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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rrxlinesc | Structured version Visualization version GIF version |
Description: Definition of lines passing through two different points in a generalized real Euclidean space of finite dimension, expressed by their coordinates. (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 |
---|---|
rrxlinesc | ⊢ (𝐼 ∈ Fin → 𝐿 = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {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 2725 | . . 3 ⊢ ( ·𝑠 ‘𝐸) = ( ·𝑠 ‘𝐸) | |
5 | eqid 2725 | . . 3 ⊢ (+g‘𝐸) = (+g‘𝐸) | |
6 | 1, 2, 3, 4, 5 | rrxlines 47997 | . 2 ⊢ (𝐼 ∈ Fin → 𝐿 = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡)( ·𝑠 ‘𝐸)𝑥)(+g‘𝐸)(𝑡( ·𝑠 ‘𝐸)𝑦))})) |
7 | eqid 2725 | . . . . . 6 ⊢ (Base‘𝐸) = (Base‘𝐸) | |
8 | simpll1 1209 | . . . . . 6 ⊢ ((((𝐼 ∈ Fin ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) ∧ 𝑡 ∈ ℝ) → 𝐼 ∈ Fin) | |
9 | 1red 11252 | . . . . . . 7 ⊢ ((((𝐼 ∈ Fin ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) ∧ 𝑡 ∈ ℝ) → 1 ∈ ℝ) | |
10 | simpr 483 | . . . . . . 7 ⊢ ((((𝐼 ∈ Fin ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) ∧ 𝑡 ∈ ℝ) → 𝑡 ∈ ℝ) | |
11 | 9, 10 | resubcld 11679 | . . . . . 6 ⊢ ((((𝐼 ∈ Fin ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) ∧ 𝑡 ∈ ℝ) → (1 − 𝑡) ∈ ℝ) |
12 | id 22 | . . . . . . . . . . . 12 ⊢ (𝐼 ∈ Fin → 𝐼 ∈ Fin) | |
13 | 12, 1, 7 | rrxbasefi 25399 | . . . . . . . . . . 11 ⊢ (𝐼 ∈ Fin → (Base‘𝐸) = (ℝ ↑m 𝐼)) |
14 | 2, 13 | eqtr4id 2784 | . . . . . . . . . 10 ⊢ (𝐼 ∈ Fin → 𝑃 = (Base‘𝐸)) |
15 | 14 | eleq2d 2811 | . . . . . . . . 9 ⊢ (𝐼 ∈ Fin → (𝑥 ∈ 𝑃 ↔ 𝑥 ∈ (Base‘𝐸))) |
16 | 15 | biimpa 475 | . . . . . . . 8 ⊢ ((𝐼 ∈ Fin ∧ 𝑥 ∈ 𝑃) → 𝑥 ∈ (Base‘𝐸)) |
17 | 16 | 3adant3 1129 | . . . . . . 7 ⊢ ((𝐼 ∈ Fin ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) → 𝑥 ∈ (Base‘𝐸)) |
18 | 17 | ad2antrr 724 | . . . . . 6 ⊢ ((((𝐼 ∈ Fin ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) ∧ 𝑡 ∈ ℝ) → 𝑥 ∈ (Base‘𝐸)) |
19 | eldifi 4123 | . . . . . . . . . 10 ⊢ (𝑦 ∈ (𝑃 ∖ {𝑥}) → 𝑦 ∈ 𝑃) | |
20 | 14 | eleq2d 2811 | . . . . . . . . . 10 ⊢ (𝐼 ∈ Fin → (𝑦 ∈ 𝑃 ↔ 𝑦 ∈ (Base‘𝐸))) |
21 | 19, 20 | imbitrid 243 | . . . . . . . . 9 ⊢ (𝐼 ∈ Fin → (𝑦 ∈ (𝑃 ∖ {𝑥}) → 𝑦 ∈ (Base‘𝐸))) |
22 | 21 | a1d 25 | . . . . . . . 8 ⊢ (𝐼 ∈ Fin → (𝑥 ∈ 𝑃 → (𝑦 ∈ (𝑃 ∖ {𝑥}) → 𝑦 ∈ (Base‘𝐸)))) |
23 | 22 | 3imp 1108 | . . . . . . 7 ⊢ ((𝐼 ∈ Fin ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) → 𝑦 ∈ (Base‘𝐸)) |
24 | 23 | ad2antrr 724 | . . . . . 6 ⊢ ((((𝐼 ∈ Fin ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) ∧ 𝑡 ∈ ℝ) → 𝑦 ∈ (Base‘𝐸)) |
25 | 14 | 3ad2ant1 1130 | . . . . . . . . 9 ⊢ ((𝐼 ∈ Fin ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) → 𝑃 = (Base‘𝐸)) |
26 | 25 | eleq2d 2811 | . . . . . . . 8 ⊢ ((𝐼 ∈ Fin ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) → (𝑝 ∈ 𝑃 ↔ 𝑝 ∈ (Base‘𝐸))) |
27 | 26 | biimpa 475 | . . . . . . 7 ⊢ (((𝐼 ∈ Fin ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → 𝑝 ∈ (Base‘𝐸)) |
28 | 27 | adantr 479 | . . . . . 6 ⊢ ((((𝐼 ∈ Fin ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) ∧ 𝑡 ∈ ℝ) → 𝑝 ∈ (Base‘𝐸)) |
29 | 1, 7, 4, 8, 11, 18, 24, 28, 5, 10 | rrxplusgvscavalb 25384 | . . . . 5 ⊢ ((((𝐼 ∈ Fin ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 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 3166 | . . . 4 ⊢ (((𝐼 ∈ Fin ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 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 3425 | . . 3 ⊢ ((𝐼 ∈ Fin ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) → {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 | 31 | mpoeq3dva 7497 | . 2 ⊢ (𝐼 ∈ Fin → (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡)( ·𝑠 ‘𝐸)𝑥)(+g‘𝐸)(𝑡( ·𝑠 ‘𝐸)𝑦))}) = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ ∀𝑖 ∈ 𝐼 (𝑝‘𝑖) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖)))})) |
33 | 6, 32 | eqtrd 2765 | 1 ⊢ (𝐼 ∈ Fin → 𝐿 = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {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 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3050 ∃wrex 3059 {crab 3418 ∖ cdif 3941 {csn 4630 ‘cfv 6549 (class class class)co 7419 ∈ cmpo 7421 ↑m cmap 8845 Fincfn 8964 ℝcr 11144 1c1 11146 + caddc 11148 · cmul 11150 − cmin 11481 Basecbs 17199 +gcplusg 17252 ·𝑠 cvsca 17256 ℝ^crrx 25372 LineMcline 47991 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 ax-pre-sup 11223 ax-addf 11224 ax-mulf 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9393 df-sup 9472 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-div 11909 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-rp 13015 df-fz 13525 df-seq 14008 df-exp 14068 df-cj 15090 df-re 15091 df-im 15092 df-sqrt 15226 df-abs 15227 df-struct 17135 df-sets 17152 df-slot 17170 df-ndx 17182 df-base 17200 df-ress 17229 df-plusg 17265 df-mulr 17266 df-starv 17267 df-sca 17268 df-vsca 17269 df-ip 17270 df-tset 17271 df-ple 17272 df-ds 17274 df-unif 17275 df-hom 17276 df-cco 17277 df-0g 17442 df-prds 17448 df-pws 17450 df-mgm 18619 df-sgrp 18698 df-mnd 18714 df-mhm 18759 df-grp 18917 df-minusg 18918 df-sbg 18919 df-subg 19103 df-ghm 19193 df-cmn 19766 df-abl 19767 df-mgp 20104 df-rng 20122 df-ur 20151 df-ring 20204 df-cring 20205 df-oppr 20302 df-dvdsr 20325 df-unit 20326 df-invr 20356 df-dvr 20369 df-rhm 20440 df-subrng 20512 df-subrg 20537 df-drng 20655 df-field 20656 df-staf 20754 df-srng 20755 df-lmod 20774 df-lss 20845 df-sra 21087 df-rgmod 21088 df-cnfld 21314 df-refld 21571 df-dsmm 21700 df-frlm 21715 df-tng 24554 df-tcph 25158 df-rrx 25374 df-line 47993 |
This theorem is referenced by: eenglngeehlnm 48003 |
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