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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 6906 | . . . 4 ⊢ (𝑋 = 𝑌 → (𝑋‘1) = (𝑌‘1)) | |
2 | 1 | necon3i 2971 | . . 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 48590 | . . 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 1164 | . 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 11244 | . . . . . . . 8 ⊢ ℝ ∈ V | |
10 | prex 5443 | . . . . . . . . 9 ⊢ {1, 2} ∈ V | |
11 | 3, 10 | eqeltri 2835 | . . . . . . . 8 ⊢ 𝐼 ∈ V |
12 | 9, 11 | elmap 8910 | . . . . . . 7 ⊢ (𝑝 ∈ (ℝ ↑m 𝐼) ↔ 𝑝:𝐼⟶ℝ) |
13 | id 22 | . . . . . . . 8 ⊢ (𝑝:𝐼⟶ℝ → 𝑝:𝐼⟶ℝ) | |
14 | 1ex 11255 | . . . . . . . . . . 11 ⊢ 1 ∈ V | |
15 | 14 | prid1 4767 | . . . . . . . . . 10 ⊢ 1 ∈ {1, 2} |
16 | 15, 3 | eleqtrri 2838 | . . . . . . . . 9 ⊢ 1 ∈ 𝐼 |
17 | 16 | a1i 11 | . . . . . . . 8 ⊢ (𝑝:𝐼⟶ℝ → 1 ∈ 𝐼) |
18 | 13, 17 | ffvelcdmd 7105 | . . . . . . 7 ⊢ (𝑝:𝐼⟶ℝ → (𝑝‘1) ∈ ℝ) |
19 | 12, 18 | sylbi 217 | . . . . . 6 ⊢ (𝑝 ∈ (ℝ ↑m 𝐼) → (𝑝‘1) ∈ ℝ) |
20 | 19, 5 | eleq2s 2857 | . . . . 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 8910 | . . . . . . . 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 7105 | . . . . . . . 8 ⊢ (𝑋:𝐼⟶ℝ → (𝑋‘1) ∈ ℝ) |
26 | 22, 25 | sylbi 217 | . . . . . . 7 ⊢ (𝑋 ∈ (ℝ ↑m 𝐼) → (𝑋‘1) ∈ ℝ) |
27 | 26, 5 | eleq2s 2857 | . . . . . 6 ⊢ (𝑋 ∈ 𝑃 → (𝑋‘1) ∈ ℝ) |
28 | 27 | 3ad2ant1 1132 | . . . . 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 8910 | . . . . . . . 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 7105 | . . . . . . . 8 ⊢ (𝑌:𝐼⟶ℝ → (𝑌‘1) ∈ ℝ) |
34 | 30, 33 | sylbi 217 | . . . . . . 7 ⊢ (𝑌 ∈ (ℝ ↑m 𝐼) → (𝑌‘1) ∈ ℝ) |
35 | 34, 5 | eleq2s 2857 | . . . . . 6 ⊢ (𝑌 ∈ 𝑃 → (𝑌‘1) ∈ ℝ) |
36 | 35 | 3ad2ant2 1133 | . . . . 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 1192 | . . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) ∧ 𝑝 ∈ 𝑃) → (𝑋‘1) ≠ (𝑌‘1)) | |
39 | 2ex 12341 | . . . . . . . . . . 11 ⊢ 2 ∈ V | |
40 | 39 | prid2 4768 | . . . . . . . . . 10 ⊢ 2 ∈ {1, 2} |
41 | 40, 3 | eleqtrri 2838 | . . . . . . . . 9 ⊢ 2 ∈ 𝐼 |
42 | 41 | a1i 11 | . . . . . . . 8 ⊢ (𝑝:𝐼⟶ℝ → 2 ∈ 𝐼) |
43 | 13, 42 | ffvelcdmd 7105 | . . . . . . 7 ⊢ (𝑝:𝐼⟶ℝ → (𝑝‘2) ∈ ℝ) |
44 | 12, 43 | sylbi 217 | . . . . . 6 ⊢ (𝑝 ∈ (ℝ ↑m 𝐼) → (𝑝‘2) ∈ ℝ) |
45 | 44, 5 | eleq2s 2857 | . . . . 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 7105 | . . . . . . . 8 ⊢ (𝑋:𝐼⟶ℝ → (𝑋‘2) ∈ ℝ) |
49 | 22, 48 | sylbi 217 | . . . . . . 7 ⊢ (𝑋 ∈ (ℝ ↑m 𝐼) → (𝑋‘2) ∈ ℝ) |
50 | 49, 5 | eleq2s 2857 | . . . . . 6 ⊢ (𝑋 ∈ 𝑃 → (𝑋‘2) ∈ ℝ) |
51 | 50 | 3ad2ant1 1132 | . . . . 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 2831 | . . . . . . . 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 7105 | . . . . . . 7 ⊢ (𝑌:𝐼⟶ℝ → (𝑌‘2) ∈ ℝ) |
57 | 54, 56 | sylbi 217 | . . . . . 6 ⊢ (𝑌 ∈ 𝑃 → (𝑌‘2) ∈ ℝ) |
58 | 57 | 3ad2ant2 1133 | . . . . 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 48552 | . . 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 3440 | . 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 2775 | 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 1537 ∈ wcel 2106 ≠ wne 2938 ∃wrex 3068 {crab 3433 Vcvv 3478 {cpr 4633 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 ℝcr 11152 1c1 11154 + caddc 11156 · cmul 11158 − cmin 11490 / cdiv 11918 2c2 12319 ℝ^crrx 25431 LineMcline 48577 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-rp 13033 df-fz 13545 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-0g 17488 df-prds 17494 df-pws 17496 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-ghm 19244 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-rhm 20489 df-subrng 20563 df-subrg 20587 df-drng 20748 df-field 20749 df-staf 20857 df-srng 20858 df-lmod 20877 df-lss 20948 df-sra 21190 df-rgmod 21191 df-cnfld 21383 df-refld 21641 df-dsmm 21770 df-frlm 21785 df-tng 24613 df-tcph 25217 df-rrx 25433 df-line 48579 |
This theorem is referenced by: line2 48602 |
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