qndenserrnopnlem - Mathbox for Glauco Siliprandi

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Theorem qndenserrnopnlem 46831

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 25461	. . . . 5 ⊢ (𝐼 ∈ Fin → 𝐷 ∈ (Met‘(ℝ ↑_m 𝐼)))
4	1, 3	syl 17	. . . 4 ⊢ (𝜑 → 𝐷 ∈ (Met‘(ℝ ↑_m 𝐼)))
5		metxmet 24381	. . . 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 2871	. . . 4 ⊢ (𝜑 → 𝑉 ∈ (TopOpen‘(ℝ^‘𝐼)))
10	1	rrxtopnfi 46821	. . . . 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 2761	. . . . . . . . 9 ⊢ (ℝ^‘𝐼) = (ℝ^‘𝐼)
13		eqid 2761	. . . . . . . . 9 ⊢ (ℝ ↑_m 𝐼) = (ℝ ↑_m 𝐼)
14	12, 13	rrxdsfi 25460	. . . . . . . 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 2797	. . . . . 6 ⊢ (𝜑 → (𝑓 ∈ (ℝ ↑_m 𝐼), 𝑔 ∈ (ℝ ↑_m 𝐼) ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2))) = 𝐷)
17	16	fveq2d 6865	. . . . 5 ⊢ (𝜑 → (MetOpen‘(𝑓 ∈ (ℝ ↑_m 𝐼), 𝑔 ∈ (ℝ ↑_m 𝐼) ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))) = (MetOpen‘𝐷))
18	10, 17	eqtrd 2796	. . . 4 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝐼)) = (MetOpen‘𝐷))
19	9, 18	eleqtrd 2863	. . 3 ⊢ (𝜑 → 𝑉 ∈ (MetOpen‘𝐷))
20		qndenserrnopnlem.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
21		eqid 2761	. . . 4 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
22	21	mopni2 24540	. . 3 ⊢ ((𝐷 ∈ (∞Met‘(ℝ ↑_m 𝐼)) ∧ 𝑉 ∈ (MetOpen‘𝐷) ∧ 𝑋 ∈ 𝑉) → ∃𝑒 ∈ ℝ⁺ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉)
23	6, 19, 20, 22	syl3anc 1389	. 2 ⊢ (𝜑 → ∃𝑒 ∈ ℝ⁺ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉)
24	1	3ad2ant1 1145	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) → 𝐼 ∈ Fin)
25		rrxtps 46820	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ Fin → (ℝ^‘𝐼) ∈ TopSp)
26	1, 25	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ^‘𝐼) ∈ TopSp)
27		eqid 2761	. . . . . . . . . . . 12 ⊢ (Base‘(ℝ^‘𝐼)) = (Base‘(ℝ^‘𝐼))
28	27, 8	istps 22981	. . . . . . . . . . 11 ⊢ ((ℝ^‘𝐼) ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘(ℝ^‘𝐼))))
29	26, 28	sylib 220	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(Base‘(ℝ^‘𝐼))))
30	1, 12, 27	rrxbasefi 25459	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘(ℝ^‘𝐼)) = (ℝ ↑_m 𝐼))
31	30	fveq2d 6865	. . . . . . . . . 10 ⊢ (𝜑 → (TopOn‘(Base‘(ℝ^‘𝐼))) = (TopOn‘(ℝ ↑_m 𝐼)))
32	29, 31	eleqtrd 2863	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(ℝ ↑_m 𝐼)))
33		toponss 22974	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘(ℝ ↑_m 𝐼)) ∧ 𝑉 ∈ 𝐽) → 𝑉 ⊆ (ℝ ↑_m 𝐼))
34	32, 7, 33	syl2anc 593	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ⊆ (ℝ ↑_m 𝐼))
35	34, 20	sseldd 3935	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (ℝ ↑_m 𝐼))
36	35	3ad2ant1 1145	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) → 𝑋 ∈ (ℝ ↑_m 𝐼))
37		simp2 1149	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) → 𝑒 ∈ ℝ⁺)
38	24, 36, 2, 37	qndenserrnbl 46829	. . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑋(ball‘𝐷)𝑒) ⊆ 𝑉) → ∃𝑦 ∈ (ℚ ↑_m 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝑒))
39		ssel 3928	. . . . . . . 8 ⊢ ((𝑋(ball‘𝐷)𝑒) ⊆ 𝑉 → (𝑦 ∈ (𝑋(ball‘𝐷)𝑒) → 𝑦 ∈ 𝑉))
40	39	adantr 484	. . . . . . 7 ⊢ (((𝑋(ball‘𝐷)𝑒) ⊆ 𝑉 ∧ 𝑦 ∈ (ℚ ↑_m 𝐼)) → (𝑦 ∈ (𝑋(ball‘𝐷)𝑒) → 𝑦 ∈ 𝑉))
41	40	3ad2antl3 1200	. . . . . 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 1131	. . 3 ⊢ (𝜑 → (𝑒 ∈ ℝ⁺ → ((𝑋(ball‘𝐷)𝑒) ⊆ 𝑉 → ∃𝑦 ∈ (ℚ ↑_m 𝐼)𝑦 ∈ 𝑉)))
45	44	rexlimdv 3160	. 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 1097 = wceq 1559 ∈ wcel 2141 ∃wrex 3085 ⊆ wss 3902 ‘cfv 6515 (class class class)co 7390 ∈ cmpo 7392 ↑_m cmap 8801 Fincfn 8920 ℝcr 11065 − cmin 11407 2c2 12265 ℚcq 12942 ℝ⁺crp 12986 ↑cexp 14067 √csqrt 15250 Σcsu 15703 Basecbs 17235 distcds 17285 TopOpenctopn 17440 ∞Metcxmet 21396 Metcmet 21397 ballcbl 21398 MetOpencmopn 21401 TopOnctopon 22957 TopSpctps 22979 ℝ^crrx 25432

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7712 ax-inf2 9589 ax-cnex 11122 ax-resscn 11123 ax-1cn 11124 ax-icn 11125 ax-addcl 11126 ax-addrcl 11127 ax-mulcl 11128 ax-mulrcl 11129 ax-mulcom 11130 ax-addass 11131 ax-mulass 11132 ax-distr 11133 ax-i2m1 11134 ax-1ne0 11135 ax-1rid 11136 ax-rnegex 11137 ax-rrecex 11138 ax-cnre 11139 ax-pre-lttri 11140 ax-pre-lttrn 11141 ax-pre-ltadd 11142 ax-pre-mulgt0 11143 ax-pre-sup 11144 ax-addf 11145 ax-mulf 11146

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-se 5597 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-isom 6524 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7654 df-om 7841 df-1st 7964 df-2nd 7965 df-supp 8134 df-tpos 8199 df-frecs 8255 df-wrecs 8286 df-recs 8335 df-rdg 8374 df-1o 8430 df-er 8671 df-map 8803 df-ixp 8873 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-fsupp 9301 df-sup 9381 df-inf 9382 df-oi 9451 df-card 9890 df-pnf 11211 df-mnf 11212 df-xr 11213 df-ltxr 11214 df-le 11215 df-sub 11409 df-neg 11410 df-div 11838 df-nn 12204 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-9 12280 df-n0 12475 df-z 12562 df-dec 12682 df-uz 12833 df-q 12943 df-rp 12987 df-xneg 13107 df-xadd 13108 df-xmul 13109 df-ioo 13346 df-ico 13348 df-fz 13506 df-fzo 13653 df-seq 14008 df-exp 14068 df-hash 14337 df-cj 15116 df-re 15117 df-im 15118 df-sqrt 15252 df-abs 15253 df-clim 15505 df-sum 15704 df-struct 17173 df-sets 17190 df-slot 17208 df-ndx 17220 df-base 17236 df-ress 17257 df-plusg 17289 df-mulr 17290 df-starv 17291 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-hom 17300 df-cco 17301 df-rest 17441 df-topn 17442 df-0g 17460 df-gsum 17461 df-topgen 17462 df-prds 17466 df-pws 17468 df-mgm 18664 df-sgrp 18743 df-mnd 18759 df-mhm 18807 df-submnd 18808 df-grp 18968 df-minusg 18969 df-sbg 18970 df-subg 19155 df-ghm 19244 df-cntz 19347 df-cmn 19812 df-abl 19813 df-mgp 20177 df-rng 20189 df-ur 20218 df-ring 20271 df-cring 20272 df-oppr 20372 df-dvdsr 20392 df-unit 20393 df-invr 20423 df-dvr 20436 df-rhm 20507 df-subrng 20582 df-subrg 20606 df-drng 20767 df-field 20768 df-abv 20845 df-staf 20875 df-srng 20876 df-lmod 20916 df-lss 20986 df-lmhm 21076 df-lvec 21157 df-sra 21227 df-rgmod 21228 df-psmet 21403 df-xmet 21404 df-met 21405 df-bl 21406 df-mopn 21407 df-cnfld 21412 df-refld 21644 df-phl 21665 df-dsmm 21771 df-frlm 21786 df-top 22941 df-topon 22958 df-topsp 22980 df-bases 22993 df-xms 24367 df-ms 24368 df-nm 24629 df-ngp 24630 df-tng 24631 df-nrg 24632 df-nlm 24633 df-clm 25112 df-cph 25217 df-tcph 25218 df-rrx 25434

This theorem is referenced by: qndenserrnopn 46832

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