rrxtopn - Mathbox for Glauco Siliprandi

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Theorem rrxtopn 46285

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 2729	. . . . . 6 ⊢ (ℝ^‘𝐼) = (ℝ^‘𝐼)
3	2	rrxval 25304	. . . . 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 6830	. . 3 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝐼)) = (TopOpen‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
6		ovex 7386	. . . . 5 ⊢ (ℝ_fld freeLMod 𝐼) ∈ V
7		eqid 2729	. . . . . 6 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
8		eqid 2729	. . . . . 6 ⊢ (dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = (dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
9		eqid 2729	. . . . . 6 ⊢ (TopOpen‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = (TopOpen‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
10	7, 8, 9	tcphtopn 25143	. . . . 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 2735	. . . . 5 ⊢ (𝜑 → (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (ℝ^‘𝐼))
14	13	fveq2d 6830	. . . 4 ⊢ (𝜑 → (dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = (dist‘(ℝ^‘𝐼)))
15	14	fveq2d 6830	. . 3 ⊢ (𝜑 → (MetOpen‘(dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))) = (MetOpen‘(dist‘(ℝ^‘𝐼))))
16	5, 12, 15	3eqtrd 2768	. 2 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝐼)) = (MetOpen‘(dist‘(ℝ^‘𝐼))))
17		eqid 2729	. . . . . 6 ⊢ (Base‘(ℝ^‘𝐼)) = (Base‘(ℝ^‘𝐼))
18	2, 17	rrxds 25310	. . . . 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 2735	. . 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 6830	. 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 2764	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 1540 ∈ wcel 2109 Vcvv 3438 ↦ cmpt 5176 ‘cfv 6486 (class class class)co 7353 ∈ cmpo 7355 − cmin 11366 2c2 12202 ↑cexp 13987 √csqrt 15159 Basecbs 17139 distcds 17189 TopOpenctopn 17344 Σ_g cgsu 17363 MetOpencmopn 21270 ℝ_fldcrefld 21530 freeLMod cfrlm 21672 toℂPreHilctcph 25084 ℝ^crrx 25300

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107 ax-mulf 11108

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 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 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7617 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-tpos 8166 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8632 df-map 8762 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9271 df-sup 9351 df-inf 9352 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-7 12215 df-8 12216 df-9 12217 df-n0 12404 df-z 12491 df-dec 12611 df-uz 12755 df-q 12869 df-rp 12913 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-fz 13430 df-seq 13928 df-exp 13988 df-cj 15025 df-re 15026 df-im 15027 df-sqrt 15161 df-abs 15162 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17140 df-ress 17161 df-plusg 17193 df-mulr 17194 df-starv 17195 df-sca 17196 df-vsca 17197 df-ip 17198 df-tset 17199 df-ple 17200 df-ds 17202 df-unif 17203 df-hom 17204 df-cco 17205 df-rest 17345 df-topn 17346 df-0g 17364 df-topgen 17366 df-prds 17370 df-pws 17372 df-mgm 18533 df-sgrp 18612 df-mnd 18628 df-mhm 18676 df-grp 18834 df-minusg 18835 df-sbg 18836 df-subg 19021 df-ghm 19111 df-cmn 19680 df-abl 19681 df-mgp 20045 df-rng 20057 df-ur 20086 df-ring 20139 df-cring 20140 df-oppr 20241 df-dvdsr 20261 df-unit 20262 df-invr 20292 df-dvr 20305 df-rhm 20376 df-subrng 20450 df-subrg 20474 df-drng 20635 df-field 20636 df-staf 20743 df-srng 20744 df-lmod 20784 df-lss 20854 df-sra 21096 df-rgmod 21097 df-psmet 21272 df-xmet 21273 df-bl 21275 df-mopn 21276 df-cnfld 21281 df-refld 21531 df-dsmm 21658 df-frlm 21673 df-top 22798 df-topon 22815 df-bases 22850 df-nm 24487 df-tng 24489 df-tcph 25086 df-rrx 25302

This theorem is referenced by: rrxtopnfi 46288

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