rrxdim - Mathbox for Thierry Arnoux

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Theorem rrxdim 33758

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 25354	. . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
3		eqid 2736	. . . . 5 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
4		eqid 2736	. . . . 5 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
5		eqid 2736	. . . . 5 ⊢ (·_𝑖‘(ℝ_fld freeLMod 𝐼)) = (·_𝑖‘(ℝ_fld freeLMod 𝐼))
6	3, 4, 5	tcphval 25185	. . . 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 2787	. . 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 6844	. 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 21588	. . . . 5 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) ∧ ℝ_fld ∈ DivRing)
10	9	simpri 485	. . . 4 ⊢ ℝ_fld ∈ DivRing
11		eqid 2736	. . . . 5 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
12	11	frlmlvec 21741	. . . 4 ⊢ ((ℝ_fld ∈ DivRing ∧ 𝐼 ∈ 𝑉) → (ℝ_fld freeLMod 𝐼) ∈ LVec)
13	10, 12	mpan 691	. . 3 ⊢ (𝐼 ∈ 𝑉 → (ℝ_fld freeLMod 𝐼) ∈ LVec)
14	4	tcphex 25184	. . 3 ⊢ (𝑥 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ↦ (√‘(𝑥(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑥))) ∈ V
15		eqid 2736	. . . 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 33757	. . 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 587	. 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 33755	. . 3 ⊢ ((ℝ_fld ∈ DivRing ∧ 𝐼 ∈ 𝑉) → (dim‘(ℝ_fld freeLMod 𝐼)) = (♯‘𝐼))
19	10, 18	mpan 691	. 2 ⊢ (𝐼 ∈ 𝑉 → (dim‘(ℝ_fld freeLMod 𝐼)) = (♯‘𝐼))
20	8, 17, 19	3eqtr2d 2777	1 ⊢ (𝐼 ∈ 𝑉 → (dim‘𝐻) = (♯‘𝐼))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 Vcvv 3429 ↦ cmpt 5166 ‘cfv 6498 (class class class)co 7367 ℝcr 11037 ♯chash 14292 √csqrt 15195 Basecbs 17179 ·_𝑖cip 17225 SubRingcsubrg 20546 DivRingcdr 20706 LVecclvec 21097 ℂ_fldccnfld 21352 ℝ_fldcrefld 21584 freeLMod cfrlm 21726 toNrmGrp ctng 24543 toℂPreHilctcph 25134 ℝ^crrx 25350 dimcldim 33743

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-reg 9507 ax-inf2 9562 ax-ac2 10385 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-rpss 7677 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-tpos 8176 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-er 8643 df-map 8775 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-sup 9355 df-oi 9425 df-r1 9688 df-rank 9689 df-dju 9825 df-card 9863 df-acn 9866 df-ac 10038 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-xnn0 12511 df-z 12525 df-dec 12645 df-uz 12789 df-rp 12943 df-fz 13462 df-fzo 13609 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 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 17548 df-mrc 17549 df-mri 17550 df-acs 17551 df-proset 18260 df-drs 18261 df-poset 18279 df-ipo 18494 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-mhm 18751 df-submnd 18752 df-grp 18912 df-minusg 18913 df-sbg 18914 df-mulg 19044 df-subg 19099 df-ghm 19188 df-cntz 19292 df-cmn 19757 df-abl 19758 df-mgp 20122 df-rng 20134 df-ur 20163 df-ring 20216 df-cring 20217 df-oppr 20317 df-dvdsr 20337 df-unit 20338 df-invr 20368 df-dvr 20381 df-nzr 20490 df-subrng 20523 df-subrg 20547 df-drng 20708 df-lmod 20857 df-lss 20927 df-lsp 20967 df-lmhm 21017 df-lbs 21070 df-lvec 21098 df-sra 21168 df-rgmod 21169 df-cnfld 21353 df-refld 21585 df-dsmm 21712 df-frlm 21727 df-uvc 21763 df-tng 24549 df-tcph 25136 df-rrx 25352 df-dim 33744

This theorem is referenced by: (None)

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