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Mirrors > Home > MPE Home > Th. List > Mathboxes > rrxlinec | Structured version Visualization version GIF version |
Description: The line passing through the two different points 𝑋 and 𝑌 in a generalized real Euclidean space of finite dimension, expressed by its coordinates. Remark: This proof is shorter and requires less distinct variables than the proof using rrxlinesc 43303. (Contributed by AV, 13-Feb-2023.) |
Ref | Expression |
---|---|
rrxlinesc.e | ⊢ 𝐸 = (ℝ^‘𝐼) |
rrxlinesc.p | ⊢ 𝑃 = (ℝ ↑𝑚 𝐼) |
rrxlinesc.l | ⊢ 𝐿 = (LineM‘𝐸) |
Ref | Expression |
---|---|
rrxlinec | ⊢ ((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) → (𝑋𝐿𝑌) = {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ ∀𝑖 ∈ 𝐼 (𝑝‘𝑖) = (((1 − 𝑡) · (𝑋‘𝑖)) + (𝑡 · (𝑌‘𝑖)))}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrxlinesc.e | . . 3 ⊢ 𝐸 = (ℝ^‘𝐼) | |
2 | rrxlinesc.p | . . 3 ⊢ 𝑃 = (ℝ ↑𝑚 𝐼) | |
3 | rrxlinesc.l | . . 3 ⊢ 𝐿 = (LineM‘𝐸) | |
4 | eqid 2825 | . . 3 ⊢ ( ·𝑠 ‘𝐸) = ( ·𝑠 ‘𝐸) | |
5 | eqid 2825 | . . 3 ⊢ (+g‘𝐸) = (+g‘𝐸) | |
6 | 1, 2, 3, 4, 5 | rrxline 43302 | . 2 ⊢ ((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) → (𝑋𝐿𝑌) = {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡)( ·𝑠 ‘𝐸)𝑋)(+g‘𝐸)(𝑡( ·𝑠 ‘𝐸)𝑌))}) |
7 | eqid 2825 | . . . . 5 ⊢ (Base‘𝐸) = (Base‘𝐸) | |
8 | simplll 791 | . . . . 5 ⊢ ((((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) ∧ 𝑝 ∈ 𝑃) ∧ 𝑡 ∈ ℝ) → 𝐼 ∈ Fin) | |
9 | 1red 10364 | . . . . . 6 ⊢ ((((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) ∧ 𝑝 ∈ 𝑃) ∧ 𝑡 ∈ ℝ) → 1 ∈ ℝ) | |
10 | simpr 479 | . . . . . 6 ⊢ ((((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) ∧ 𝑝 ∈ 𝑃) ∧ 𝑡 ∈ ℝ) → 𝑡 ∈ ℝ) | |
11 | 9, 10 | resubcld 10789 | . . . . 5 ⊢ ((((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) ∧ 𝑝 ∈ 𝑃) ∧ 𝑡 ∈ ℝ) → (1 − 𝑡) ∈ ℝ) |
12 | id 22 | . . . . . . . . . . . 12 ⊢ (𝐼 ∈ Fin → 𝐼 ∈ Fin) | |
13 | 12, 1, 7 | rrxbasefi 23585 | . . . . . . . . . . 11 ⊢ (𝐼 ∈ Fin → (Base‘𝐸) = (ℝ ↑𝑚 𝐼)) |
14 | 13, 2 | syl6reqr 2880 | . . . . . . . . . 10 ⊢ (𝐼 ∈ Fin → 𝑃 = (Base‘𝐸)) |
15 | 14 | eleq2d 2892 | . . . . . . . . 9 ⊢ (𝐼 ∈ Fin → (𝑋 ∈ 𝑃 ↔ 𝑋 ∈ (Base‘𝐸))) |
16 | 15 | biimpcd 241 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝑃 → (𝐼 ∈ Fin → 𝑋 ∈ (Base‘𝐸))) |
17 | 16 | 3ad2ant1 1167 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (𝐼 ∈ Fin → 𝑋 ∈ (Base‘𝐸))) |
18 | 17 | impcom 398 | . . . . . 6 ⊢ ((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) → 𝑋 ∈ (Base‘𝐸)) |
19 | 18 | ad2antrr 717 | . . . . 5 ⊢ ((((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) ∧ 𝑝 ∈ 𝑃) ∧ 𝑡 ∈ ℝ) → 𝑋 ∈ (Base‘𝐸)) |
20 | 14 | eleq2d 2892 | . . . . . . . . 9 ⊢ (𝐼 ∈ Fin → (𝑌 ∈ 𝑃 ↔ 𝑌 ∈ (Base‘𝐸))) |
21 | 20 | biimpcd 241 | . . . . . . . 8 ⊢ (𝑌 ∈ 𝑃 → (𝐼 ∈ Fin → 𝑌 ∈ (Base‘𝐸))) |
22 | 21 | 3ad2ant2 1168 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (𝐼 ∈ Fin → 𝑌 ∈ (Base‘𝐸))) |
23 | 22 | impcom 398 | . . . . . 6 ⊢ ((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) → 𝑌 ∈ (Base‘𝐸)) |
24 | 23 | ad2antrr 717 | . . . . 5 ⊢ ((((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) ∧ 𝑝 ∈ 𝑃) ∧ 𝑡 ∈ ℝ) → 𝑌 ∈ (Base‘𝐸)) |
25 | 14 | adantr 474 | . . . . . . . 8 ⊢ ((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) → 𝑃 = (Base‘𝐸)) |
26 | 25 | eleq2d 2892 | . . . . . . 7 ⊢ ((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) → (𝑝 ∈ 𝑃 ↔ 𝑝 ∈ (Base‘𝐸))) |
27 | 26 | biimpa 470 | . . . . . 6 ⊢ (((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) ∧ 𝑝 ∈ 𝑃) → 𝑝 ∈ (Base‘𝐸)) |
28 | 27 | adantr 474 | . . . . 5 ⊢ ((((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) ∧ 𝑝 ∈ 𝑃) ∧ 𝑡 ∈ ℝ) → 𝑝 ∈ (Base‘𝐸)) |
29 | 1, 7, 4, 8, 11, 19, 24, 28, 5, 10 | rrxplusgvscavalb 23570 | . . . 4 ⊢ ((((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) ∧ 𝑝 ∈ 𝑃) ∧ 𝑡 ∈ ℝ) → (𝑝 = (((1 − 𝑡)( ·𝑠 ‘𝐸)𝑋)(+g‘𝐸)(𝑡( ·𝑠 ‘𝐸)𝑌)) ↔ ∀𝑖 ∈ 𝐼 (𝑝‘𝑖) = (((1 − 𝑡) · (𝑋‘𝑖)) + (𝑡 · (𝑌‘𝑖))))) |
30 | 29 | rexbidva 3259 | . . 3 ⊢ (((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) ∧ 𝑝 ∈ 𝑃) → (∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡)( ·𝑠 ‘𝐸)𝑋)(+g‘𝐸)(𝑡( ·𝑠 ‘𝐸)𝑌)) ↔ ∃𝑡 ∈ ℝ ∀𝑖 ∈ 𝐼 (𝑝‘𝑖) = (((1 − 𝑡) · (𝑋‘𝑖)) + (𝑡 · (𝑌‘𝑖))))) |
31 | 30 | rabbidva 3401 | . 2 ⊢ ((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) → {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡)( ·𝑠 ‘𝐸)𝑋)(+g‘𝐸)(𝑡( ·𝑠 ‘𝐸)𝑌))} = {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ ∀𝑖 ∈ 𝐼 (𝑝‘𝑖) = (((1 − 𝑡) · (𝑋‘𝑖)) + (𝑡 · (𝑌‘𝑖)))}) |
32 | 6, 31 | eqtrd 2861 | 1 ⊢ ((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) → (𝑋𝐿𝑌) = {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ ∀𝑖 ∈ 𝐼 (𝑝‘𝑖) = (((1 − 𝑡) · (𝑋‘𝑖)) + (𝑡 · (𝑌‘𝑖)))}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 ≠ wne 2999 ∀wral 3117 ∃wrex 3118 {crab 3121 ‘cfv 6127 (class class class)co 6910 ↑𝑚 cmap 8127 Fincfn 8228 ℝcr 10258 1c1 10260 + caddc 10262 · cmul 10264 − cmin 10592 Basecbs 16229 +gcplusg 16312 ·𝑠 cvsca 16316 ℝ^crrx 23558 LineMcline 43295 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-pre-sup 10337 ax-addf 10338 ax-mulf 10339 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-of 7162 df-om 7332 df-1st 7433 df-2nd 7434 df-supp 7565 df-tpos 7622 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-map 8129 df-ixp 8182 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-fsupp 8551 df-sup 8623 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-dec 11829 df-uz 11976 df-rp 12120 df-fz 12627 df-seq 13103 df-exp 13162 df-cj 14223 df-re 14224 df-im 14225 df-sqrt 14359 df-abs 14360 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-starv 16327 df-sca 16328 df-vsca 16329 df-ip 16330 df-tset 16331 df-ple 16332 df-ds 16334 df-unif 16335 df-hom 16336 df-cco 16337 df-0g 16462 df-prds 16468 df-pws 16470 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-mhm 17695 df-grp 17786 df-minusg 17787 df-sbg 17788 df-subg 17949 df-ghm 18016 df-cmn 18555 df-mgp 18851 df-ur 18863 df-ring 18910 df-cring 18911 df-oppr 18984 df-dvdsr 19002 df-unit 19003 df-invr 19033 df-dvr 19044 df-rnghom 19078 df-drng 19112 df-field 19113 df-subrg 19141 df-staf 19208 df-srng 19209 df-lmod 19228 df-lss 19296 df-sra 19540 df-rgmod 19541 df-cnfld 20114 df-refld 20319 df-dsmm 20446 df-frlm 20461 df-tng 22766 df-tcph 23345 df-rrx 23560 df-line 43297 |
This theorem is referenced by: rrx2line 43308 |
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