rrxdim - Mathbox for Thierry Arnoux

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Theorem rrxdim 33948

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 25514	. . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
3		eqid 2769	. . . . 5 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
4		eqid 2769	. . . . 5 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
5		eqid 2769	. . . . 5 ⊢ (·_𝑖‘(ℝ_fld freeLMod 𝐼)) = (·_𝑖‘(ℝ_fld freeLMod 𝐼))
6	3, 4, 5	tcphval 25345	. . . 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 2820	. . 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 6886	. 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 21726	. . . . 5 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) ∧ ℝ_fld ∈ DivRing)
10	9	simpri 490	. . . 4 ⊢ ℝ_fld ∈ DivRing
11		eqid 2769	. . . . 5 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
12	11	frlmlvec 21879	. . . 4 ⊢ ((ℝ_fld ∈ DivRing ∧ 𝐼 ∈ 𝑉) → (ℝ_fld freeLMod 𝐼) ∈ LVec)
13	10, 12	mpan 702	. . 3 ⊢ (𝐼 ∈ 𝑉 → (ℝ_fld freeLMod 𝐼) ∈ LVec)
14	4	tcphex 25344	. . 3 ⊢ (𝑥 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ↦ (√‘(𝑥(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑥))) ∈ V
15		eqid 2769	. . . 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 33947	. . 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 597	. 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 33945	. . 3 ⊢ ((ℝ_fld ∈ DivRing ∧ 𝐼 ∈ 𝑉) → (dim‘(ℝ_fld freeLMod 𝐼)) = (♯‘𝐼))
19	10, 18	mpan 702	. 2 ⊢ (𝐼 ∈ 𝑉 → (dim‘(ℝ_fld freeLMod 𝐼)) = (♯‘𝐼))
20	8, 17, 19	3eqtr2d 2810	1 ⊢ (𝐼 ∈ 𝑉 → (dim‘𝐻) = (♯‘𝐼))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1567 ∈ wcel 2149 Vcvv 3463 ↦ cmpt 5196 ‘cfv 6537 (class class class)co 7411 ℝcr 11098 ♯chash 14365 √csqrt 15283 Basecbs 17268 ·_𝑖cip 17314 SubRingcsubrg 20653 DivRingcdr 20812 LVecclvec 21200 ℂ_fldccnfld 21490 ℝ_fldcrefld 21722 freeLMod cfrlm 21864 toNrmGrp ctng 24703 toℂPreHilctcph 25294 ℝ^crrx 25510 dimcldim 33933

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-reg 9553 ax-inf2 9609 ax-ac2 10446 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176 ax-pre-sup 11177 ax-addf 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-iin 4963 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-rpss 7721 df-om 7862 df-1st 7985 df-2nd 7986 df-supp 8156 df-tpos 8221 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-2o 8453 df-oadd 8456 df-er 8693 df-map 8825 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9321 df-sup 9401 df-oi 9471 df-r1 9735 df-rank 9736 df-dju 9886 df-card 9924 df-acn 9927 df-ac 10099 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-div 11871 df-nn 12233 df-2 12302 df-3 12303 df-4 12304 df-5 12305 df-6 12306 df-7 12307 df-8 12308 df-9 12309 df-n0 12504 df-xnn0 12577 df-z 12591 df-dec 12711 df-uz 12862 df-rp 13016 df-fz 13535 df-fzo 13682 df-seq 14037 df-exp 14097 df-hash 14366 df-cj 15149 df-re 15150 df-im 15151 df-sqrt 15285 df-abs 15286 df-struct 17206 df-sets 17223 df-slot 17241 df-ndx 17253 df-base 17269 df-ress 17290 df-plusg 17322 df-mulr 17323 df-starv 17324 df-sca 17325 df-vsca 17326 df-ip 17327 df-tset 17328 df-ple 17329 df-ocomp 17330 df-ds 17331 df-unif 17332 df-hom 17333 df-cco 17334 df-0g 17493 df-gsum 17494 df-prds 17499 df-pws 17501 df-mre 17637 df-mrc 17638 df-mri 17639 df-acs 17640 df-proset 18349 df-drs 18350 df-poset 18368 df-ipo 18583 df-mgm 18697 df-sgrp 18776 df-mnd 18792 df-mhm 18840 df-submnd 18841 df-grp 19002 df-minusg 19003 df-sbg 19004 df-mulg 19133 df-subg 19188 df-ghm 19283 df-cntz 19386 df-cmn 19851 df-abl 19852 df-mgp 20216 df-rng 20230 df-ur 20263 df-ring 20316 df-cring 20317 df-oppr 20418 df-dvdsr 20438 df-unit 20439 df-invr 20469 df-dvr 20482 df-nzr 20595 df-subrng 20630 df-subrg 20654 df-drng 20814 df-lmod 20960 df-lss 21030 df-lsp 21070 df-lmhm 21120 df-lbs 21173 df-lvec 21201 df-sra 21271 df-rgmod 21272 df-cnfld 21491 df-refld 21723 df-dsmm 21850 df-frlm 21865 df-uvc 21901 df-tng 24709 df-tcph 25296 df-rrx 25512 df-dim 33934

This theorem is referenced by: (None)

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