rrxdim - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > rrxdim		Structured version Visualization version GIF version

Theorem rrxdim 33641

Description: Dimension of the generalized Euclidean space. (Contributed by Thierry Arnoux, 20-May-2023.)

**Hypothesis**
Ref	Expression
rrxdim.1	⊢ 𝐻 = (ℝ^‘𝐼)

**Assertion**
Ref	Expression
rrxdim	⊢ (𝐼 ∈ 𝑉 → (dim‘𝐻) = (♯‘𝐼))

Proof of Theorem rrxdim Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rrxdim.1	. . . . 5 ⊢ 𝐻 = (ℝ^‘𝐼)
2	1	rrxval 25434	. . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
3		eqid 2734	. . . . 5 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
4		eqid 2734	. . . . 5 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
5		eqid 2734	. . . . 5 ⊢ (·_𝑖‘(ℝ_fld freeLMod 𝐼)) = (·_𝑖‘(ℝ_fld freeLMod 𝐼))
6	3, 4, 5	tcphval 25265	. . . 4 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = ((ℝ_fld freeLMod 𝐼) toNrmGrp (𝑥 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ↦ (√‘(𝑥(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑥))))
7	2, 6	eqtrdi 2790	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = ((ℝ_fld freeLMod 𝐼) toNrmGrp (𝑥 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ↦ (√‘(𝑥(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑥)))))
8	7	fveq2d 6910	. 2 ⊢ (𝐼 ∈ 𝑉 → (dim‘𝐻) = (dim‘((ℝ_fld freeLMod 𝐼) toNrmGrp (𝑥 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ↦ (√‘(𝑥(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑥))))))
9		resubdrg 21643	. . . . 5 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) ∧ ℝ_fld ∈ DivRing)
10	9	simpri 485	. . . 4 ⊢ ℝ_fld ∈ DivRing
11		eqid 2734	. . . . 5 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
12	11	frlmlvec 21798	. . . 4 ⊢ ((ℝ_fld ∈ DivRing ∧ 𝐼 ∈ 𝑉) → (ℝ_fld freeLMod 𝐼) ∈ LVec)
13	10, 12	mpan 690	. . 3 ⊢ (𝐼 ∈ 𝑉 → (ℝ_fld freeLMod 𝐼) ∈ LVec)
14	4	tcphex 25264	. . 3 ⊢ (𝑥 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ↦ (√‘(𝑥(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑥))) ∈ V
15		eqid 2734	. . . 4 ⊢ ((ℝ_fld freeLMod 𝐼) toNrmGrp (𝑥 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ↦ (√‘(𝑥(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑥)))) = ((ℝ_fld freeLMod 𝐼) toNrmGrp (𝑥 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ↦ (√‘(𝑥(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑥))))
16	15	tngdim 33640	. . 3 ⊢ (((ℝ_fld freeLMod 𝐼) ∈ LVec ∧ (𝑥 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ↦ (√‘(𝑥(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑥))) ∈ V) → (dim‘(ℝ_fld freeLMod 𝐼)) = (dim‘((ℝ_fld freeLMod 𝐼) toNrmGrp (𝑥 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ↦ (√‘(𝑥(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑥))))))
17	13, 14, 16	sylancl 586	. 2 ⊢ (𝐼 ∈ 𝑉 → (dim‘(ℝ_fld freeLMod 𝐼)) = (dim‘((ℝ_fld freeLMod 𝐼) toNrmGrp (𝑥 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ↦ (√‘(𝑥(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑥))))))
18	11	frlmdim 33638	. . 3 ⊢ ((ℝ_fld ∈ DivRing ∧ 𝐼 ∈ 𝑉) → (dim‘(ℝ_fld freeLMod 𝐼)) = (♯‘𝐼))
19	10, 18	mpan 690	. 2 ⊢ (𝐼 ∈ 𝑉 → (dim‘(ℝ_fld freeLMod 𝐼)) = (♯‘𝐼))
20	8, 17, 19	3eqtr2d 2780	1 ⊢ (𝐼 ∈ 𝑉 → (dim‘𝐻) = (♯‘𝐼))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 Vcvv 3477 ↦ cmpt 5230 ‘cfv 6562 (class class class)co 7430 ℝcr 11151 ♯chash 14365 √csqrt 15268 Basecbs 17244 ·_𝑖cip 17302 SubRingcsubrg 20585 DivRingcdr 20745 LVecclvec 21118 ℂ_fldccnfld 21381 ℝ_fldcrefld 21639 freeLMod cfrlm 21783 toNrmGrp ctng 24606 toℂPreHilctcph 25214 ℝ^crrx 25430 dimcldim 33625

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-reg 9629 ax-inf2 9678 ax-ac2 10500 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-rpss 7741 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-tpos 8249 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-oadd 8508 df-er 8743 df-map 8866 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-sup 9479 df-oi 9547 df-r1 9801 df-rank 9802 df-dju 9938 df-card 9976 df-acn 9979 df-ac 10153 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-xnn0 12597 df-z 12611 df-dec 12731 df-uz 12876 df-rp 13032 df-fz 13544 df-fzo 13691 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ocomp 17318 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-0g 17487 df-gsum 17488 df-prds 17493 df-pws 17495 df-mre 17630 df-mrc 17631 df-mri 17632 df-acs 17633 df-proset 18351 df-drs 18352 df-poset 18370 df-ipo 18585 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-mhm 18808 df-submnd 18809 df-grp 18966 df-minusg 18967 df-sbg 18968 df-mulg 19098 df-subg 19153 df-ghm 19243 df-cntz 19347 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-cring 20253 df-oppr 20350 df-dvdsr 20373 df-unit 20374 df-invr 20404 df-dvr 20417 df-nzr 20529 df-subrng 20562 df-subrg 20586 df-drng 20747 df-lmod 20876 df-lss 20947 df-lsp 20987 df-lmhm 21038 df-lbs 21091 df-lvec 21119 df-sra 21189 df-rgmod 21190 df-cnfld 21382 df-refld 21640 df-dsmm 21769 df-frlm 21784 df-uvc 21820 df-tng 24612 df-tcph 25216 df-rrx 25432 df-dim 33626

This theorem is referenced by: (None)

W3C validator