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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 6669 | . . . 4 ⊢ (𝑋 = 𝑌 → (𝑋‘1) = (𝑌‘1)) | |
2 | 1 | necon3i 3048 | . . 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 44776 | . . 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 1161 | . 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 10628 | . . . . . . . 8 ⊢ ℝ ∈ V | |
10 | prex 5333 | . . . . . . . . 9 ⊢ {1, 2} ∈ V | |
11 | 3, 10 | eqeltri 2909 | . . . . . . . 8 ⊢ 𝐼 ∈ V |
12 | 9, 11 | elmap 8435 | . . . . . . 7 ⊢ (𝑝 ∈ (ℝ ↑m 𝐼) ↔ 𝑝:𝐼⟶ℝ) |
13 | id 22 | . . . . . . . 8 ⊢ (𝑝:𝐼⟶ℝ → 𝑝:𝐼⟶ℝ) | |
14 | 1ex 10637 | . . . . . . . . . . 11 ⊢ 1 ∈ V | |
15 | 14 | prid1 4698 | . . . . . . . . . 10 ⊢ 1 ∈ {1, 2} |
16 | 15, 3 | eleqtrri 2912 | . . . . . . . . 9 ⊢ 1 ∈ 𝐼 |
17 | 16 | a1i 11 | . . . . . . . 8 ⊢ (𝑝:𝐼⟶ℝ → 1 ∈ 𝐼) |
18 | 13, 17 | ffvelrnd 6852 | . . . . . . 7 ⊢ (𝑝:𝐼⟶ℝ → (𝑝‘1) ∈ ℝ) |
19 | 12, 18 | sylbi 219 | . . . . . 6 ⊢ (𝑝 ∈ (ℝ ↑m 𝐼) → (𝑝‘1) ∈ ℝ) |
20 | 19, 5 | eleq2s 2931 | . . . . 5 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘1) ∈ ℝ) |
21 | 20 | adantl 484 | . . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) ∧ 𝑝 ∈ 𝑃) → (𝑝‘1) ∈ ℝ) |
22 | 9, 11 | elmap 8435 | . . . . . . . 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 6852 | . . . . . . . 8 ⊢ (𝑋:𝐼⟶ℝ → (𝑋‘1) ∈ ℝ) |
26 | 22, 25 | sylbi 219 | . . . . . . 7 ⊢ (𝑋 ∈ (ℝ ↑m 𝐼) → (𝑋‘1) ∈ ℝ) |
27 | 26, 5 | eleq2s 2931 | . . . . . 6 ⊢ (𝑋 ∈ 𝑃 → (𝑋‘1) ∈ ℝ) |
28 | 27 | 3ad2ant1 1129 | . . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) → (𝑋‘1) ∈ ℝ) |
29 | 28 | adantr 483 | . . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) ∧ 𝑝 ∈ 𝑃) → (𝑋‘1) ∈ ℝ) |
30 | 9, 11 | elmap 8435 | . . . . . . . 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 6852 | . . . . . . . 8 ⊢ (𝑌:𝐼⟶ℝ → (𝑌‘1) ∈ ℝ) |
34 | 30, 33 | sylbi 219 | . . . . . . 7 ⊢ (𝑌 ∈ (ℝ ↑m 𝐼) → (𝑌‘1) ∈ ℝ) |
35 | 34, 5 | eleq2s 2931 | . . . . . 6 ⊢ (𝑌 ∈ 𝑃 → (𝑌‘1) ∈ ℝ) |
36 | 35 | 3ad2ant2 1130 | . . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) → (𝑌‘1) ∈ ℝ) |
37 | 36 | adantr 483 | . . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) ∧ 𝑝 ∈ 𝑃) → (𝑌‘1) ∈ ℝ) |
38 | simpl3 1189 | . . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) ∧ 𝑝 ∈ 𝑃) → (𝑋‘1) ≠ (𝑌‘1)) | |
39 | 2ex 11715 | . . . . . . . . . . 11 ⊢ 2 ∈ V | |
40 | 39 | prid2 4699 | . . . . . . . . . 10 ⊢ 2 ∈ {1, 2} |
41 | 40, 3 | eleqtrri 2912 | . . . . . . . . 9 ⊢ 2 ∈ 𝐼 |
42 | 41 | a1i 11 | . . . . . . . 8 ⊢ (𝑝:𝐼⟶ℝ → 2 ∈ 𝐼) |
43 | 13, 42 | ffvelrnd 6852 | . . . . . . 7 ⊢ (𝑝:𝐼⟶ℝ → (𝑝‘2) ∈ ℝ) |
44 | 12, 43 | sylbi 219 | . . . . . 6 ⊢ (𝑝 ∈ (ℝ ↑m 𝐼) → (𝑝‘2) ∈ ℝ) |
45 | 44, 5 | eleq2s 2931 | . . . . 5 ⊢ (𝑝 ∈ 𝑃 → (𝑝‘2) ∈ ℝ) |
46 | 45 | adantl 484 | . . . 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 6852 | . . . . . . . 8 ⊢ (𝑋:𝐼⟶ℝ → (𝑋‘2) ∈ ℝ) |
49 | 22, 48 | sylbi 219 | . . . . . . 7 ⊢ (𝑋 ∈ (ℝ ↑m 𝐼) → (𝑋‘2) ∈ ℝ) |
50 | 49, 5 | eleq2s 2931 | . . . . . 6 ⊢ (𝑋 ∈ 𝑃 → (𝑋‘2) ∈ ℝ) |
51 | 50 | 3ad2ant1 1129 | . . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) → (𝑋‘2) ∈ ℝ) |
52 | 51 | adantr 483 | . . . 4 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) ∧ 𝑝 ∈ 𝑃) → (𝑋‘2) ∈ ℝ) |
53 | 5 | eleq2i 2904 | . . . . . . . 8 ⊢ (𝑌 ∈ 𝑃 ↔ 𝑌 ∈ (ℝ ↑m 𝐼)) |
54 | 53, 30 | bitri 277 | . . . . . . 7 ⊢ (𝑌 ∈ 𝑃 ↔ 𝑌:𝐼⟶ℝ) |
55 | 41 | a1i 11 | . . . . . . . 8 ⊢ (𝑌:𝐼⟶ℝ → 2 ∈ 𝐼) |
56 | 31, 55 | ffvelrnd 6852 | . . . . . . 7 ⊢ (𝑌:𝐼⟶ℝ → (𝑌‘2) ∈ ℝ) |
57 | 54, 56 | sylbi 219 | . . . . . 6 ⊢ (𝑌 ∈ 𝑃 → (𝑌‘2) ∈ ℝ) |
58 | 57 | 3ad2ant2 1130 | . . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ (𝑋‘1) ≠ (𝑌‘1)) → (𝑌‘2) ∈ ℝ) |
59 | 58 | adantr 483 | . . . 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 44738 | . . 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 3478 | . 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 2856 | 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 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∃wrex 3139 {crab 3142 Vcvv 3494 {cpr 4569 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ↑m cmap 8406 ℝcr 10536 1c1 10538 + caddc 10540 · cmul 10542 − cmin 10870 / cdiv 11297 2c2 11693 ℝ^crrx 23986 LineMcline 44763 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-rp 12391 df-fz 12894 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-0g 16715 df-prds 16721 df-pws 16723 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-ghm 18356 df-cmn 18908 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-dvr 19433 df-rnghom 19467 df-drng 19504 df-field 19505 df-subrg 19533 df-staf 19616 df-srng 19617 df-lmod 19636 df-lss 19704 df-sra 19944 df-rgmod 19945 df-cnfld 20546 df-refld 20749 df-dsmm 20876 df-frlm 20891 df-tng 23194 df-tcph 23773 df-rrx 23988 df-line 44765 |
This theorem is referenced by: line2 44788 |
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