) ∈ ℝ) |
27 | 26, 5 | eleq2s 2908 | . . . . . 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 484 | . . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) ∧ 𝑝 ∈ 𝑃) → (𝑋‘1) ∈ ℝ) |
30 | 9, 11 | elmap 8418 | . . . . . . . 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 6829 | . . . . . . . 8 ⊢ (𝑌:𝐼⟶ℝ → (𝑌‘1) ∈ ℝ) |
34 | 30, 33 | sylbi 220 | . . . . . . 7 ⊢ (𝑌 ∈ (ℝ ↑m 𝐼) → (𝑌‘1) ∈ ℝ) |
35 | 34, 5 | eleq2s 2908 | . . . . . 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 484 | . . . 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 11702 | . . . . . . . . . . 11 ⊢ 2 ∈ V | |
40 | 39 | prid2 4659 | . . . . . . . . . 10 ⊢ 2 ∈ {1, 2} |
41 | 40, 3 | eleqtrri 2889 | . . . . . . . . 9 ⊢ 2 ∈ 𝐼 |
42 | 41 | a1i 11 | . . . . . . . 8 ⊢ (𝑝:𝐼⟶ℝ → 2 ∈ 𝐼) |
43 | 13, 42 | ffvelrnd 6829 | . . . . . . 7 ⊢ (𝑝:𝐼⟶ℝ → (𝑝‘2) ∈ ℝ) |
44 | 12, 43 | sylbi 220 | . . . . . 6 ⊢ (𝑝 ∈ (ℝ ↑m 𝐼) → (𝑝‘2) ∈ ℝ) |
45 | 44, 5 | eleq2s 2908 | . . . . 5 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘2) ∈ ℝ) |
46 | 45 | adantl 485 | . . . 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 6829 | . . . . . . . 8 ⊢ (𝑋:𝐼⟶ℝ → (𝑋‘2) ∈ ℝ) |
49 | 22, 48 | sylbi 220 | . . . . . . 7 ⊢ (𝑋 ∈ (ℝ ↑m 𝐼) → (𝑋‘2) ∈ ℝ) |
50 | 49, 5 | eleq2s 2908 | . . . . . 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 484 | . . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) ∧ 𝑝 ∈ 𝑃) → (𝑋‘2) ∈ ℝ) |
53 | 5 | eleq2i 2881 | . . . . . . . 8 ⊢ (𝑌 ∈ 𝑃 ↔ 𝑌 ∈ (ℝ ↑m 𝐼)) |
54 | 53, 30 | bitri 278 | . . . . . . 7 ⊢ (𝑌 ∈ 𝑃 ↔ 𝑌:𝐼⟶ℝ) |
55 | 41 | a1i 11 | . . . . . . . 8 ⊢ (𝑌:𝐼⟶ℝ → 2 ∈ 𝐼) |
56 | 31, 55 | ffvelrnd 6829 | . . . . . . 7 ⊢ (𝑌:𝐼⟶ℝ → (𝑌‘2) ∈ ℝ) |
57 | 54, 56 | sylbi 220 | . . . . . 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 484 | . . . 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 45116 | . . 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 3425 | . 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 2833 | 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 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∃wrex 3107 {crab 3110 Vcvv 3441 {cpr 4527 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ↑m cmap 8389 ℝcr 10525 1c1 10527 + caddc 10529 · cmul 10531 − cmin 10859 / cdiv 11286 2c2 11680 ℝ^crrx 23987 LineMcline 45141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-rp 12378 df-fz 12886 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-0g 16707 df-prds 16713 df-pws 16715 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-ghm 18348 df-cmn 18900 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-rnghom 19463 df-drng 19497 df-field 19498 df-subrg 19526 df-staf 19609 df-srng 19610 df-lmod 19629 df-lss 19697 df-sra 19937 df-rgmod 19938 df-cnfld 20092 df-refld 20294 df-dsmm 20421 df-frlm 20436 df-tng 23191 df-tcph 23774 df-rrx 23989 df-line 45143 |
This theorem is referenced by: line2 45166 |
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