![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
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 6905 | . . 3 ⊢ 𝐸 ∈ V |
3 | eqid 2731 | . . . 4 ⊢ (Base‘𝐸) = (Base‘𝐸) | |
4 | rrxlines.l | . . . 4 ⊢ 𝐿 = (LineM‘𝐸) | |
5 | eqid 2731 | . . . 4 ⊢ (Scalar‘𝐸) = (Scalar‘𝐸) | |
6 | eqid 2731 | . . . 4 ⊢ (Base‘(Scalar‘𝐸)) = (Base‘(Scalar‘𝐸)) | |
7 | rrxlines.m | . . . 4 ⊢ · = ( ·𝑠 ‘𝐸) | |
8 | rrxlines.a | . . . 4 ⊢ + = (+g‘𝐸) | |
9 | eqid 2731 | . . . 4 ⊢ (-g‘(Scalar‘𝐸)) = (-g‘(Scalar‘𝐸)) | |
10 | eqid 2731 | . . . 4 ⊢ (1r‘(Scalar‘𝐸)) = (1r‘(Scalar‘𝐸)) | |
11 | 3, 4, 5, 6, 7, 8, 9, 10 | lines 47579 | . . 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 25258 | . . . 4 ⊢ (𝐼 ∈ Fin → (Base‘𝐸) = (ℝ ↑m 𝐼)) |
15 | rrxlines.p | . . . 4 ⊢ 𝑃 = (ℝ ↑m 𝐼) | |
16 | 14, 15 | eqtr4di 2789 | . . 3 ⊢ (𝐼 ∈ Fin → (Base‘𝐸) = 𝑃) |
17 | 16 | difeq1d 4121 | . . 3 ⊢ (𝐼 ∈ Fin → ((Base‘𝐸) ∖ {𝑥}) = (𝑃 ∖ {𝑥})) |
18 | 1 | rrxsca 25244 | . . . . . . 7 ⊢ (𝐼 ∈ Fin → (Scalar‘𝐸) = ℝfld) |
19 | 18 | fveq2d 6895 | . . . . . 6 ⊢ (𝐼 ∈ Fin → (Base‘(Scalar‘𝐸)) = (Base‘ℝfld)) |
20 | rebase 21469 | . . . . . 6 ⊢ ℝ = (Base‘ℝfld) | |
21 | 19, 20 | eqtr4di 2789 | . . . . 5 ⊢ (𝐼 ∈ Fin → (Base‘(Scalar‘𝐸)) = ℝ) |
22 | 18 | fveq2d 6895 | . . . . . . . . . . . 12 ⊢ (𝐼 ∈ Fin → (1r‘(Scalar‘𝐸)) = (1r‘ℝfld)) |
23 | re1r 21476 | . . . . . . . . . . . 12 ⊢ 1 = (1r‘ℝfld) | |
24 | 22, 23 | eqtr4di 2789 | . . . . . . . . . . 11 ⊢ (𝐼 ∈ Fin → (1r‘(Scalar‘𝐸)) = 1) |
25 | 24 | oveq1d 7427 | . . . . . . . . . 10 ⊢ (𝐼 ∈ Fin → ((1r‘(Scalar‘𝐸))(-g‘(Scalar‘𝐸))𝑡) = (1(-g‘(Scalar‘𝐸))𝑡)) |
26 | 25 | adantr 480 | . . . . . . . . 9 ⊢ ((𝐼 ∈ Fin ∧ 𝑡 ∈ (Base‘(Scalar‘𝐸))) → ((1r‘(Scalar‘𝐸))(-g‘(Scalar‘𝐸))𝑡) = (1(-g‘(Scalar‘𝐸))𝑡)) |
27 | 18 | fveq2d 6895 | . . . . . . . . . . 11 ⊢ (𝐼 ∈ Fin → (-g‘(Scalar‘𝐸)) = (-g‘ℝfld)) |
28 | 27 | oveqd 7429 | . . . . . . . . . 10 ⊢ (𝐼 ∈ Fin → (1(-g‘(Scalar‘𝐸))𝑡) = (1(-g‘ℝfld)𝑡)) |
29 | 28 | adantr 480 | . . . . . . . . 9 ⊢ ((𝐼 ∈ Fin ∧ 𝑡 ∈ (Base‘(Scalar‘𝐸))) → (1(-g‘(Scalar‘𝐸))𝑡) = (1(-g‘ℝfld)𝑡)) |
30 | 21 | eleq2d 2818 | . . . . . . . . . . 11 ⊢ (𝐼 ∈ Fin → (𝑡 ∈ (Base‘(Scalar‘𝐸)) ↔ 𝑡 ∈ ℝ)) |
31 | 1re 11221 | . . . . . . . . . . . 12 ⊢ 1 ∈ ℝ | |
32 | eqid 2731 | . . . . . . . . . . . . . 14 ⊢ (-g‘ℝfld) = (-g‘ℝfld) | |
33 | 32 | resubgval 21472 | . . . . . . . . . . . . 13 ⊢ ((1 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (1 − 𝑡) = (1(-g‘ℝfld)𝑡)) |
34 | 33 | eqcomd 2737 | . . . . . . . . . . . 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 253 | . . . . . . . . . 10 ⊢ (𝐼 ∈ Fin → (𝑡 ∈ (Base‘(Scalar‘𝐸)) → (1(-g‘ℝfld)𝑡) = (1 − 𝑡))) |
37 | 36 | imp 406 | . . . . . . . . 9 ⊢ ((𝐼 ∈ Fin ∧ 𝑡 ∈ (Base‘(Scalar‘𝐸))) → (1(-g‘ℝfld)𝑡) = (1 − 𝑡)) |
38 | 26, 29, 37 | 3eqtrd 2775 | . . . . . . . 8 ⊢ ((𝐼 ∈ Fin ∧ 𝑡 ∈ (Base‘(Scalar‘𝐸))) → ((1r‘(Scalar‘𝐸))(-g‘(Scalar‘𝐸))𝑡) = (1 − 𝑡)) |
39 | 38 | oveq1d 7427 | . . . . . . 7 ⊢ ((𝐼 ∈ Fin ∧ 𝑡 ∈ (Base‘(Scalar‘𝐸))) → (((1r‘(Scalar‘𝐸))(-g‘(Scalar‘𝐸))𝑡) · 𝑥) = ((1 − 𝑡) · 𝑥)) |
40 | 39 | oveq1d 7427 | . . . . . 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 2742 | . . . . 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 3327 | . . . 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 3448 | . . 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 7487 | . 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 2771 | 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 395 = wceq 1540 ∈ wcel 2105 ∃wrex 3069 {crab 3431 Vcvv 3473 ∖ cdif 3945 {csn 4628 ‘cfv 6543 (class class class)co 7412 ∈ cmpo 7414 ↑m cmap 8826 Fincfn 8945 ℝcr 11115 1c1 11117 − cmin 11451 Basecbs 17151 +gcplusg 17204 Scalarcsca 17207 ·𝑠 cvsca 17208 -gcsg 18863 1rcur 20082 ℝfldcrefld 21467 ℝ^crrx 25231 LineMcline 47575 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 ax-addf 11195 ax-mulf 11196 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-tpos 8217 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-sup 9443 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-rp 12982 df-fz 13492 df-seq 13974 df-exp 14035 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-0g 17394 df-prds 17400 df-pws 17402 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19046 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-cring 20137 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-dvr 20299 df-subrng 20442 df-subrg 20467 df-drng 20585 df-field 20586 df-sra 21019 df-rgmod 21020 df-cnfld 21234 df-refld 21468 df-dsmm 21597 df-frlm 21612 df-tng 24413 df-tcph 25017 df-rrx 25233 df-line 47577 |
This theorem is referenced by: rrxline 47582 rrxlinesc 47583 |
Copyright terms: Public domain | W3C validator |