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Mirrors > Home > MPE Home > Th. List > Mathboxes > rrx2line | 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. (Contributed by AV, 22-Jan-2023.) (Proof shortened by AV, 13-Feb-2023.) |
Ref | Expression |
---|---|
rrx2line.i | ⊢ 𝐼 = {1, 2} |
rrx2line.e | ⊢ 𝐸 = (ℝ^‘𝐼) |
rrx2line.b | ⊢ 𝑃 = (ℝ ↑m 𝐼) |
rrx2line.l | ⊢ 𝐿 = (LineM‘𝐸) |
Ref | Expression |
---|---|
rrx2line | ⊢ ((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))))}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrx2line.i | . . . 4 ⊢ 𝐼 = {1, 2} | |
2 | prfi 8781 | . . . 4 ⊢ {1, 2} ∈ Fin | |
3 | 1, 2 | eqeltri 2906 | . . 3 ⊢ 𝐼 ∈ Fin |
4 | rrx2line.e | . . . 4 ⊢ 𝐸 = (ℝ^‘𝐼) | |
5 | rrx2line.b | . . . 4 ⊢ 𝑃 = (ℝ ↑m 𝐼) | |
6 | rrx2line.l | . . . 4 ⊢ 𝐿 = (LineM‘𝐸) | |
7 | 4, 5, 6 | rrxlinec 44651 | . . 3 ⊢ ((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) → (𝑋𝐿𝑌) = {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ ∀𝑖 ∈ 𝐼 (𝑝‘𝑖) = (((1 − 𝑡) · (𝑋‘𝑖)) + (𝑡 · (𝑌‘𝑖)))}) |
8 | 3, 7 | mpan 686 | . 2 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (𝑋𝐿𝑌) = {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ ∀𝑖 ∈ 𝐼 (𝑝‘𝑖) = (((1 − 𝑡) · (𝑋‘𝑖)) + (𝑡 · (𝑌‘𝑖)))}) |
9 | 1 | a1i 11 | . . . . . 6 ⊢ ((((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) ∧ 𝑝 ∈ 𝑃) ∧ 𝑡 ∈ ℝ) → 𝐼 = {1, 2}) |
10 | 9 | raleqdv 3413 | . . . . 5 ⊢ ((((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) ∧ 𝑝 ∈ 𝑃) ∧ 𝑡 ∈ ℝ) → (∀𝑖 ∈ 𝐼 (𝑝‘𝑖) = (((1 − 𝑡) · (𝑋‘𝑖)) + (𝑡 · (𝑌‘𝑖))) ↔ ∀𝑖 ∈ {1, 2} (𝑝‘𝑖) = (((1 − 𝑡) · (𝑋‘𝑖)) + (𝑡 · (𝑌‘𝑖))))) |
11 | 1ex 10625 | . . . . . 6 ⊢ 1 ∈ V | |
12 | 2ex 11702 | . . . . . 6 ⊢ 2 ∈ V | |
13 | fveq2 6663 | . . . . . . 7 ⊢ (𝑖 = 1 → (𝑝‘𝑖) = (𝑝‘1)) | |
14 | fveq2 6663 | . . . . . . . . 9 ⊢ (𝑖 = 1 → (𝑋‘𝑖) = (𝑋‘1)) | |
15 | 14 | oveq2d 7161 | . . . . . . . 8 ⊢ (𝑖 = 1 → ((1 − 𝑡) · (𝑋‘𝑖)) = ((1 − 𝑡) · (𝑋‘1))) |
16 | fveq2 6663 | . . . . . . . . 9 ⊢ (𝑖 = 1 → (𝑌‘𝑖) = (𝑌‘1)) | |
17 | 16 | oveq2d 7161 | . . . . . . . 8 ⊢ (𝑖 = 1 → (𝑡 · (𝑌‘𝑖)) = (𝑡 · (𝑌‘1))) |
18 | 15, 17 | oveq12d 7163 | . . . . . . 7 ⊢ (𝑖 = 1 → (((1 − 𝑡) · (𝑋‘𝑖)) + (𝑡 · (𝑌‘𝑖))) = (((1 − 𝑡) · (𝑋‘1)) + (𝑡 · (𝑌‘1)))) |
19 | 13, 18 | eqeq12d 2834 | . . . . . 6 ⊢ (𝑖 = 1 → ((𝑝‘𝑖) = (((1 − 𝑡) · (𝑋‘𝑖)) + (𝑡 · (𝑌‘𝑖))) ↔ (𝑝‘1) = (((1 − 𝑡) · (𝑋‘1)) + (𝑡 · (𝑌‘1))))) |
20 | fveq2 6663 | . . . . . . 7 ⊢ (𝑖 = 2 → (𝑝‘𝑖) = (𝑝‘2)) | |
21 | fveq2 6663 | . . . . . . . . 9 ⊢ (𝑖 = 2 → (𝑋‘𝑖) = (𝑋‘2)) | |
22 | 21 | oveq2d 7161 | . . . . . . . 8 ⊢ (𝑖 = 2 → ((1 − 𝑡) · (𝑋‘𝑖)) = ((1 − 𝑡) · (𝑋‘2))) |
23 | fveq2 6663 | . . . . . . . . 9 ⊢ (𝑖 = 2 → (𝑌‘𝑖) = (𝑌‘2)) | |
24 | 23 | oveq2d 7161 | . . . . . . . 8 ⊢ (𝑖 = 2 → (𝑡 · (𝑌‘𝑖)) = (𝑡 · (𝑌‘2))) |
25 | 22, 24 | oveq12d 7163 | . . . . . . 7 ⊢ (𝑖 = 2 → (((1 − 𝑡) · (𝑋‘𝑖)) + (𝑡 · (𝑌‘𝑖))) = (((1 − 𝑡) · (𝑋‘2)) + (𝑡 · (𝑌‘2)))) |
26 | 20, 25 | eqeq12d 2834 | . . . . . 6 ⊢ (𝑖 = 2 → ((𝑝‘𝑖) = (((1 − 𝑡) · (𝑋‘𝑖)) + (𝑡 · (𝑌‘𝑖))) ↔ (𝑝‘2) = (((1 − 𝑡) · (𝑋‘2)) + (𝑡 · (𝑌‘2))))) |
27 | 11, 12, 19, 26 | ralpr 4628 | . . . . 5 ⊢ (∀𝑖 ∈ {1, 2} (𝑝‘𝑖) = (((1 − 𝑡) · (𝑋‘𝑖)) + (𝑡 · (𝑌‘𝑖))) ↔ ((𝑝‘1) = (((1 − 𝑡) · (𝑋‘1)) + (𝑡 · (𝑌‘1))) ∧ (𝑝‘2) = (((1 − 𝑡) · (𝑋‘2)) + (𝑡 · (𝑌‘2))))) |
28 | 10, 27 | syl6bb 288 | . . . 4 ⊢ ((((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) ∧ 𝑝 ∈ 𝑃) ∧ 𝑡 ∈ ℝ) → (∀𝑖 ∈ 𝐼 (𝑝‘𝑖) = (((1 − 𝑡) · (𝑋‘𝑖)) + (𝑡 · (𝑌‘𝑖))) ↔ ((𝑝‘1) = (((1 − 𝑡) · (𝑋‘1)) + (𝑡 · (𝑌‘1))) ∧ (𝑝‘2) = (((1 − 𝑡) · (𝑋‘2)) + (𝑡 · (𝑌‘2)))))) |
29 | 28 | rexbidva 3293 | . . 3 ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) ∧ 𝑝 ∈ 𝑃) → (∃𝑡 ∈ ℝ ∀𝑖 ∈ 𝐼 (𝑝‘𝑖) = (((1 − 𝑡) · (𝑋‘𝑖)) + (𝑡 · (𝑌‘𝑖))) ↔ ∃𝑡 ∈ ℝ ((𝑝‘1) = (((1 − 𝑡) · (𝑋‘1)) + (𝑡 · (𝑌‘1))) ∧ (𝑝‘2) = (((1 − 𝑡) · (𝑋‘2)) + (𝑡 · (𝑌‘2)))))) |
30 | 29 | rabbidva 3476 | . 2 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ ∀𝑖 ∈ 𝐼 (𝑝‘𝑖) = (((1 − 𝑡) · (𝑋‘𝑖)) + (𝑡 · (𝑌‘𝑖)))} = {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ ((𝑝‘1) = (((1 − 𝑡) · (𝑋‘1)) + (𝑡 · (𝑌‘1))) ∧ (𝑝‘2) = (((1 − 𝑡) · (𝑋‘2)) + (𝑡 · (𝑌‘2))))}) |
31 | 8, 30 | eqtrd 2853 | 1 ⊢ ((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))))}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 ∃wrex 3136 {crab 3139 {cpr 4559 ‘cfv 6348 (class class class)co 7145 ↑m cmap 8395 Fincfn 8497 ℝcr 10524 1c1 10526 + caddc 10528 · cmul 10530 − cmin 10858 2c2 11680 ℝ^crrx 23913 LineMcline 44642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 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 12881 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-0g 16703 df-prds 16709 df-pws 16711 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-grp 18044 df-minusg 18045 df-sbg 18046 df-subg 18214 df-ghm 18294 df-cmn 18837 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-dvr 19362 df-rnghom 19396 df-drng 19433 df-field 19434 df-subrg 19462 df-staf 19545 df-srng 19546 df-lmod 19565 df-lss 19633 df-sra 19873 df-rgmod 19874 df-cnfld 20474 df-refld 20677 df-dsmm 20804 df-frlm 20819 df-tng 23121 df-tcph 23700 df-rrx 23915 df-line 44644 |
This theorem is referenced by: rrx2vlinest 44656 rrx2linest 44657 rrx2linesl 44658 |
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