Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > rrxdim | Structured version Visualization version GIF version |
Description: Dimension of the generalized Euclidean space. (Contributed by Thierry Arnoux, 20-May-2023.) |
Ref | Expression |
---|---|
rrxdim.1 | ⊢ 𝐻 = (ℝ^‘𝐼) |
Ref | Expression |
---|---|
rrxdim | ⊢ (𝐼 ∈ 𝑉 → (dim‘𝐻) = (♯‘𝐼)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrxdim.1 | . . . . 5 ⊢ 𝐻 = (ℝ^‘𝐼) | |
2 | 1 | rrxval 24284 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼))) |
3 | eqid 2737 | . . . . 5 ⊢ (toℂPreHil‘(ℝfld freeLMod 𝐼)) = (toℂPreHil‘(ℝfld freeLMod 𝐼)) | |
4 | eqid 2737 | . . . . 5 ⊢ (Base‘(ℝfld freeLMod 𝐼)) = (Base‘(ℝfld freeLMod 𝐼)) | |
5 | eqid 2737 | . . . . 5 ⊢ (·𝑖‘(ℝfld freeLMod 𝐼)) = (·𝑖‘(ℝfld freeLMod 𝐼)) | |
6 | 3, 4, 5 | tcphval 24115 | . . . 4 ⊢ (toℂPreHil‘(ℝfld freeLMod 𝐼)) = ((ℝfld freeLMod 𝐼) toNrmGrp (𝑥 ∈ (Base‘(ℝfld freeLMod 𝐼)) ↦ (√‘(𝑥(·𝑖‘(ℝfld freeLMod 𝐼))𝑥)))) |
7 | 2, 6 | eqtrdi 2794 | . . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = ((ℝfld freeLMod 𝐼) toNrmGrp (𝑥 ∈ (Base‘(ℝfld freeLMod 𝐼)) ↦ (√‘(𝑥(·𝑖‘(ℝfld freeLMod 𝐼))𝑥))))) |
8 | 7 | fveq2d 6721 | . 2 ⊢ (𝐼 ∈ 𝑉 → (dim‘𝐻) = (dim‘((ℝfld freeLMod 𝐼) toNrmGrp (𝑥 ∈ (Base‘(ℝfld freeLMod 𝐼)) ↦ (√‘(𝑥(·𝑖‘(ℝfld freeLMod 𝐼))𝑥)))))) |
9 | resubdrg 20570 | . . . . 5 ⊢ (ℝ ∈ (SubRing‘ℂfld) ∧ ℝfld ∈ DivRing) | |
10 | 9 | simpri 489 | . . . 4 ⊢ ℝfld ∈ DivRing |
11 | eqid 2737 | . . . . 5 ⊢ (ℝfld freeLMod 𝐼) = (ℝfld freeLMod 𝐼) | |
12 | 11 | frlmlvec 20723 | . . . 4 ⊢ ((ℝfld ∈ DivRing ∧ 𝐼 ∈ 𝑉) → (ℝfld freeLMod 𝐼) ∈ LVec) |
13 | 10, 12 | mpan 690 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (ℝfld freeLMod 𝐼) ∈ LVec) |
14 | 4 | tcphex 24114 | . . 3 ⊢ (𝑥 ∈ (Base‘(ℝfld freeLMod 𝐼)) ↦ (√‘(𝑥(·𝑖‘(ℝfld freeLMod 𝐼))𝑥))) ∈ V |
15 | eqid 2737 | . . . 4 ⊢ ((ℝfld freeLMod 𝐼) toNrmGrp (𝑥 ∈ (Base‘(ℝfld freeLMod 𝐼)) ↦ (√‘(𝑥(·𝑖‘(ℝfld freeLMod 𝐼))𝑥)))) = ((ℝfld freeLMod 𝐼) toNrmGrp (𝑥 ∈ (Base‘(ℝfld freeLMod 𝐼)) ↦ (√‘(𝑥(·𝑖‘(ℝfld freeLMod 𝐼))𝑥)))) | |
16 | 15 | tngdim 31410 | . . 3 ⊢ (((ℝfld freeLMod 𝐼) ∈ LVec ∧ (𝑥 ∈ (Base‘(ℝfld freeLMod 𝐼)) ↦ (√‘(𝑥(·𝑖‘(ℝfld freeLMod 𝐼))𝑥))) ∈ V) → (dim‘(ℝfld freeLMod 𝐼)) = (dim‘((ℝfld freeLMod 𝐼) toNrmGrp (𝑥 ∈ (Base‘(ℝfld freeLMod 𝐼)) ↦ (√‘(𝑥(·𝑖‘(ℝfld freeLMod 𝐼))𝑥)))))) |
17 | 13, 14, 16 | sylancl 589 | . 2 ⊢ (𝐼 ∈ 𝑉 → (dim‘(ℝfld freeLMod 𝐼)) = (dim‘((ℝfld freeLMod 𝐼) toNrmGrp (𝑥 ∈ (Base‘(ℝfld freeLMod 𝐼)) ↦ (√‘(𝑥(·𝑖‘(ℝfld freeLMod 𝐼))𝑥)))))) |
18 | 11 | frlmdim 31408 | . . 3 ⊢ ((ℝfld ∈ DivRing ∧ 𝐼 ∈ 𝑉) → (dim‘(ℝfld freeLMod 𝐼)) = (♯‘𝐼)) |
19 | 10, 18 | mpan 690 | . 2 ⊢ (𝐼 ∈ 𝑉 → (dim‘(ℝfld freeLMod 𝐼)) = (♯‘𝐼)) |
20 | 8, 17, 19 | 3eqtr2d 2783 | 1 ⊢ (𝐼 ∈ 𝑉 → (dim‘𝐻) = (♯‘𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 Vcvv 3408 ↦ cmpt 5135 ‘cfv 6380 (class class class)co 7213 ℝcr 10728 ♯chash 13896 √csqrt 14796 Basecbs 16760 ·𝑖cip 16807 DivRingcdr 19767 SubRingcsubrg 19796 LVecclvec 20139 ℂfldccnfld 20363 ℝfldcrefld 20566 freeLMod cfrlm 20708 toNrmGrp ctng 23476 toℂPreHilctcph 24064 ℝ^crrx 24280 dimcldim 31398 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-reg 9208 ax-inf2 9256 ax-ac2 10077 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 ax-addf 10808 ax-mulf 10809 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-rpss 7511 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-tpos 7968 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-oadd 8206 df-er 8391 df-map 8510 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-sup 9058 df-oi 9126 df-r1 9380 df-rank 9381 df-dju 9517 df-card 9555 df-acn 9558 df-ac 9730 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-xnn0 12163 df-z 12177 df-dec 12294 df-uz 12439 df-rp 12587 df-fz 13096 df-fzo 13239 df-seq 13575 df-exp 13636 df-hash 13897 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-starv 16817 df-sca 16818 df-vsca 16819 df-ip 16820 df-tset 16821 df-ple 16822 df-ocomp 16823 df-ds 16824 df-unif 16825 df-hom 16826 df-cco 16827 df-0g 16946 df-gsum 16947 df-prds 16952 df-pws 16954 df-mre 17089 df-mrc 17090 df-mri 17091 df-acs 17092 df-proset 17802 df-drs 17803 df-poset 17820 df-ipo 18034 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-mhm 18218 df-submnd 18219 df-grp 18368 df-minusg 18369 df-sbg 18370 df-mulg 18489 df-subg 18540 df-ghm 18620 df-cntz 18711 df-cmn 19172 df-abl 19173 df-mgp 19505 df-ur 19517 df-ring 19564 df-cring 19565 df-oppr 19641 df-dvdsr 19659 df-unit 19660 df-invr 19690 df-dvr 19701 df-drng 19769 df-subrg 19798 df-lmod 19901 df-lss 19969 df-lsp 20009 df-lmhm 20059 df-lbs 20112 df-lvec 20140 df-sra 20209 df-rgmod 20210 df-nzr 20296 df-cnfld 20364 df-refld 20567 df-dsmm 20694 df-frlm 20709 df-uvc 20745 df-tng 23482 df-tcph 24066 df-rrx 24282 df-dim 31399 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |