Mathbox for Thierry Arnoux |
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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 23989 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼))) |
3 | eqid 2821 | . . . . 5 ⊢ (toℂPreHil‘(ℝfld freeLMod 𝐼)) = (toℂPreHil‘(ℝfld freeLMod 𝐼)) | |
4 | eqid 2821 | . . . . 5 ⊢ (Base‘(ℝfld freeLMod 𝐼)) = (Base‘(ℝfld freeLMod 𝐼)) | |
5 | eqid 2821 | . . . . 5 ⊢ (·𝑖‘(ℝfld freeLMod 𝐼)) = (·𝑖‘(ℝfld freeLMod 𝐼)) | |
6 | 3, 4, 5 | tcphval 23820 | . . . 4 ⊢ (toℂPreHil‘(ℝfld freeLMod 𝐼)) = ((ℝfld freeLMod 𝐼) toNrmGrp (𝑥 ∈ (Base‘(ℝfld freeLMod 𝐼)) ↦ (√‘(𝑥(·𝑖‘(ℝfld freeLMod 𝐼))𝑥)))) |
7 | 2, 6 | syl6eq 2872 | . . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = ((ℝfld freeLMod 𝐼) toNrmGrp (𝑥 ∈ (Base‘(ℝfld freeLMod 𝐼)) ↦ (√‘(𝑥(·𝑖‘(ℝfld freeLMod 𝐼))𝑥))))) |
8 | 7 | fveq2d 6673 | . 2 ⊢ (𝐼 ∈ 𝑉 → (dim‘𝐻) = (dim‘((ℝfld freeLMod 𝐼) toNrmGrp (𝑥 ∈ (Base‘(ℝfld freeLMod 𝐼)) ↦ (√‘(𝑥(·𝑖‘(ℝfld freeLMod 𝐼))𝑥)))))) |
9 | resubdrg 20751 | . . . . 5 ⊢ (ℝ ∈ (SubRing‘ℂfld) ∧ ℝfld ∈ DivRing) | |
10 | 9 | simpri 488 | . . . 4 ⊢ ℝfld ∈ DivRing |
11 | eqid 2821 | . . . . 5 ⊢ (ℝfld freeLMod 𝐼) = (ℝfld freeLMod 𝐼) | |
12 | 11 | frlmlvec 20904 | . . . 4 ⊢ ((ℝfld ∈ DivRing ∧ 𝐼 ∈ 𝑉) → (ℝfld freeLMod 𝐼) ∈ LVec) |
13 | 10, 12 | mpan 688 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (ℝfld freeLMod 𝐼) ∈ LVec) |
14 | 4 | tcphex 23819 | . . 3 ⊢ (𝑥 ∈ (Base‘(ℝfld freeLMod 𝐼)) ↦ (√‘(𝑥(·𝑖‘(ℝfld freeLMod 𝐼))𝑥))) ∈ V |
15 | eqid 2821 | . . . 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 31011 | . . 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 588 | . 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 31009 | . . 3 ⊢ ((ℝfld ∈ DivRing ∧ 𝐼 ∈ 𝑉) → (dim‘(ℝfld freeLMod 𝐼)) = (♯‘𝐼)) |
19 | 10, 18 | mpan 688 | . 2 ⊢ (𝐼 ∈ 𝑉 → (dim‘(ℝfld freeLMod 𝐼)) = (♯‘𝐼)) |
20 | 8, 17, 19 | 3eqtr2d 2862 | 1 ⊢ (𝐼 ∈ 𝑉 → (dim‘𝐻) = (♯‘𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ↦ cmpt 5145 ‘cfv 6354 (class class class)co 7155 ℝcr 10535 ♯chash 13689 √csqrt 14591 Basecbs 16482 ·𝑖cip 16569 DivRingcdr 19501 SubRingcsubrg 19530 LVecclvec 19873 ℂfldccnfld 20544 ℝfldcrefld 20747 freeLMod cfrlm 20889 toNrmGrp ctng 23187 toℂPreHilctcph 23770 ℝ^crrx 23985 dimcldim 30999 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-reg 9055 ax-inf2 9103 ax-ac2 9884 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 ax-addf 10615 ax-mulf 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-rpss 7448 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-tpos 7891 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-sup 8905 df-oi 8973 df-r1 9192 df-rank 9193 df-dju 9329 df-card 9367 df-acn 9370 df-ac 9541 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-xnn0 11967 df-z 11981 df-dec 12098 df-uz 12243 df-rp 12389 df-fz 12892 df-fzo 13033 df-seq 13369 df-exp 13429 df-hash 13690 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-starv 16579 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ocomp 16585 df-ds 16586 df-unif 16587 df-hom 16588 df-cco 16589 df-0g 16714 df-gsum 16715 df-prds 16720 df-pws 16722 df-mre 16856 df-mrc 16857 df-mri 16858 df-acs 16859 df-proset 17537 df-drs 17538 df-poset 17555 df-ipo 17761 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-mhm 17955 df-submnd 17956 df-grp 18105 df-minusg 18106 df-sbg 18107 df-mulg 18224 df-subg 18275 df-ghm 18355 df-cntz 18446 df-cmn 18907 df-abl 18908 df-mgp 19239 df-ur 19251 df-ring 19298 df-cring 19299 df-oppr 19372 df-dvdsr 19390 df-unit 19391 df-invr 19421 df-dvr 19432 df-drng 19503 df-subrg 19532 df-lmod 19635 df-lss 19703 df-lsp 19743 df-lmhm 19793 df-lbs 19846 df-lvec 19874 df-sra 19943 df-rgmod 19944 df-nzr 20030 df-cnfld 20545 df-refld 20748 df-dsmm 20875 df-frlm 20890 df-uvc 20926 df-tng 23193 df-tcph 23772 df-rrx 23987 df-dim 31000 |
This theorem is referenced by: (None) |
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