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Mirrors > Home > MPE Home > Th. List > rrxbasefi | Structured version Visualization version GIF version |
Description: The base of the generalized real Euclidean space, when the dimension of the space is finite. This justifies the use of (ℝ ↑𝑚 𝑋) for the development of the Lebesgue measure theory for n-dimensional real numbers. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
Ref | Expression |
---|---|
rrxbasefi.x | ⊢ (𝜑 → 𝑋 ∈ Fin) |
rrxbasefi.h | ⊢ 𝐻 = (ℝ^‘𝑋) |
rrxbasefi.b | ⊢ 𝐵 = (Base‘𝐻) |
Ref | Expression |
---|---|
rrxbasefi | ⊢ (𝜑 → 𝐵 = (ℝ ↑𝑚 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrxbasefi.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
2 | rrxbasefi.h | . . . . 5 ⊢ 𝐻 = (ℝ^‘𝑋) | |
3 | rrxbasefi.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐻) | |
4 | 2, 3 | rrxbase 23594 | . . . 4 ⊢ (𝑋 ∈ Fin → 𝐵 = {𝑓 ∈ (ℝ ↑𝑚 𝑋) ∣ 𝑓 finSupp 0}) |
5 | 1, 4 | syl 17 | . . 3 ⊢ (𝜑 → 𝐵 = {𝑓 ∈ (ℝ ↑𝑚 𝑋) ∣ 𝑓 finSupp 0}) |
6 | ssrab2 3908 | . . . 4 ⊢ {𝑓 ∈ (ℝ ↑𝑚 𝑋) ∣ 𝑓 finSupp 0} ⊆ (ℝ ↑𝑚 𝑋) | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → {𝑓 ∈ (ℝ ↑𝑚 𝑋) ∣ 𝑓 finSupp 0} ⊆ (ℝ ↑𝑚 𝑋)) |
8 | 5, 7 | eqsstrd 3858 | . 2 ⊢ (𝜑 → 𝐵 ⊆ (ℝ ↑𝑚 𝑋)) |
9 | simpr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) → 𝑓 ∈ (ℝ ↑𝑚 𝑋)) | |
10 | elmapi 8162 | . . . . . . 7 ⊢ (𝑓 ∈ (ℝ ↑𝑚 𝑋) → 𝑓:𝑋⟶ℝ) | |
11 | 10 | adantl 475 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) → 𝑓:𝑋⟶ℝ) |
12 | 1 | adantr 474 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) → 𝑋 ∈ Fin) |
13 | c0ex 10370 | . . . . . . 7 ⊢ 0 ∈ V | |
14 | 13 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) → 0 ∈ V) |
15 | 11, 12, 14 | fdmfifsupp 8573 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) → 𝑓 finSupp 0) |
16 | 9, 15 | jca 507 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) → (𝑓 ∈ (ℝ ↑𝑚 𝑋) ∧ 𝑓 finSupp 0)) |
17 | rabid 3302 | . . . 4 ⊢ (𝑓 ∈ {𝑓 ∈ (ℝ ↑𝑚 𝑋) ∣ 𝑓 finSupp 0} ↔ (𝑓 ∈ (ℝ ↑𝑚 𝑋) ∧ 𝑓 finSupp 0)) | |
18 | 16, 17 | sylibr 226 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) → 𝑓 ∈ {𝑓 ∈ (ℝ ↑𝑚 𝑋) ∣ 𝑓 finSupp 0}) |
19 | 5 | eqcomd 2784 | . . . 4 ⊢ (𝜑 → {𝑓 ∈ (ℝ ↑𝑚 𝑋) ∣ 𝑓 finSupp 0} = 𝐵) |
20 | 19 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) → {𝑓 ∈ (ℝ ↑𝑚 𝑋) ∣ 𝑓 finSupp 0} = 𝐵) |
21 | 18, 20 | eleqtrd 2861 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) → 𝑓 ∈ 𝐵) |
22 | 8, 21 | eqelssd 3841 | 1 ⊢ (𝜑 → 𝐵 = (ℝ ↑𝑚 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 {crab 3094 Vcvv 3398 ⊆ wss 3792 class class class wbr 4886 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 ↑𝑚 cmap 8140 Fincfn 8241 finSupp cfsupp 8563 ℝcr 10271 0cc0 10272 Basecbs 16255 ℝ^crrx 23589 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-addf 10351 ax-mulf 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-tpos 7634 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-sup 8636 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-rp 12138 df-fz 12644 df-seq 13120 df-exp 13179 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-hom 16362 df-cco 16363 df-0g 16488 df-prds 16494 df-pws 16496 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-grp 17812 df-minusg 17813 df-subg 17975 df-cmn 18581 df-mgp 18877 df-ur 18889 df-ring 18936 df-cring 18937 df-oppr 19010 df-dvdsr 19028 df-unit 19029 df-invr 19059 df-dvr 19070 df-drng 19141 df-field 19142 df-subrg 19170 df-sra 19569 df-rgmod 19570 df-cnfld 20143 df-refld 20348 df-dsmm 20475 df-frlm 20490 df-tng 22797 df-tcph 23376 df-rrx 23591 |
This theorem is referenced by: rrxdsfi 23617 rrxmetfi 23618 rrxtopnfi 41435 rrxtoponfi 41439 qndenserrnopnlem 41445 qndenserrn 41447 rrnprjdstle 41449 rrxlines 43473 rrxlinesc 43475 rrxlinec 43476 rrxsphere 43488 |
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