qndenserrnopnlem - Mathbox for Glauco Siliprandi

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Theorem qndenserrnopnlem 46218

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 25465	. . . . 5 ⊢ (𝐼 ∈ Fin → 𝐷 ∈ (Met‘(ℝ ↑_m 𝐼)))
4	1, 3	syl 17	. . . 4 ⊢ (𝜑 → 𝐷 ∈ (Met‘(ℝ ↑_m 𝐼)))
5		metxmet 24365	. . . 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 2854	. . . 4 ⊢ (𝜑 → 𝑉 ∈ (TopOpen‘(ℝ^‘𝐼)))
10	1	rrxtopnfi 46208	. . . . 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 2740	. . . . . . . . 9 ⊢ (ℝ^‘𝐼) = (ℝ^‘𝐼)
13		eqid 2740	. . . . . . . . 9 ⊢ (ℝ ↑_m 𝐼) = (ℝ ↑_m 𝐼)
14	12, 13	rrxdsfi 25464	. . . . . . . 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 2781	. . . . . 6 ⊢ (𝜑 → (𝑓 ∈ (ℝ ↑_m 𝐼), 𝑔 ∈ (ℝ ↑_m 𝐼) ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))) = 𝐷)
17	16	fveq2d 6924	. . . . 5 ⊢ (𝜑 → (MetOpen‘(𝑓 ∈ (ℝ ↑_m 𝐼), 𝑔 ∈ (ℝ ↑_m 𝐼) ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) = (MetOpen‘𝐷))
18	10, 17	eqtrd 2780	. . . 4 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝐼)) = (MetOpen‘𝐷))
19	9, 18	eleqtrd 2846	. . 3 ⊢ (𝜑 → 𝑉 ∈ (MetOpen‘𝐷))
20		qndenserrnopnlem.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
21		eqid 2740	. . . 4 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
22	21	mopni2 24527	. . 3 ⊢ ((𝐷 ∈ (∞Met‘(ℝ ↑_m 𝐼)) ∧ 𝑉 ∈ (MetOpen‘𝐷) ∧ 𝑋 ∈ 𝑉) → ∃𝑒 ∈ ℝ⁺ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉)
23	6, 19, 20, 22	syl3anc 1371	. 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 46207	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ Fin → (ℝ^‘𝐼) ∈ TopSp)
26	1, 25	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ^‘𝐼) ∈ TopSp)
27		eqid 2740	. . . . . . . . . . . 12 ⊢ (Base‘(ℝ^‘𝐼)) = (Base‘(ℝ^‘𝐼))
28	27, 8	istps 22961	. . . . . . . . . . 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 25463	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘(ℝ^‘𝐼)) = (ℝ ↑_m 𝐼))
31	30	fveq2d 6924	. . . . . . . . . 10 ⊢ (𝜑 → (TopOn‘(Base‘(ℝ^‘𝐼))) = (TopOn‘(ℝ ↑_m 𝐼)))
32	29, 31	eleqtrd 2846	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(ℝ ↑_m 𝐼)))
33		toponss 22954	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘(ℝ ↑_m 𝐼)) ∧ 𝑉 ∈ 𝐽) → 𝑉 ⊆ (ℝ ↑_m 𝐼))
34	32, 7, 33	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ⊆ (ℝ ↑_m 𝐼))
35	34, 20	sseldd 4009	. . . . . . 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 46216	. . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) → ∃𝑦 ∈ (ℚ ↑_m 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝑒))
39		ssel 4002	. . . . . . . 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 1187	. . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) ∧ 𝑦 ∈ (ℚ ↑_m 𝐼)) → (𝑦 ∈ (𝑋(ball‘𝐷)𝑒) → 𝑦 ∈ 𝑉))
42	41	reximdva 3174	. . . . 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 3159	. 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 1087 = wceq 1537 ∈ wcel 2108 ∃wrex 3076 ⊆ wss 3976 ‘cfv 6573 (class class class)co 7448 ∈ cmpo 7450 ↑_m cmap 8884 Fincfn 9003 ℝcr 11183 − cmin 11520 2c2 12348 ℚcq 13013 ℝ⁺crp 13057 ↑cexp 14112 √csqrt 15282 Σcsu 15734 Basecbs 17258 distcds 17320 TopOpenctopn 17481 ∞Metcxmet 21372 Metcmet 21373 ballcbl 21374 MetOpencmopn 21377 TopOnctopon 22937 TopSpctps 22959 ℝ^crrx 25436

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263 ax-mulf 11264

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-tpos 8267 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-er 8763 df-map 8886 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ico 13413 df-fz 13568 df-fzo 13712 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-clim 15534 df-sum 15735 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-prds 17507 df-pws 17509 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-subg 19163 df-ghm 19253 df-cntz 19357 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-cring 20263 df-oppr 20360 df-dvdsr 20383 df-unit 20384 df-invr 20414 df-dvr 20427 df-rhm 20498 df-subrng 20572 df-subrg 20597 df-drng 20753 df-field 20754 df-abv 20832 df-staf 20862 df-srng 20863 df-lmod 20882 df-lss 20953 df-lmhm 21044 df-lvec 21125 df-sra 21195 df-rgmod 21196 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-cnfld 21388 df-refld 21646 df-phl 21667 df-dsmm 21775 df-frlm 21790 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-xms 24351 df-ms 24352 df-nm 24616 df-ngp 24617 df-tng 24618 df-nrg 24619 df-nlm 24620 df-clm 25115 df-cph 25221 df-tcph 25222 df-rrx 25438

This theorem is referenced by: qndenserrnopn 46219

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