rrxdim - Mathbox for Thierry Arnoux

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Theorem rrxdim 31599

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 24456	. . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
3		eqid 2738	. . . . 5 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
4		eqid 2738	. . . . 5 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
5		eqid 2738	. . . . 5 ⊢ (·_𝑖‘(ℝ_fld freeLMod 𝐼)) = (·_𝑖‘(ℝ_fld freeLMod 𝐼))
6	3, 4, 5	tcphval 24287	. . . 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 2795	. . 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 6760	. 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 20725	. . . . 5 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) ∧ ℝ_fld ∈ DivRing)
10	9	simpri 485	. . . 4 ⊢ ℝ_fld ∈ DivRing
11		eqid 2738	. . . . 5 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
12	11	frlmlvec 20878	. . . 4 ⊢ ((ℝ_fld ∈ DivRing ∧ 𝐼 ∈ 𝑉) → (ℝ_fld freeLMod 𝐼) ∈ LVec)
13	10, 12	mpan 686	. . 3 ⊢ (𝐼 ∈ 𝑉 → (ℝ_fld freeLMod 𝐼) ∈ LVec)
14	4	tcphex 24286	. . 3 ⊢ (𝑥 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ↦ (√‘(𝑥(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑥))) ∈ V
15		eqid 2738	. . . 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 31598	. . 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 585	. 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 31596	. . 3 ⊢ ((ℝ_fld ∈ DivRing ∧ 𝐼 ∈ 𝑉) → (dim‘(ℝ_fld freeLMod 𝐼)) = (♯‘𝐼))
19	10, 18	mpan 686	. 2 ⊢ (𝐼 ∈ 𝑉 → (dim‘(ℝ_fld freeLMod 𝐼)) = (♯‘𝐼))
20	8, 17, 19	3eqtr2d 2784	1 ⊢ (𝐼 ∈ 𝑉 → (dim‘𝐻) = (♯‘𝐼))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2108 Vcvv 3422 ↦ cmpt 5153 ‘cfv 6418 (class class class)co 7255 ℝcr 10801 ♯chash 13972 √csqrt 14872 Basecbs 16840 ·_𝑖cip 16893 DivRingcdr 19906 SubRingcsubrg 19935 LVecclvec 20279 ℂ_fldccnfld 20510 ℝ_fldcrefld 20721 freeLMod cfrlm 20863 toNrmGrp ctng 23640 toℂPreHilctcph 24236 ℝ^crrx 24452 dimcldim 31586

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-reg 9281 ax-inf2 9329 ax-ac2 10150 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-rpss 7554 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-tpos 8013 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-oadd 8271 df-er 8456 df-map 8575 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-sup 9131 df-oi 9199 df-r1 9453 df-rank 9454 df-dju 9590 df-card 9628 df-acn 9631 df-ac 9803 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-xnn0 12236 df-z 12250 df-dec 12367 df-uz 12512 df-rp 12660 df-fz 13169 df-fzo 13312 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ocomp 16909 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-0g 17069 df-gsum 17070 df-prds 17075 df-pws 17077 df-mre 17212 df-mrc 17213 df-mri 17214 df-acs 17215 df-proset 17928 df-drs 17929 df-poset 17946 df-ipo 18161 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mhm 18345 df-submnd 18346 df-grp 18495 df-minusg 18496 df-sbg 18497 df-mulg 18616 df-subg 18667 df-ghm 18747 df-cntz 18838 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-dvr 19840 df-drng 19908 df-subrg 19937 df-lmod 20040 df-lss 20109 df-lsp 20149 df-lmhm 20199 df-lbs 20252 df-lvec 20280 df-sra 20349 df-rgmod 20350 df-nzr 20442 df-cnfld 20511 df-refld 20722 df-dsmm 20849 df-frlm 20864 df-uvc 20900 df-tng 23646 df-tcph 24238 df-rrx 24454 df-dim 31587

This theorem is referenced by: (None)

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