rrxtopn - Mathbox for Glauco Siliprandi

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Theorem rrxtopn 46998

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 2763	. . . . . 6 ⊢ (ℝ^‘𝐼) = (ℝ^‘𝐼)
3	2	rrxval 25546	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (ℝ^‘𝐼) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
4	1, 3	syl 18	. . . 4 ⊢ (𝜑 → (ℝ^‘𝐼) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
5	4	fveq2d 6885	. . 3 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝐼)) = (TopOpen‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
6		ovex 7443	. . . . 5 ⊢ (ℝ_fld freeLMod 𝐼) ∈ V
7		eqid 2763	. . . . . 6 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
8		eqid 2763	. . . . . 6 ⊢ (dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = (dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
9		eqid 2763	. . . . . 6 ⊢ (TopOpen‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = (TopOpen‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
10	7, 8, 9	tcphtopn 25385	. . . . 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 2769	. . . . 5 ⊢ (𝜑 → (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (ℝ^‘𝐼))
14	13	fveq2d 6885	. . . 4 ⊢ (𝜑 → (dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = (dist‘(ℝ^‘𝐼)))
15	14	fveq2d 6885	. . 3 ⊢ (𝜑 → (MetOpen‘(dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))) = (MetOpen‘(dist‘(ℝ^‘𝐼))))
16	5, 12, 15	3eqtrd 2802	. 2 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝐼)) = (MetOpen‘(dist‘(ℝ^‘𝐼))))
17		eqid 2763	. . . . . 6 ⊢ (Base‘(ℝ^‘𝐼)) = (Base‘(ℝ^‘𝐼))
18	2, 17	rrxds 25552	. . . . 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 18	. . . 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 2769	. . 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 6885	. 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 2798	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 1570 ∈ wcel 2143 Vcvv 3455 ↦ cmpt 5192 ‘cfv 6536 (class class class)co 7410 ∈ cmpo 7412 − cmin 11436 2c2 12290 ↑cexp 14093 √csqrt 15280 Basecbs 17264 distcds 17314 TopOpenctopn 17469 Σ_g cgsu 17488 MetOpencmopn 21512 ℝ_fldcrefld 21754 freeLMod cfrlm 21896 toℂPreHilctcph 25326 ℝ^crrx 25542

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173 ax-addf 11174 ax-mulf 11175

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-tpos 8218 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-er 8690 df-map 8822 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-sup 9398 df-inf 9399 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-7 12303 df-8 12304 df-9 12305 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12858 df-q 12968 df-rp 13012 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-fz 13531 df-seq 14034 df-exp 14094 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-struct 17202 df-sets 17219 df-slot 17237 df-ndx 17249 df-base 17265 df-ress 17286 df-plusg 17318 df-mulr 17319 df-starv 17320 df-sca 17321 df-vsca 17322 df-ip 17323 df-tset 17324 df-ple 17325 df-ds 17327 df-unif 17328 df-hom 17329 df-cco 17330 df-rest 17470 df-topn 17471 df-0g 17489 df-topgen 17491 df-prds 17495 df-pws 17497 df-mgm 18693 df-sgrp 18772 df-mnd 18788 df-mhm 18836 df-grp 18998 df-minusg 18999 df-sbg 19000 df-subg 19184 df-ghm 19279 df-cmn 19847 df-abl 19848 df-mgp 20212 df-rng 20226 df-ur 20259 df-ring 20312 df-cring 20313 df-oppr 20415 df-dvdsr 20435 df-unit 20436 df-invr 20466 df-dvr 20479 df-rhm 20550 df-subrng 20645 df-subrg 20669 df-drng 20829 df-field 20830 df-staf 20942 df-srng 20943 df-lmod 20983 df-lss 21053 df-sra 21294 df-rgmod 21295 df-psmet 21514 df-xmet 21515 df-bl 21517 df-mopn 21518 df-cnfld 21523 df-refld 21755 df-dsmm 21882 df-frlm 21897 df-top 23051 df-topon 23068 df-bases 23103 df-nm 24739 df-tng 24741 df-tcph 25328 df-rrx 25544

This theorem is referenced by: rrxtopnfi 47001

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