qndenserrnopnlem - Mathbox for Glauco Siliprandi

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Theorem qndenserrnopnlem 46279

Description: n-dimensional rational numbers are dense in the space of n-dimensional real numbers, with respect to the n-dimensional standard topology. (Contributed by Glauco Siliprandi, 24-Dec-2020.)

**Hypotheses**
Ref	Expression
qndenserrnopnlem.i	⊢ (𝜑 → 𝐼 ∈ Fin)
qndenserrnopnlem.j	⊢ 𝐽 = (TopOpen‘(ℝ^‘𝐼))
qndenserrnopnlem.v	⊢ (𝜑 → 𝑉 ∈ 𝐽)
qndenserrnopnlem.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
qndenserrnopnlem.d	⊢ 𝐷 = (dist‘(ℝ^‘𝐼))

**Assertion**
Ref	Expression
qndenserrnopnlem	⊢ (𝜑 → ∃𝑦 ∈ (ℚ ↑_m 𝐼)𝑦 ∈ 𝑉)

Distinct variable groups: 𝑦,𝐷 𝑦,𝐼 𝑦,𝑉 𝑦,𝑋 𝜑,𝑦 Allowed substitution hint: 𝐽(𝑦)

Proof of Theorem qndenserrnopnlem Dummy variables 𝑒 𝑓 𝑔 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		qndenserrnopnlem.i	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ Fin)
2		qndenserrnopnlem.d	. . . . . 6 ⊢ 𝐷 = (dist‘(ℝ^‘𝐼))
3	2	rrxmetfi 25328	. . . . 5 ⊢ (𝐼 ∈ Fin → 𝐷 ∈ (Met‘(ℝ ↑_m 𝐼)))
4	1, 3	syl 17	. . . 4 ⊢ (𝜑 → 𝐷 ∈ (Met‘(ℝ ↑_m 𝐼)))
5		metxmet 24238	. . . 4 ⊢ (𝐷 ∈ (Met‘(ℝ ↑_m 𝐼)) → 𝐷 ∈ (∞Met‘(ℝ ↑_m 𝐼)))
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘(ℝ ↑_m 𝐼)))
7		qndenserrnopnlem.v	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ 𝐽)
8		qndenserrnopnlem.j	. . . . 5 ⊢ 𝐽 = (TopOpen‘(ℝ^‘𝐼))
9	7, 8	eleqtrdi 2838	. . . 4 ⊢ (𝜑 → 𝑉 ∈ (TopOpen‘(ℝ^‘𝐼)))
10	1	rrxtopnfi 46269	. . . . 5 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝐼)) = (MetOpen‘(𝑓 ∈ (ℝ ↑_m 𝐼), 𝑔 ∈ (ℝ ↑_m 𝐼) ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))))
11	2	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐷 = (dist‘(ℝ^‘𝐼)))
12		eqid 2729	. . . . . . . . 9 ⊢ (ℝ^‘𝐼) = (ℝ^‘𝐼)
13		eqid 2729	. . . . . . . . 9 ⊢ (ℝ ↑_m 𝐼) = (ℝ ↑_m 𝐼)
14	12, 13	rrxdsfi 25327	. . . . . . . 8 ⊢ (𝐼 ∈ Fin → (dist‘(ℝ^‘𝐼)) = (𝑓 ∈ (ℝ ↑_m 𝐼), 𝑔 ∈ (ℝ ↑_m 𝐼) ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))))
15	1, 14	syl 17	. . . . . . 7 ⊢ (𝜑 → (dist‘(ℝ^‘𝐼)) = (𝑓 ∈ (ℝ ↑_m 𝐼), 𝑔 ∈ (ℝ ↑_m 𝐼) ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))))
16	11, 15	eqtr2d 2765	. . . . . 6 ⊢ (𝜑 → (𝑓 ∈ (ℝ ↑_m 𝐼), 𝑔 ∈ (ℝ ↑_m 𝐼) ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))) = 𝐷)
17	16	fveq2d 6830	. . . . 5 ⊢ (𝜑 → (MetOpen‘(𝑓 ∈ (ℝ ↑_m 𝐼), 𝑔 ∈ (ℝ ↑_m 𝐼) ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) = (MetOpen‘𝐷))
18	10, 17	eqtrd 2764	. . . 4 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝐼)) = (MetOpen‘𝐷))
19	9, 18	eleqtrd 2830	. . 3 ⊢ (𝜑 → 𝑉 ∈ (MetOpen‘𝐷))
20		qndenserrnopnlem.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
21		eqid 2729	. . . 4 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
22	21	mopni2 24397	. . 3 ⊢ ((𝐷 ∈ (∞Met‘(ℝ ↑_m 𝐼)) ∧ 𝑉 ∈ (MetOpen‘𝐷) ∧ 𝑋 ∈ 𝑉) → ∃𝑒 ∈ ℝ⁺ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉)
23	6, 19, 20, 22	syl3anc 1373	. 2 ⊢ (𝜑 → ∃𝑒 ∈ ℝ⁺ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉)
24	1	3ad2ant1 1133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) → 𝐼 ∈ Fin)
25		rrxtps 46268	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ Fin → (ℝ^‘𝐼) ∈ TopSp)
26	1, 25	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ^‘𝐼) ∈ TopSp)
27		eqid 2729	. . . . . . . . . . . 12 ⊢ (Base‘(ℝ^‘𝐼)) = (Base‘(ℝ^‘𝐼))
28	27, 8	istps 22837	. . . . . . . . . . 11 ⊢ ((ℝ^‘𝐼) ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘(ℝ^‘𝐼))))
29	26, 28	sylib 218	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(Base‘(ℝ^‘𝐼))))
30	1, 12, 27	rrxbasefi 25326	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘(ℝ^‘𝐼)) = (ℝ ↑_m 𝐼))
31	30	fveq2d 6830	. . . . . . . . . 10 ⊢ (𝜑 → (TopOn‘(Base‘(ℝ^‘𝐼))) = (TopOn‘(ℝ ↑_m 𝐼)))
32	29, 31	eleqtrd 2830	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(ℝ ↑_m 𝐼)))
33		toponss 22830	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘(ℝ ↑_m 𝐼)) ∧ 𝑉 ∈ 𝐽) → 𝑉 ⊆ (ℝ ↑_m 𝐼))
34	32, 7, 33	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ⊆ (ℝ ↑_m 𝐼))
35	34, 20	sseldd 3938	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (ℝ ↑_m 𝐼))
36	35	3ad2ant1 1133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) → 𝑋 ∈ (ℝ ↑_m 𝐼))
37		simp2 1137	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) → 𝑒 ∈ ℝ⁺)
38	24, 36, 2, 37	qndenserrnbl 46277	. . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) → ∃𝑦 ∈ (ℚ ↑_m 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝑒))
39		ssel 3931	. . . . . . . 8 ⊢ ((𝑋(ball‘𝐷)𝑒) ⊆ 𝑉 → (𝑦 ∈ (𝑋(ball‘𝐷)𝑒) → 𝑦 ∈ 𝑉))
40	39	adantr 480	. . . . . . 7 ⊢ (((𝑋(ball‘𝐷)𝑒) ⊆ 𝑉 ∧ 𝑦 ∈ (ℚ ↑_m 𝐼)) → (𝑦 ∈ (𝑋(ball‘𝐷)𝑒) → 𝑦 ∈ 𝑉))
41	40	3ad2antl3 1188	. . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) ∧ 𝑦 ∈ (ℚ ↑_m 𝐼)) → (𝑦 ∈ (𝑋(ball‘𝐷)𝑒) → 𝑦 ∈ 𝑉))
42	41	reximdva 3142	. . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) → (∃𝑦 ∈ (ℚ ↑_m 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝑒) → ∃𝑦 ∈ (ℚ ↑_m 𝐼)𝑦 ∈ 𝑉))
43	38, 42	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) → ∃𝑦 ∈ (ℚ ↑_m 𝐼)𝑦 ∈ 𝑉)
44	43	3exp 1119	. . 3 ⊢ (𝜑 → (𝑒 ∈ ℝ⁺ → ((𝑋(ball‘𝐷)𝑒) ⊆ 𝑉 → ∃𝑦 ∈ (ℚ ↑_m 𝐼)𝑦 ∈ 𝑉)))
45	44	rexlimdv 3128	. 2 ⊢ (𝜑 → (∃𝑒 ∈ ℝ⁺ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉 → ∃𝑦 ∈ (ℚ ↑_m 𝐼)𝑦 ∈ 𝑉))
46	23, 45	mpd 15	1 ⊢ (𝜑 → ∃𝑦 ∈ (ℚ ↑_m 𝐼)𝑦 ∈ 𝑉)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∃wrex 3053 ⊆ wss 3905 ‘cfv 6486 (class class class)co 7353 ∈ cmpo 7355 ↑_m cmap 8760 Fincfn 8879 ℝcr 11027 − cmin 11365 2c2 12201 ℚcq 12867 ℝ⁺crp 12911 ↑cexp 13986 √csqrt 15158 Σcsu 15611 Basecbs 17138 distcds 17188 TopOpenctopn 17343 ∞Metcxmet 21264 Metcmet 21265 ballcbl 21266 MetOpencmopn 21269 TopOnctopon 22813 TopSpctps 22835 ℝ^crrx 25299

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-inf2 9556 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-int 4900 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-se 5577 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-isom 6495 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-oi 9421 df-card 9854 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12610 df-uz 12754 df-q 12868 df-rp 12912 df-xneg 13032 df-xadd 13033 df-xmul 13034 df-ioo 13270 df-ico 13272 df-fz 13429 df-fzo 13576 df-seq 13927 df-exp 13987 df-hash 14256 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-clim 15413 df-sum 15612 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-hom 17203 df-cco 17204 df-rest 17344 df-topn 17345 df-0g 17363 df-gsum 17364 df-topgen 17365 df-prds 17369 df-pws 17371 df-mgm 18532 df-sgrp 18611 df-mnd 18627 df-mhm 18675 df-submnd 18676 df-grp 18833 df-minusg 18834 df-sbg 18835 df-subg 19020 df-ghm 19110 df-cntz 19214 df-cmn 19679 df-abl 19680 df-mgp 20044 df-rng 20056 df-ur 20085 df-ring 20138 df-cring 20139 df-oppr 20240 df-dvdsr 20260 df-unit 20261 df-invr 20291 df-dvr 20304 df-rhm 20375 df-subrng 20449 df-subrg 20473 df-drng 20634 df-field 20635 df-abv 20712 df-staf 20742 df-srng 20743 df-lmod 20783 df-lss 20853 df-lmhm 20944 df-lvec 21025 df-sra 21095 df-rgmod 21096 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-cnfld 21280 df-refld 21530 df-phl 21551 df-dsmm 21657 df-frlm 21672 df-top 22797 df-topon 22814 df-topsp 22836 df-bases 22849 df-xms 24224 df-ms 24225 df-nm 24486 df-ngp 24487 df-tng 24488 df-nrg 24489 df-nlm 24490 df-clm 24979 df-cph 25084 df-tcph 25085 df-rrx 25301

This theorem is referenced by: qndenserrnopn 46280

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