Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > rrxbase | Structured version Visualization version GIF version |
Description: The base of the generalized real Euclidean space is the set of functions with finite support. (Contributed by Thierry Arnoux, 16-Jun-2019.) (Proof shortened by AV, 22-Jul-2019.) |
Ref | Expression |
---|---|
rrxval.r | ⊢ 𝐻 = (ℝ^‘𝐼) |
rrxbase.b | ⊢ 𝐵 = (Base‘𝐻) |
Ref | Expression |
---|---|
rrxbase | ⊢ (𝐼 ∈ 𝑉 → 𝐵 = {𝑓 ∈ (ℝ ↑m 𝐼) ∣ 𝑓 finSupp 0}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrxval.r | . . . . 5 ⊢ 𝐻 = (ℝ^‘𝐼) | |
2 | 1 | rrxval 23917 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼))) |
3 | 2 | fveq2d 6667 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (Base‘𝐻) = (Base‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))) |
4 | eqid 2818 | . . . 4 ⊢ (toℂPreHil‘(ℝfld freeLMod 𝐼)) = (toℂPreHil‘(ℝfld freeLMod 𝐼)) | |
5 | eqid 2818 | . . . 4 ⊢ (Base‘(ℝfld freeLMod 𝐼)) = (Base‘(ℝfld freeLMod 𝐼)) | |
6 | 4, 5 | tcphbas 23749 | . . 3 ⊢ (Base‘(ℝfld freeLMod 𝐼)) = (Base‘(toℂPreHil‘(ℝfld freeLMod 𝐼))) |
7 | 3, 6 | syl6eqr 2871 | . 2 ⊢ (𝐼 ∈ 𝑉 → (Base‘𝐻) = (Base‘(ℝfld freeLMod 𝐼))) |
8 | rrxbase.b | . . 3 ⊢ 𝐵 = (Base‘𝐻) | |
9 | 8 | a1i 11 | . 2 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = (Base‘𝐻)) |
10 | refld 20691 | . . 3 ⊢ ℝfld ∈ Field | |
11 | eqid 2818 | . . . 4 ⊢ (ℝfld freeLMod 𝐼) = (ℝfld freeLMod 𝐼) | |
12 | rebase 20678 | . . . 4 ⊢ ℝ = (Base‘ℝfld) | |
13 | re0g 20684 | . . . 4 ⊢ 0 = (0g‘ℝfld) | |
14 | eqid 2818 | . . . 4 ⊢ {𝑓 ∈ (ℝ ↑m 𝐼) ∣ 𝑓 finSupp 0} = {𝑓 ∈ (ℝ ↑m 𝐼) ∣ 𝑓 finSupp 0} | |
15 | 11, 12, 13, 14 | frlmbas 20827 | . . 3 ⊢ ((ℝfld ∈ Field ∧ 𝐼 ∈ 𝑉) → {𝑓 ∈ (ℝ ↑m 𝐼) ∣ 𝑓 finSupp 0} = (Base‘(ℝfld freeLMod 𝐼))) |
16 | 10, 15 | mpan 686 | . 2 ⊢ (𝐼 ∈ 𝑉 → {𝑓 ∈ (ℝ ↑m 𝐼) ∣ 𝑓 finSupp 0} = (Base‘(ℝfld freeLMod 𝐼))) |
17 | 7, 9, 16 | 3eqtr4d 2863 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = {𝑓 ∈ (ℝ ↑m 𝐼) ∣ 𝑓 finSupp 0}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 {crab 3139 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 ↑m cmap 8395 finSupp cfsupp 8821 ℝcr 10524 0cc0 10525 Basecbs 16471 Fieldcfield 19432 ℝfldcrefld 20676 freeLMod cfrlm 20818 toℂPreHilctcph 23698 ℝ^crrx 23913 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 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 12881 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-0g 16703 df-prds 16709 df-pws 16711 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-subg 18214 df-cmn 18837 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-dvr 19362 df-drng 19433 df-field 19434 df-subrg 19462 df-sra 19873 df-rgmod 19874 df-cnfld 20474 df-refld 20677 df-dsmm 20804 df-frlm 20819 df-tng 23121 df-tcph 23700 df-rrx 23915 |
This theorem is referenced by: rrxnm 23921 rrxds 23923 rrxmval 23935 rrxmfval 23936 rrxbasefi 23940 rrxmetfi 23942 ehlbase 23945 k0004ss2 40380 rrnprjdstle 42463 |
Copyright terms: Public domain | W3C validator |