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 6785 | . . 3 ⊢ 𝐸 ∈ V |
3 | eqid 2740 | . . . 4 ⊢ (Base‘𝐸) = (Base‘𝐸) | |
4 | rrxlines.l | . . . 4 ⊢ 𝐿 = (LineM‘𝐸) | |
5 | eqid 2740 | . . . 4 ⊢ (Scalar‘𝐸) = (Scalar‘𝐸) | |
6 | eqid 2740 | . . . 4 ⊢ (Base‘(Scalar‘𝐸)) = (Base‘(Scalar‘𝐸)) | |
7 | rrxlines.m | . . . 4 ⊢ · = ( ·𝑠 ‘𝐸) | |
8 | rrxlines.a | . . . 4 ⊢ + = (+g‘𝐸) | |
9 | eqid 2740 | . . . 4 ⊢ (-g‘(Scalar‘𝐸)) = (-g‘(Scalar‘𝐸)) | |
10 | eqid 2740 | . . . 4 ⊢ (1r‘(Scalar‘𝐸)) = (1r‘(Scalar‘𝐸)) | |
11 | 3, 4, 5, 6, 7, 8, 9, 10 | lines 46056 | . . 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 24585 | . . . 4 ⊢ (𝐼 ∈ Fin → (Base‘𝐸) = (ℝ ↑m 𝐼)) |
15 | rrxlines.p | . . . 4 ⊢ 𝑃 = (ℝ ↑m 𝐼) | |
16 | 14, 15 | eqtr4di 2798 | . . 3 ⊢ (𝐼 ∈ Fin → (Base‘𝐸) = 𝑃) |
17 | 16 | difeq1d 4061 | . . 3 ⊢ (𝐼 ∈ Fin → ((Base‘𝐸) ∖ {𝑥}) = (𝑃 ∖ {𝑥})) |
18 | 1 | rrxsca 24571 | . . . . . . 7 ⊢ (𝐼 ∈ Fin → (Scalar‘𝐸) = ℝfld) |
19 | 18 | fveq2d 6775 | . . . . . 6 ⊢ (𝐼 ∈ Fin → (Base‘(Scalar‘𝐸)) = (Base‘ℝfld)) |
20 | rebase 20822 | . . . . . 6 ⊢ ℝ = (Base‘ℝfld) | |
21 | 19, 20 | eqtr4di 2798 | . . . . 5 ⊢ (𝐼 ∈ Fin → (Base‘(Scalar‘𝐸)) = ℝ) |
22 | 18 | fveq2d 6775 | . . . . . . . . . . . 12 ⊢ (𝐼 ∈ Fin → (1r‘(Scalar‘𝐸)) = (1r‘ℝfld)) |
23 | re1r 20829 | . . . . . . . . . . . 12 ⊢ 1 = (1r‘ℝfld) | |
24 | 22, 23 | eqtr4di 2798 | . . . . . . . . . . 11 ⊢ (𝐼 ∈ Fin → (1r‘(Scalar‘𝐸)) = 1) |
25 | 24 | oveq1d 7287 | . . . . . . . . . 10 ⊢ (𝐼 ∈ Fin → ((1r‘(Scalar‘𝐸))(-g‘(Scalar‘𝐸))𝑡) = (1(-g‘(Scalar‘𝐸))𝑡)) |
26 | 25 | adantr 481 | . . . . . . . . 9 ⊢ ((𝐼 ∈ Fin ∧ 𝑡 ∈ (Base‘(Scalar‘𝐸))) → ((1r‘(Scalar‘𝐸))(-g‘(Scalar‘𝐸))𝑡) = (1(-g‘(Scalar‘𝐸))𝑡)) |
27 | 18 | fveq2d 6775 | . . . . . . . . . . 11 ⊢ (𝐼 ∈ Fin → (-g‘(Scalar‘𝐸)) = (-g‘ℝfld)) |
28 | 27 | oveqd 7289 | . . . . . . . . . 10 ⊢ (𝐼 ∈ Fin → (1(-g‘(Scalar‘𝐸))𝑡) = (1(-g‘ℝfld)𝑡)) |
29 | 28 | adantr 481 | . . . . . . . . 9 ⊢ ((𝐼 ∈ Fin ∧ 𝑡 ∈ (Base‘(Scalar‘𝐸))) → (1(-g‘(Scalar‘𝐸))𝑡) = (1(-g‘ℝfld)𝑡)) |
30 | 21 | eleq2d 2826 | . . . . . . . . . . 11 ⊢ (𝐼 ∈ Fin → (𝑡 ∈ (Base‘(Scalar‘𝐸)) ↔ 𝑡 ∈ ℝ)) |
31 | 1re 10986 | . . . . . . . . . . . 12 ⊢ 1 ∈ ℝ | |
32 | eqid 2740 | . . . . . . . . . . . . . 14 ⊢ (-g‘ℝfld) = (-g‘ℝfld) | |
33 | 32 | resubgval 20825 | . . . . . . . . . . . . 13 ⊢ ((1 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (1 − 𝑡) = (1(-g‘ℝfld)𝑡)) |
34 | 33 | eqcomd 2746 | . . . . . . . . . . . 12 ⊢ ((1 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (1(-g‘ℝfld)𝑡) = (1 − 𝑡)) |
35 | 31, 34 | mpan 687 | . . . . . . . . . . 11 ⊢ (𝑡 ∈ ℝ → (1(-g‘ℝfld)𝑡) = (1 − 𝑡)) |
36 | 30, 35 | syl6bi 252 | . . . . . . . . . 10 ⊢ (𝐼 ∈ Fin → (𝑡 ∈ (Base‘(Scalar‘𝐸)) → (1(-g‘ℝfld)𝑡) = (1 − 𝑡))) |
37 | 36 | imp 407 | . . . . . . . . 9 ⊢ ((𝐼 ∈ Fin ∧ 𝑡 ∈ (Base‘(Scalar‘𝐸))) → (1(-g‘ℝfld)𝑡) = (1 − 𝑡)) |
38 | 26, 29, 37 | 3eqtrd 2784 | . . . . . . . 8 ⊢ ((𝐼 ∈ Fin ∧ 𝑡 ∈ (Base‘(Scalar‘𝐸))) → ((1r‘(Scalar‘𝐸))(-g‘(Scalar‘𝐸))𝑡) = (1 − 𝑡)) |
39 | 38 | oveq1d 7287 | . . . . . . 7 ⊢ ((𝐼 ∈ Fin ∧ 𝑡 ∈ (Base‘(Scalar‘𝐸))) → (((1r‘(Scalar‘𝐸))(-g‘(Scalar‘𝐸))𝑡) · 𝑥) = ((1 − 𝑡) · 𝑥)) |
40 | 39 | oveq1d 7287 | . . . . . 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 2751 | . . . . 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 3354 | . . . 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 3419 | . . 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 7345 | . 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 2780 | 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 396 = wceq 1542 ∈ wcel 2110 ∃wrex 3067 {crab 3070 Vcvv 3431 ∖ cdif 3889 {csn 4567 ‘cfv 6432 (class class class)co 7272 ∈ cmpo 7274 ↑m cmap 8607 Fincfn 8725 ℝcr 10881 1c1 10883 − cmin 11216 Basecbs 16923 +gcplusg 16973 Scalarcsca 16976 ·𝑠 cvsca 16977 -gcsg 18590 1rcur 19748 ℝfldcrefld 20820 ℝ^crrx 24558 LineMcline 46052 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-cnex 10938 ax-resscn 10939 ax-1cn 10940 ax-icn 10941 ax-addcl 10942 ax-addrcl 10943 ax-mulcl 10944 ax-mulrcl 10945 ax-mulcom 10946 ax-addass 10947 ax-mulass 10948 ax-distr 10949 ax-i2m1 10950 ax-1ne0 10951 ax-1rid 10952 ax-rnegex 10953 ax-rrecex 10954 ax-cnre 10955 ax-pre-lttri 10956 ax-pre-lttrn 10957 ax-pre-ltadd 10958 ax-pre-mulgt0 10959 ax-pre-sup 10960 ax-addf 10961 ax-mulf 10962 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-om 7708 df-1st 7825 df-2nd 7826 df-supp 7970 df-tpos 8034 df-frecs 8089 df-wrecs 8120 df-recs 8194 df-rdg 8233 df-1o 8289 df-er 8490 df-map 8609 df-ixp 8678 df-en 8726 df-dom 8727 df-sdom 8728 df-fin 8729 df-fsupp 9117 df-sup 9189 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-sub 11218 df-neg 11219 df-div 11644 df-nn 11985 df-2 12047 df-3 12048 df-4 12049 df-5 12050 df-6 12051 df-7 12052 df-8 12053 df-9 12054 df-n0 12245 df-z 12331 df-dec 12449 df-uz 12594 df-rp 12742 df-fz 13251 df-seq 13733 df-exp 13794 df-cj 14821 df-re 14822 df-im 14823 df-sqrt 14957 df-abs 14958 df-struct 16859 df-sets 16876 df-slot 16894 df-ndx 16906 df-base 16924 df-ress 16953 df-plusg 16986 df-mulr 16987 df-starv 16988 df-sca 16989 df-vsca 16990 df-ip 16991 df-tset 16992 df-ple 16993 df-ds 16995 df-unif 16996 df-hom 16997 df-cco 16998 df-0g 17163 df-prds 17169 df-pws 17171 df-mgm 18337 df-sgrp 18386 df-mnd 18397 df-grp 18591 df-minusg 18592 df-sbg 18593 df-subg 18763 df-cmn 19399 df-mgp 19732 df-ur 19749 df-ring 19796 df-cring 19797 df-oppr 19873 df-dvdsr 19894 df-unit 19895 df-invr 19925 df-dvr 19936 df-drng 20004 df-field 20005 df-subrg 20033 df-sra 20445 df-rgmod 20446 df-cnfld 20609 df-refld 20821 df-dsmm 20950 df-frlm 20965 df-tng 23751 df-tcph 24344 df-rrx 24560 df-line 46054 |
This theorem is referenced by: rrxline 46059 rrxlinesc 46060 |
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