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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 6921 | . . 3 ⊢ 𝐸 ∈ V |
3 | eqid 2735 | . . . 4 ⊢ (Base‘𝐸) = (Base‘𝐸) | |
4 | rrxlines.l | . . . 4 ⊢ 𝐿 = (LineM‘𝐸) | |
5 | eqid 2735 | . . . 4 ⊢ (Scalar‘𝐸) = (Scalar‘𝐸) | |
6 | eqid 2735 | . . . 4 ⊢ (Base‘(Scalar‘𝐸)) = (Base‘(Scalar‘𝐸)) | |
7 | rrxlines.m | . . . 4 ⊢ · = ( ·𝑠 ‘𝐸) | |
8 | rrxlines.a | . . . 4 ⊢ + = (+g‘𝐸) | |
9 | eqid 2735 | . . . 4 ⊢ (-g‘(Scalar‘𝐸)) = (-g‘(Scalar‘𝐸)) | |
10 | eqid 2735 | . . . 4 ⊢ (1r‘(Scalar‘𝐸)) = (1r‘(Scalar‘𝐸)) | |
11 | 3, 4, 5, 6, 7, 8, 9, 10 | lines 48581 | . . 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 25458 | . . . 4 ⊢ (𝐼 ∈ Fin → (Base‘𝐸) = (ℝ ↑m 𝐼)) |
15 | rrxlines.p | . . . 4 ⊢ 𝑃 = (ℝ ↑m 𝐼) | |
16 | 14, 15 | eqtr4di 2793 | . . 3 ⊢ (𝐼 ∈ Fin → (Base‘𝐸) = 𝑃) |
17 | 16 | difeq1d 4135 | . . 3 ⊢ (𝐼 ∈ Fin → ((Base‘𝐸) ∖ {𝑥}) = (𝑃 ∖ {𝑥})) |
18 | 1 | rrxsca 25444 | . . . . . . 7 ⊢ (𝐼 ∈ Fin → (Scalar‘𝐸) = ℝfld) |
19 | 18 | fveq2d 6911 | . . . . . 6 ⊢ (𝐼 ∈ Fin → (Base‘(Scalar‘𝐸)) = (Base‘ℝfld)) |
20 | rebase 21642 | . . . . . 6 ⊢ ℝ = (Base‘ℝfld) | |
21 | 19, 20 | eqtr4di 2793 | . . . . 5 ⊢ (𝐼 ∈ Fin → (Base‘(Scalar‘𝐸)) = ℝ) |
22 | 18 | fveq2d 6911 | . . . . . . . . . . . 12 ⊢ (𝐼 ∈ Fin → (1r‘(Scalar‘𝐸)) = (1r‘ℝfld)) |
23 | re1r 21649 | . . . . . . . . . . . 12 ⊢ 1 = (1r‘ℝfld) | |
24 | 22, 23 | eqtr4di 2793 | . . . . . . . . . . 11 ⊢ (𝐼 ∈ Fin → (1r‘(Scalar‘𝐸)) = 1) |
25 | 24 | oveq1d 7446 | . . . . . . . . . 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 6911 | . . . . . . . . . . 11 ⊢ (𝐼 ∈ Fin → (-g‘(Scalar‘𝐸)) = (-g‘ℝfld)) |
28 | 27 | oveqd 7448 | . . . . . . . . . 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 2825 | . . . . . . . . . . 11 ⊢ (𝐼 ∈ Fin → (𝑡 ∈ (Base‘(Scalar‘𝐸)) ↔ 𝑡 ∈ ℝ)) |
31 | 1re 11259 | . . . . . . . . . . . 12 ⊢ 1 ∈ ℝ | |
32 | eqid 2735 | . . . . . . . . . . . . . 14 ⊢ (-g‘ℝfld) = (-g‘ℝfld) | |
33 | 32 | resubgval 21645 | . . . . . . . . . . . . 13 ⊢ ((1 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (1 − 𝑡) = (1(-g‘ℝfld)𝑡)) |
34 | 33 | eqcomd 2741 | . . . . . . . . . . . 12 ⊢ ((1 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (1(-g‘ℝfld)𝑡) = (1 − 𝑡)) |
35 | 31, 34 | mpan 690 | . . . . . . . . . . 11 ⊢ (𝑡 ∈ ℝ → (1(-g‘ℝfld)𝑡) = (1 − 𝑡)) |
36 | 30, 35 | biimtrdi 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 2779 | . . . . . . . 8 ⊢ ((𝐼 ∈ Fin ∧ 𝑡 ∈ (Base‘(Scalar‘𝐸))) → ((1r‘(Scalar‘𝐸))(-g‘(Scalar‘𝐸))𝑡) = (1 − 𝑡)) |
39 | 38 | oveq1d 7446 | . . . . . . 7 ⊢ ((𝐼 ∈ Fin ∧ 𝑡 ∈ (Base‘(Scalar‘𝐸))) → (((1r‘(Scalar‘𝐸))(-g‘(Scalar‘𝐸))𝑡) · 𝑥) = ((1 − 𝑡) · 𝑥)) |
40 | 39 | oveq1d 7446 | . . . . . 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 2746 | . . . . 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 3331 | . . . 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 3452 | . . 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 7508 | . 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 2775 | 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 1537 ∈ wcel 2106 ∃wrex 3068 {crab 3433 Vcvv 3478 ∖ cdif 3960 {csn 4631 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 ↑m cmap 8865 Fincfn 8984 ℝcr 11152 1c1 11154 − cmin 11490 Basecbs 17245 +gcplusg 17298 Scalarcsca 17301 ·𝑠 cvsca 17302 -gcsg 18966 1rcur 20199 ℝfldcrefld 21640 ℝ^crrx 25431 LineMcline 48577 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-rp 13033 df-fz 13545 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-0g 17488 df-prds 17494 df-pws 17496 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-subrng 20563 df-subrg 20587 df-drng 20748 df-field 20749 df-sra 21190 df-rgmod 21191 df-cnfld 21383 df-refld 21641 df-dsmm 21770 df-frlm 21785 df-tng 24613 df-tcph 25217 df-rrx 25433 df-line 48579 |
This theorem is referenced by: rrxline 48584 rrxlinesc 48585 |
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