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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 44650. (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 2818 | . . 3 ⊢ ( ·𝑠 ‘𝐸) = ( ·𝑠 ‘𝐸) | |
5 | eqid 2818 | . . 3 ⊢ (+g‘𝐸) = (+g‘𝐸) | |
6 | 1, 2, 3, 4, 5 | rrxline 44649 | . 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 2818 | . . . . 5 ⊢ (Base‘𝐸) = (Base‘𝐸) | |
8 | simplll 771 | . . . . 5 ⊢ ((((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) ∧ 𝑝 ∈ 𝑃) ∧ 𝑡 ∈ ℝ) → 𝐼 ∈ Fin) | |
9 | 1red 10630 | . . . . . 6 ⊢ ((((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) ∧ 𝑝 ∈ 𝑃) ∧ 𝑡 ∈ ℝ) → 1 ∈ ℝ) | |
10 | simpr 485 | . . . . . 6 ⊢ ((((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) ∧ 𝑝 ∈ 𝑃) ∧ 𝑡 ∈ ℝ) → 𝑡 ∈ ℝ) | |
11 | 9, 10 | resubcld 11056 | . . . . 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 23940 | . . . . . . . . . . 11 ⊢ (𝐼 ∈ Fin → (Base‘𝐸) = (ℝ ↑m 𝐼)) |
14 | 13, 2 | syl6reqr 2872 | . . . . . . . . . 10 ⊢ (𝐼 ∈ Fin → 𝑃 = (Base‘𝐸)) |
15 | 14 | eleq2d 2895 | . . . . . . . . 9 ⊢ (𝐼 ∈ Fin → (𝑋 ∈ 𝑃 ↔ 𝑋 ∈ (Base‘𝐸))) |
16 | 15 | biimpcd 250 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝑃 → (𝐼 ∈ Fin → 𝑋 ∈ (Base‘𝐸))) |
17 | 16 | 3ad2ant1 1125 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (𝐼 ∈ Fin → 𝑋 ∈ (Base‘𝐸))) |
18 | 17 | impcom 408 | . . . . . 6 ⊢ ((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) → 𝑋 ∈ (Base‘𝐸)) |
19 | 18 | ad2antrr 722 | . . . . 5 ⊢ ((((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) ∧ 𝑝 ∈ 𝑃) ∧ 𝑡 ∈ ℝ) → 𝑋 ∈ (Base‘𝐸)) |
20 | 14 | eleq2d 2895 | . . . . . . . . 9 ⊢ (𝐼 ∈ Fin → (𝑌 ∈ 𝑃 ↔ 𝑌 ∈ (Base‘𝐸))) |
21 | 20 | biimpcd 250 | . . . . . . . 8 ⊢ (𝑌 ∈ 𝑃 → (𝐼 ∈ Fin → 𝑌 ∈ (Base‘𝐸))) |
22 | 21 | 3ad2ant2 1126 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → (𝐼 ∈ Fin → 𝑌 ∈ (Base‘𝐸))) |
23 | 22 | impcom 408 | . . . . . 6 ⊢ ((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) → 𝑌 ∈ (Base‘𝐸)) |
24 | 23 | ad2antrr 722 | . . . . 5 ⊢ ((((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) ∧ 𝑝 ∈ 𝑃) ∧ 𝑡 ∈ ℝ) → 𝑌 ∈ (Base‘𝐸)) |
25 | 14 | adantr 481 | . . . . . . . 8 ⊢ ((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) → 𝑃 = (Base‘𝐸)) |
26 | 25 | eleq2d 2895 | . . . . . . 7 ⊢ ((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) → (𝑝 ∈ 𝑃 ↔ 𝑝 ∈ (Base‘𝐸))) |
27 | 26 | biimpa 477 | . . . . . 6 ⊢ (((𝐼 ∈ Fin ∧ (𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌)) ∧ 𝑝 ∈ 𝑃) → 𝑝 ∈ (Base‘𝐸)) |
28 | 27 | adantr 481 | . . . . 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 23925 | . . . 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 3293 | . . 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 3476 | . 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 2853 | 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 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 ∃wrex 3136 {crab 3139 ‘cfv 6348 (class class class)co 7145 ↑m cmap 8395 Fincfn 8497 ℝcr 10524 1c1 10526 + caddc 10528 · cmul 10530 − cmin 10858 Basecbs 16471 +gcplusg 16553 ·𝑠 cvsca 16557 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: rrx2line 44655 |
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