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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 6905 | . . . 4 ⊢ (𝑋 = 𝑌 → (𝑋‘1) = (𝑌‘1)) | |
| 2 | 1 | necon3i 2973 | . . 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 48661 | . . 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 11246 | . . . . . . . 8 ⊢ ℝ ∈ V | |
| 10 | prex 5437 | . . . . . . . . 9 ⊢ {1, 2} ∈ V | |
| 11 | 3, 10 | eqeltri 2837 | . . . . . . . 8 ⊢ 𝐼 ∈ V |
| 12 | 9, 11 | elmap 8911 | . . . . . . 7 ⊢ (𝑝 ∈ (ℝ ↑m 𝐼) ↔ 𝑝:𝐼⟶ℝ) |
| 13 | id 22 | . . . . . . . 8 ⊢ (𝑝:𝐼⟶ℝ → 𝑝:𝐼⟶ℝ) | |
| 14 | 1ex 11257 | . . . . . . . . . . 11 ⊢ 1 ∈ V | |
| 15 | 14 | prid1 4762 | . . . . . . . . . 10 ⊢ 1 ∈ {1, 2} |
| 16 | 15, 3 | eleqtrri 2840 | . . . . . . . . 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 2859 | . . . . 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 8911 | . . . . . . . 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 2859 | . . . . . 6 ⊢ (𝑋 ∈ 𝑃 → (𝑋‘1) ∈ ℝ) |
| 28 | 27 | 3ad2ant1 1134 | . . . . 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 8911 | . . . . . . . 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 2859 | . . . . . 6 ⊢ (𝑌 ∈ 𝑃 → (𝑌‘1) ∈ ℝ) |
| 36 | 35 | 3ad2ant2 1135 | . . . . 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 12343 | . . . . . . . . . . 11 ⊢ 2 ∈ V | |
| 40 | 39 | prid2 4763 | . . . . . . . . . 10 ⊢ 2 ∈ {1, 2} |
| 41 | 40, 3 | eleqtrri 2840 | . . . . . . . . 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 2859 | . . . . 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 2859 | . . . . . 6 ⊢ (𝑋 ∈ 𝑃 → (𝑋‘2) ∈ ℝ) |
| 51 | 50 | 3ad2ant1 1134 | . . . . 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 2833 | . . . . . . . 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 1135 | . . . . 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 48623 | . . 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 3443 | . 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 2777 | 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 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 {crab 3436 Vcvv 3480 {cpr 4628 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 ℝcr 11154 1c1 11156 + caddc 11158 · cmul 11160 − cmin 11492 / cdiv 11920 2c2 12321 ℝ^crrx 25417 LineMcline 48648 |
| 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 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-fz 13548 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-0g 17486 df-prds 17492 df-pws 17494 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-ghm 19231 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-rhm 20472 df-subrng 20546 df-subrg 20570 df-drng 20731 df-field 20732 df-staf 20840 df-srng 20841 df-lmod 20860 df-lss 20930 df-sra 21172 df-rgmod 21173 df-cnfld 21365 df-refld 21623 df-dsmm 21752 df-frlm 21767 df-tng 24597 df-tcph 25203 df-rrx 25419 df-line 48650 |
| This theorem is referenced by: line2 48673 |
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