qndenserrnopnlem - Mathbox for Glauco Siliprandi

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Theorem qndenserrnopnlem 43728

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 24481	. . . . 5 ⊢ (𝐼 ∈ Fin → 𝐷 ∈ (Met‘(ℝ ↑_m 𝐼)))
4	1, 3	syl 17	. . . 4 ⊢ (𝜑 → 𝐷 ∈ (Met‘(ℝ ↑_m 𝐼)))
5		metxmet 23395	. . . 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 2849	. . . 4 ⊢ (𝜑 → 𝑉 ∈ (TopOpen‘(ℝ^‘𝐼)))
10	1	rrxtopnfi 43718	. . . . 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 2738	. . . . . . . . 9 ⊢ (ℝ^‘𝐼) = (ℝ^‘𝐼)
13		eqid 2738	. . . . . . . . 9 ⊢ (ℝ ↑_m 𝐼) = (ℝ ↑_m 𝐼)
14	12, 13	rrxdsfi 24480	. . . . . . . 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 2779	. . . . . 6 ⊢ (𝜑 → (𝑓 ∈ (ℝ ↑_m 𝐼), 𝑔 ∈ (ℝ ↑_m 𝐼) ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))) = 𝐷)
17	16	fveq2d 6760	. . . . 5 ⊢ (𝜑 → (MetOpen‘(𝑓 ∈ (ℝ ↑_m 𝐼), 𝑔 ∈ (ℝ ↑_m 𝐼) ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) = (MetOpen‘𝐷))
18	10, 17	eqtrd 2778	. . . 4 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝐼)) = (MetOpen‘𝐷))
19	9, 18	eleqtrd 2841	. . 3 ⊢ (𝜑 → 𝑉 ∈ (MetOpen‘𝐷))
20		qndenserrnopnlem.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
21		eqid 2738	. . . 4 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
22	21	mopni2 23555	. . 3 ⊢ ((𝐷 ∈ (∞Met‘(ℝ ↑_m 𝐼)) ∧ 𝑉 ∈ (MetOpen‘𝐷) ∧ 𝑋 ∈ 𝑉) → ∃𝑒 ∈ ℝ⁺ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉)
23	6, 19, 20, 22	syl3anc 1369	. 2 ⊢ (𝜑 → ∃𝑒 ∈ ℝ⁺ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉)
24	1	3ad2ant1 1131	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) → 𝐼 ∈ Fin)
25		rrxtps 43717	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ Fin → (ℝ^‘𝐼) ∈ TopSp)
26	1, 25	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ^‘𝐼) ∈ TopSp)
27		eqid 2738	. . . . . . . . . . . 12 ⊢ (Base‘(ℝ^‘𝐼)) = (Base‘(ℝ^‘𝐼))
28	27, 8	istps 21991	. . . . . . . . . . 11 ⊢ ((ℝ^‘𝐼) ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘(ℝ^‘𝐼))))
29	26, 28	sylib 217	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(Base‘(ℝ^‘𝐼))))
30	1, 12, 27	rrxbasefi 24479	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘(ℝ^‘𝐼)) = (ℝ ↑_m 𝐼))
31	30	fveq2d 6760	. . . . . . . . . 10 ⊢ (𝜑 → (TopOn‘(Base‘(ℝ^‘𝐼))) = (TopOn‘(ℝ ↑_m 𝐼)))
32	29, 31	eleqtrd 2841	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(ℝ ↑_m 𝐼)))
33		toponss 21984	. . . . . . . . 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 3918	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (ℝ ↑_m 𝐼))
36	35	3ad2ant1 1131	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) → 𝑋 ∈ (ℝ ↑_m 𝐼))
37		simp2 1135	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) → 𝑒 ∈ ℝ⁺)
38	24, 36, 2, 37	qndenserrnbl 43726	. . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) → ∃𝑦 ∈ (ℚ ↑_m 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝑒))
39		ssel 3910	. . . . . . . 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 1185	. . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) ∧ 𝑦 ∈ (ℚ ↑_m 𝐼)) → (𝑦 ∈ (𝑋(ball‘𝐷)𝑒) → 𝑦 ∈ 𝑉))
42	41	reximdva 3202	. . . . 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 1117	. . 3 ⊢ (𝜑 → (𝑒 ∈ ℝ⁺ → ((𝑋(ball‘𝐷)𝑒) ⊆ 𝑉 → ∃𝑦 ∈ (ℚ ↑_m 𝐼)𝑦 ∈ 𝑉)))
45	44	rexlimdv 3211	. 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 1085 = wceq 1539 ∈ wcel 2108 ∃wrex 3064 ⊆ wss 3883 ‘cfv 6418 (class class class)co 7255 ∈ cmpo 7257 ↑_m cmap 8573 Fincfn 8691 ℝcr 10801 − cmin 11135 2c2 11958 ℚcq 12617 ℝ⁺crp 12659 ↑cexp 13710 √csqrt 14872 Σcsu 15325 Basecbs 16840 distcds 16897 TopOpenctopn 17049 ∞Metcxmet 20495 Metcmet 20496 ballcbl 20497 MetOpencmopn 20500 TopOnctopon 21967 TopSpctps 21989 ℝ^crrx 24452

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-tpos 8013 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ico 13014 df-fz 13169 df-fzo 13312 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-sum 15326 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-prds 17075 df-pws 17077 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mhm 18345 df-submnd 18346 df-grp 18495 df-minusg 18496 df-sbg 18497 df-subg 18667 df-ghm 18747 df-cntz 18838 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-dvr 19840 df-rnghom 19874 df-drng 19908 df-field 19909 df-subrg 19937 df-abv 19992 df-staf 20020 df-srng 20021 df-lmod 20040 df-lss 20109 df-lmhm 20199 df-lvec 20280 df-sra 20349 df-rgmod 20350 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-cnfld 20511 df-refld 20722 df-phl 20743 df-dsmm 20849 df-frlm 20864 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-xms 23381 df-ms 23382 df-nm 23644 df-ngp 23645 df-tng 23646 df-nrg 23647 df-nlm 23648 df-clm 24132 df-cph 24237 df-tcph 24238 df-rrx 24454

This theorem is referenced by: qndenserrnopn 43729

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