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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rrxlines | Structured version Visualization version GIF version |
Description: Definition of lines passing through two different points in a generalized real Euclidean space of finite dimension. (Contributed by AV, 14-Jan-2023.) |
Ref | Expression |
---|---|
rrxlines.e | ⊢ 𝐸 = (ℝ^‘𝐼) |
rrxlines.p | ⊢ 𝑃 = (ℝ ↑m 𝐼) |
rrxlines.l | ⊢ 𝐿 = (LineM‘𝐸) |
rrxlines.m | ⊢ · = ( ·𝑠 ‘𝐸) |
rrxlines.a | ⊢ + = (+g‘𝐸) |
Ref | Expression |
---|---|
rrxlines | ⊢ (𝐼 ∈ Fin → 𝐿 = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrxlines.e | . . . 4 ⊢ 𝐸 = (ℝ^‘𝐼) | |
2 | 1 | fvexi 6659 | . . 3 ⊢ 𝐸 ∈ V |
3 | eqid 2798 | . . . 4 ⊢ (Base‘𝐸) = (Base‘𝐸) | |
4 | rrxlines.l | . . . 4 ⊢ 𝐿 = (LineM‘𝐸) | |
5 | eqid 2798 | . . . 4 ⊢ (Scalar‘𝐸) = (Scalar‘𝐸) | |
6 | eqid 2798 | . . . 4 ⊢ (Base‘(Scalar‘𝐸)) = (Base‘(Scalar‘𝐸)) | |
7 | rrxlines.m | . . . 4 ⊢ · = ( ·𝑠 ‘𝐸) | |
8 | rrxlines.a | . . . 4 ⊢ + = (+g‘𝐸) | |
9 | eqid 2798 | . . . 4 ⊢ (-g‘(Scalar‘𝐸)) = (-g‘(Scalar‘𝐸)) | |
10 | eqid 2798 | . . . 4 ⊢ (1r‘(Scalar‘𝐸)) = (1r‘(Scalar‘𝐸)) | |
11 | 3, 4, 5, 6, 7, 8, 9, 10 | lines 45145 | . . 3 ⊢ (𝐸 ∈ V → 𝐿 = (𝑥 ∈ (Base‘𝐸), 𝑦 ∈ ((Base‘𝐸) ∖ {𝑥}) ↦ {𝑝 ∈ (Base‘𝐸) ∣ ∃𝑡 ∈ (Base‘(Scalar‘𝐸))𝑝 = ((((1r‘(Scalar‘𝐸))(-g‘(Scalar‘𝐸))𝑡) · 𝑥) + (𝑡 · 𝑦))})) |
12 | 2, 11 | mp1i 13 | . 2 ⊢ (𝐼 ∈ Fin → 𝐿 = (𝑥 ∈ (Base‘𝐸), 𝑦 ∈ ((Base‘𝐸) ∖ {𝑥}) ↦ {𝑝 ∈ (Base‘𝐸) ∣ ∃𝑡 ∈ (Base‘(Scalar‘𝐸))𝑝 = ((((1r‘(Scalar‘𝐸))(-g‘(Scalar‘𝐸))𝑡) · 𝑥) + (𝑡 · 𝑦))})) |
13 | id 22 | . . . . 5 ⊢ (𝐼 ∈ Fin → 𝐼 ∈ Fin) | |
14 | 13, 1, 3 | rrxbasefi 24014 | . . . 4 ⊢ (𝐼 ∈ Fin → (Base‘𝐸) = (ℝ ↑m 𝐼)) |
15 | rrxlines.p | . . . 4 ⊢ 𝑃 = (ℝ ↑m 𝐼) | |
16 | 14, 15 | eqtr4di 2851 | . . 3 ⊢ (𝐼 ∈ Fin → (Base‘𝐸) = 𝑃) |
17 | 16 | difeq1d 4049 | . . 3 ⊢ (𝐼 ∈ Fin → ((Base‘𝐸) ∖ {𝑥}) = (𝑃 ∖ {𝑥})) |
18 | 1 | rrxsca 24000 | . . . . . . 7 ⊢ (𝐼 ∈ Fin → (Scalar‘𝐸) = ℝfld) |
19 | 18 | fveq2d 6649 | . . . . . 6 ⊢ (𝐼 ∈ Fin → (Base‘(Scalar‘𝐸)) = (Base‘ℝfld)) |
20 | rebase 20295 | . . . . . 6 ⊢ ℝ = (Base‘ℝfld) | |
21 | 19, 20 | eqtr4di 2851 | . . . . 5 ⊢ (𝐼 ∈ Fin → (Base‘(Scalar‘𝐸)) = ℝ) |
22 | 18 | fveq2d 6649 | . . . . . . . . . . . 12 ⊢ (𝐼 ∈ Fin → (1r‘(Scalar‘𝐸)) = (1r‘ℝfld)) |
23 | re1r 20302 | . . . . . . . . . . . 12 ⊢ 1 = (1r‘ℝfld) | |
24 | 22, 23 | eqtr4di 2851 | . . . . . . . . . . 11 ⊢ (𝐼 ∈ Fin → (1r‘(Scalar‘𝐸)) = 1) |
25 | 24 | oveq1d 7150 | . . . . . . . . . 10 ⊢ (𝐼 ∈ Fin → ((1r‘(Scalar‘𝐸))(-g‘(Scalar‘𝐸))𝑡) = (1(-g‘(Scalar‘𝐸))𝑡)) |
26 | 25 | adantr 484 | . . . . . . . . 9 ⊢ ((𝐼 ∈ Fin ∧ 𝑡 ∈ (Base‘(Scalar‘𝐸))) → ((1r‘(Scalar‘𝐸))(-g‘(Scalar‘𝐸))𝑡) = (1(-g‘(Scalar‘𝐸))𝑡)) |
27 | 18 | fveq2d 6649 | . . . . . . . . . . 11 ⊢ (𝐼 ∈ Fin → (-g‘(Scalar‘𝐸)) = (-g‘ℝfld)) |
28 | 27 | oveqd 7152 | . . . . . . . . . 10 ⊢ (𝐼 ∈ Fin → (1(-g‘(Scalar‘𝐸))𝑡) = (1(-g‘ℝfld)𝑡)) |
29 | 28 | adantr 484 | . . . . . . . . 9 ⊢ ((𝐼 ∈ Fin ∧ 𝑡 ∈ (Base‘(Scalar‘𝐸))) → (1(-g‘(Scalar‘𝐸))𝑡) = (1(-g‘ℝfld)𝑡)) |
30 | 21 | eleq2d 2875 | . . . . . . . . . . 11 ⊢ (𝐼 ∈ Fin → (𝑡 ∈ (Base‘(Scalar‘𝐸)) ↔ 𝑡 ∈ ℝ)) |
31 | 1re 10630 | . . . . . . . . . . . 12 ⊢ 1 ∈ ℝ | |
32 | eqid 2798 | . . . . . . . . . . . . . 14 ⊢ (-g‘ℝfld) = (-g‘ℝfld) | |
33 | 32 | resubgval 20298 | . . . . . . . . . . . . 13 ⊢ ((1 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (1 − 𝑡) = (1(-g‘ℝfld)𝑡)) |
34 | 33 | eqcomd 2804 | . . . . . . . . . . . 12 ⊢ ((1 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (1(-g‘ℝfld)𝑡) = (1 − 𝑡)) |
35 | 31, 34 | mpan 689 | . . . . . . . . . . 11 ⊢ (𝑡 ∈ ℝ → (1(-g‘ℝfld)𝑡) = (1 − 𝑡)) |
36 | 30, 35 | syl6bi 256 | . . . . . . . . . 10 ⊢ (𝐼 ∈ Fin → (𝑡 ∈ (Base‘(Scalar‘𝐸)) → (1(-g‘ℝfld)𝑡) = (1 − 𝑡))) |
37 | 36 | imp 410 | . . . . . . . . 9 ⊢ ((𝐼 ∈ Fin ∧ 𝑡 ∈ (Base‘(Scalar‘𝐸))) → (1(-g‘ℝfld)𝑡) = (1 − 𝑡)) |
38 | 26, 29, 37 | 3eqtrd 2837 | . . . . . . . 8 ⊢ ((𝐼 ∈ Fin ∧ 𝑡 ∈ (Base‘(Scalar‘𝐸))) → ((1r‘(Scalar‘𝐸))(-g‘(Scalar‘𝐸))𝑡) = (1 − 𝑡)) |
39 | 38 | oveq1d 7150 | . . . . . . 7 ⊢ ((𝐼 ∈ Fin ∧ 𝑡 ∈ (Base‘(Scalar‘𝐸))) → (((1r‘(Scalar‘𝐸))(-g‘(Scalar‘𝐸))𝑡) · 𝑥) = ((1 − 𝑡) · 𝑥)) |
40 | 39 | oveq1d 7150 | . . . . . 6 ⊢ ((𝐼 ∈ Fin ∧ 𝑡 ∈ (Base‘(Scalar‘𝐸))) → ((((1r‘(Scalar‘𝐸))(-g‘(Scalar‘𝐸))𝑡) · 𝑥) + (𝑡 · 𝑦)) = (((1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))) |
41 | 40 | eqeq2d 2809 | . . . . 5 ⊢ ((𝐼 ∈ Fin ∧ 𝑡 ∈ (Base‘(Scalar‘𝐸))) → (𝑝 = ((((1r‘(Scalar‘𝐸))(-g‘(Scalar‘𝐸))𝑡) · 𝑥) + (𝑡 · 𝑦)) ↔ 𝑝 = (((1 − 𝑡) · 𝑥) + (𝑡 · 𝑦)))) |
42 | 21, 41 | rexeqbidva 3371 | . . . 4 ⊢ (𝐼 ∈ Fin → (∃𝑡 ∈ (Base‘(Scalar‘𝐸))𝑝 = ((((1r‘(Scalar‘𝐸))(-g‘(Scalar‘𝐸))𝑡) · 𝑥) + (𝑡 · 𝑦)) ↔ ∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡) · 𝑥) + (𝑡 · 𝑦)))) |
43 | 16, 42 | rabeqbidv 3433 | . . 3 ⊢ (𝐼 ∈ Fin → {𝑝 ∈ (Base‘𝐸) ∣ ∃𝑡 ∈ (Base‘(Scalar‘𝐸))𝑝 = ((((1r‘(Scalar‘𝐸))(-g‘(Scalar‘𝐸))𝑡) · 𝑥) + (𝑡 · 𝑦))} = {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))}) |
44 | 16, 17, 43 | mpoeq123dv 7208 | . 2 ⊢ (𝐼 ∈ Fin → (𝑥 ∈ (Base‘𝐸), 𝑦 ∈ ((Base‘𝐸) ∖ {𝑥}) ↦ {𝑝 ∈ (Base‘𝐸) ∣ ∃𝑡 ∈ (Base‘(Scalar‘𝐸))𝑝 = ((((1r‘(Scalar‘𝐸))(-g‘(Scalar‘𝐸))𝑡) · 𝑥) + (𝑡 · 𝑦))}) = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))})) |
45 | 12, 44 | eqtrd 2833 | 1 ⊢ (𝐼 ∈ Fin → 𝐿 = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑝 ∈ 𝑃 ∣ ∃𝑡 ∈ ℝ 𝑝 = (((1 − 𝑡) · 𝑥) + (𝑡 · 𝑦))})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 {crab 3110 Vcvv 3441 ∖ cdif 3878 {csn 4525 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 ↑m cmap 8389 Fincfn 8492 ℝcr 10525 1c1 10527 − cmin 10859 Basecbs 16475 +gcplusg 16557 Scalarcsca 16560 ·𝑠 cvsca 16561 -gcsg 18097 1rcur 19244 ℝfldcrefld 20293 ℝ^crrx 23987 LineMcline 45141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 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 12886 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-0g 16707 df-prds 16713 df-pws 16715 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-cmn 18900 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-drng 19497 df-field 19498 df-subrg 19526 df-sra 19937 df-rgmod 19938 df-cnfld 20092 df-refld 20294 df-dsmm 20421 df-frlm 20436 df-tng 23191 df-tcph 23774 df-rrx 23989 df-line 45143 |
This theorem is referenced by: rrxline 45148 rrxlinesc 45149 |
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