rrxtopn - Mathbox for Glauco Siliprandi

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Theorem rrxtopn 41438

Description: The topology of the generalized real Euclidean space. (Contributed by Glauco Siliprandi, 24-Dec-2020.)

**Hypothesis**
Ref	Expression
rrxtopn.1	⊢ (𝜑 → 𝐼 ∈ 𝑉)

**Assertion**
Ref	Expression
rrxtopn	⊢ (𝜑 → (TopOpen‘(ℝ^‘𝐼)) = (MetOpen‘(𝑓 ∈ (Base‘(ℝ^‘𝐼)), 𝑔 ∈ (Base‘(ℝ^‘𝐼)) ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)))))))

Distinct variable groups: 𝑓,𝐼,𝑔,𝑥 𝑓,𝑉,𝑔,𝑥 Allowed substitution hints: 𝜑(𝑥,𝑓,𝑔)

**Proof of Theorem rrxtopn**
Step	Hyp	Ref	Expression
1		rrxtopn.1	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
2		eqid 2778	. . . . . 6 ⊢ (ℝ^‘𝐼) = (ℝ^‘𝐼)
3	2	rrxval 23597	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (ℝ^‘𝐼) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
4	1, 3	syl 17	. . . 4 ⊢ (𝜑 → (ℝ^‘𝐼) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
5	4	fveq2d 6452	. . 3 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝐼)) = (TopOpen‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
6		ovex 6956	. . . . 5 ⊢ (ℝ_fld freeLMod 𝐼) ∈ V
7		eqid 2778	. . . . . 6 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
8		eqid 2778	. . . . . 6 ⊢ (dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = (dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
9		eqid 2778	. . . . . 6 ⊢ (TopOpen‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = (TopOpen‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
10	7, 8, 9	tcphtopn 23436	. . . . 5 ⊢ ((ℝ_fld freeLMod 𝐼) ∈ V → (TopOpen‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = (MetOpen‘(dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))))
11	6, 10	ax-mp 5	. . . 4 ⊢ (TopOpen‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = (MetOpen‘(dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
12	11	a1i 11	. . 3 ⊢ (𝜑 → (TopOpen‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = (MetOpen‘(dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))))
13	4	eqcomd 2784	. . . . 5 ⊢ (𝜑 → (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (ℝ^‘𝐼))
14	13	fveq2d 6452	. . . 4 ⊢ (𝜑 → (dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = (dist‘(ℝ^‘𝐼)))
15	14	fveq2d 6452	. . 3 ⊢ (𝜑 → (MetOpen‘(dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))) = (MetOpen‘(dist‘(ℝ^‘𝐼))))
16	5, 12, 15	3eqtrd 2818	. 2 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝐼)) = (MetOpen‘(dist‘(ℝ^‘𝐼))))
17		eqid 2778	. . . . . 6 ⊢ (Base‘(ℝ^‘𝐼)) = (Base‘(ℝ^‘𝐼))
18	2, 17	rrxds 23603	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (𝑓 ∈ (Base‘(ℝ^‘𝐼)), 𝑔 ∈ (Base‘(ℝ^‘𝐼)) ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))))) = (dist‘(ℝ^‘𝐼)))
19	1, 18	syl 17	. . . 4 ⊢ (𝜑 → (𝑓 ∈ (Base‘(ℝ^‘𝐼)), 𝑔 ∈ (Base‘(ℝ^‘𝐼)) ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))))) = (dist‘(ℝ^‘𝐼)))
20	19	eqcomd 2784	. . 3 ⊢ (𝜑 → (dist‘(ℝ^‘𝐼)) = (𝑓 ∈ (Base‘(ℝ^‘𝐼)), 𝑔 ∈ (Base‘(ℝ^‘𝐼)) ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))))))
21	20	fveq2d 6452	. 2 ⊢ (𝜑 → (MetOpen‘(dist‘(ℝ^‘𝐼))) = (MetOpen‘(𝑓 ∈ (Base‘(ℝ^‘𝐼)), 𝑔 ∈ (Base‘(ℝ^‘𝐼)) ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)))))))
22	16, 21	eqtrd 2814	1 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝐼)) = (MetOpen‘(𝑓 ∈ (Base‘(ℝ^‘𝐼)), 𝑔 ∈ (Base‘(ℝ^‘𝐼)) ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)))))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 Vcvv 3398 ↦ cmpt 4967 ‘cfv 6137 (class class class)co 6924 ↦ cmpt2 6926 − cmin 10608 2c2 11434 ↑cexp 13182 √csqrt 14384 Basecbs 16259 distcds 16351 TopOpenctopn 16472 Σ_g cgsu 16491 MetOpencmopn 20136 ℝ_fldcrefld 20351 freeLMod cfrlm 20493 toℂPreHilctcph 23378 ℝ^crrx 23593

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 ax-addf 10353 ax-mulf 10354

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-of 7176 df-om 7346 df-1st 7447 df-2nd 7448 df-supp 7579 df-tpos 7636 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-map 8144 df-ixp 8197 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fsupp 8566 df-sup 8638 df-inf 8639 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11035 df-nn 11379 df-2 11442 df-3 11443 df-4 11444 df-5 11445 df-6 11446 df-7 11447 df-8 11448 df-9 11449 df-n0 11647 df-z 11733 df-dec 11850 df-uz 11997 df-q 12100 df-rp 12142 df-xneg 12261 df-xadd 12262 df-xmul 12263 df-fz 12648 df-seq 13124 df-exp 13183 df-cj 14250 df-re 14251 df-im 14252 df-sqrt 14386 df-abs 14387 df-struct 16261 df-ndx 16262 df-slot 16263 df-base 16265 df-sets 16266 df-ress 16267 df-plusg 16355 df-mulr 16356 df-starv 16357 df-sca 16358 df-vsca 16359 df-ip 16360 df-tset 16361 df-ple 16362 df-ds 16364 df-unif 16365 df-hom 16366 df-cco 16367 df-rest 16473 df-topn 16474 df-0g 16492 df-topgen 16494 df-prds 16498 df-pws 16500 df-mgm 17632 df-sgrp 17674 df-mnd 17685 df-mhm 17725 df-grp 17816 df-minusg 17817 df-sbg 17818 df-subg 17979 df-ghm 18046 df-cmn 18585 df-mgp 18881 df-ur 18893 df-ring 18940 df-cring 18941 df-oppr 19014 df-dvdsr 19032 df-unit 19033 df-invr 19063 df-dvr 19074 df-rnghom 19108 df-drng 19145 df-field 19146 df-subrg 19174 df-staf 19241 df-srng 19242 df-lmod 19261 df-lss 19329 df-sra 19573 df-rgmod 19574 df-psmet 20138 df-xmet 20139 df-bl 20141 df-mopn 20142 df-cnfld 20147 df-refld 20352 df-dsmm 20479 df-frlm 20494 df-top 21110 df-topon 21127 df-bases 21162 df-nm 22799 df-tng 22801 df-tcph 23380 df-rrx 23595

This theorem is referenced by: rrxtopnfi 41441

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