rrxdim - Mathbox for Thierry Arnoux

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Theorem rrxdim 33610

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 25287	. . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
3		eqid 2729	. . . . 5 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
4		eqid 2729	. . . . 5 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
5		eqid 2729	. . . . 5 ⊢ (·_𝑖‘(ℝ_fld freeLMod 𝐼)) = (·_𝑖‘(ℝ_fld freeLMod 𝐼))
6	3, 4, 5	tcphval 25118	. . . 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 2780	. . 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 6862	. 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 21517	. . . . 5 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) ∧ ℝ_fld ∈ DivRing)
10	9	simpri 485	. . . 4 ⊢ ℝ_fld ∈ DivRing
11		eqid 2729	. . . . 5 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
12	11	frlmlvec 21670	. . . 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 25117	. . 3 ⊢ (𝑥 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ↦ (√‘(𝑥(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑥))) ∈ V
15		eqid 2729	. . . 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 33609	. . 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 33607	. . 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 2770	1 ⊢ (𝐼 ∈ 𝑉 → (dim‘𝐻) = (♯‘𝐼))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 Vcvv 3447 ↦ cmpt 5188 ‘cfv 6511 (class class class)co 7387 ℝcr 11067 ♯chash 14295 √csqrt 15199 Basecbs 17179 ·_𝑖cip 17225 SubRingcsubrg 20478 DivRingcdr 20638 LVecclvec 21009 ℂ_fldccnfld 21264 ℝ_fldcrefld 21513 freeLMod cfrlm 21655 toNrmGrp ctng 24466 toℂPreHilctcph 25067 ℝ^crrx 25283 dimcldim 33594

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-reg 9545 ax-inf2 9594 ax-ac2 10416 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-rpss 7699 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-tpos 8205 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-oadd 8438 df-er 8671 df-map 8801 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-sup 9393 df-oi 9463 df-r1 9717 df-rank 9718 df-dju 9854 df-card 9892 df-acn 9895 df-ac 10069 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-xnn0 12516 df-z 12530 df-dec 12650 df-uz 12794 df-rp 12952 df-fz 13469 df-fzo 13616 df-seq 13967 df-exp 14027 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ocomp 17241 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-0g 17404 df-gsum 17405 df-prds 17410 df-pws 17412 df-mre 17547 df-mrc 17548 df-mri 17549 df-acs 17550 df-proset 18255 df-drs 18256 df-poset 18274 df-ipo 18487 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18710 df-submnd 18711 df-grp 18868 df-minusg 18869 df-sbg 18870 df-mulg 19000 df-subg 19055 df-ghm 19145 df-cntz 19249 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-ring 20144 df-cring 20145 df-oppr 20246 df-dvdsr 20266 df-unit 20267 df-invr 20297 df-dvr 20310 df-nzr 20422 df-subrng 20455 df-subrg 20479 df-drng 20640 df-lmod 20768 df-lss 20838 df-lsp 20878 df-lmhm 20929 df-lbs 20982 df-lvec 21010 df-sra 21080 df-rgmod 21081 df-cnfld 21265 df-refld 21514 df-dsmm 21641 df-frlm 21656 df-uvc 21692 df-tng 24472 df-tcph 25069 df-rrx 25285 df-dim 33595

This theorem is referenced by: (None)

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