qndenserrnbl - Mathbox for Glauco Siliprandi

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Theorem qndenserrnbl 41432

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
qndenserrnbl.i	⊢ (𝜑 → 𝐼 ∈ Fin)
qndenserrnbl.x	⊢ (𝜑 → 𝑋 ∈ (ℝ ↑_𝑚 𝐼))
qndenserrnbl.d	⊢ 𝐷 = (dist‘(ℝ^‘𝐼))
qndenserrnbl.e	⊢ (𝜑 → 𝐸 ∈ ℝ⁺)

**Assertion**
Ref	Expression
qndenserrnbl	⊢ (𝜑 → ∃𝑦 ∈ (ℚ ↑_𝑚 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸))

Distinct variable groups: 𝑦,𝐷 𝑦,𝐸 𝑦,𝐼 𝑦,𝑋 𝜑,𝑦

**Proof of Theorem qndenserrnbl**
Step	Hyp	Ref	Expression
1		0ex 5026	. . . . . 6 ⊢ ∅ ∈ V
2	1	snid 4429	. . . . 5 ⊢ ∅ ∈ {∅}
3	2	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∅ ∈ {∅})
4		oveq2 6930	. . . . . 6 ⊢ (𝐼 = ∅ → (ℚ ↑_𝑚 𝐼) = (ℚ ↑_𝑚 ∅))
5		qex 12108	. . . . . . . 8 ⊢ ℚ ∈ V
6		mapdm0 8155	. . . . . . . 8 ⊢ (ℚ ∈ V → (ℚ ↑_𝑚 ∅) = {∅})
7	5, 6	ax-mp 5	. . . . . . 7 ⊢ (ℚ ↑_𝑚 ∅) = {∅}
8	7	a1i 11	. . . . . 6 ⊢ (𝐼 = ∅ → (ℚ ↑_𝑚 ∅) = {∅})
9	4, 8	eqtr2d 2814	. . . . 5 ⊢ (𝐼 = ∅ → {∅} = (ℚ ↑_𝑚 𝐼))
10	9	adantl 475	. . . 4 ⊢ ((𝜑 ∧ 𝐼 = ∅) → {∅} = (ℚ ↑_𝑚 𝐼))
11	3, 10	eleqtrd 2860	. . 3 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∅ ∈ (ℚ ↑_𝑚 𝐼))
12		qndenserrnbl.i	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ Fin)
13		qndenserrnbl.d	. . . . . . . . 9 ⊢ 𝐷 = (dist‘(ℝ^‘𝐼))
14	13	rrxmetfi 23618	. . . . . . . 8 ⊢ (𝐼 ∈ Fin → 𝐷 ∈ (Met‘(ℝ ↑_𝑚 𝐼)))
15	12, 14	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (Met‘(ℝ ↑_𝑚 𝐼)))
16		metxmet 22547	. . . . . . 7 ⊢ (𝐷 ∈ (Met‘(ℝ ↑_𝑚 𝐼)) → 𝐷 ∈ (∞Met‘(ℝ ↑_𝑚 𝐼)))
17	15, 16	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘(ℝ ↑_𝑚 𝐼)))
18	17	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝐼 = ∅) → 𝐷 ∈ (∞Met‘(ℝ ↑_𝑚 𝐼)))
19		qndenserrnbl.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (ℝ ↑_𝑚 𝐼))
20	19	adantr 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐼 = ∅) → 𝑋 ∈ (ℝ ↑_𝑚 𝐼))
21		oveq2 6930	. . . . . . . . . . 11 ⊢ (𝐼 = ∅ → (ℝ ↑_𝑚 𝐼) = (ℝ ↑_𝑚 ∅))
22		reex 10363	. . . . . . . . . . . . 13 ⊢ ℝ ∈ V
23		mapdm0 8155	. . . . . . . . . . . . 13 ⊢ (ℝ ∈ V → (ℝ ↑_𝑚 ∅) = {∅})
24	22, 23	ax-mp 5	. . . . . . . . . . . 12 ⊢ (ℝ ↑_𝑚 ∅) = {∅}
25	24	a1i 11	. . . . . . . . . . 11 ⊢ (𝐼 = ∅ → (ℝ ↑_𝑚 ∅) = {∅})
26	21, 25	eqtrd 2813	. . . . . . . . . 10 ⊢ (𝐼 = ∅ → (ℝ ↑_𝑚 𝐼) = {∅})
27	26	adantl 475	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐼 = ∅) → (ℝ ↑_𝑚 𝐼) = {∅})
28	20, 27	eleqtrd 2860	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐼 = ∅) → 𝑋 ∈ {∅})
29		elsng 4411	. . . . . . . . . 10 ⊢ (𝑋 ∈ (ℝ ↑_𝑚 𝐼) → (𝑋 ∈ {∅} ↔ 𝑋 = ∅))
30	19, 29	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 ∈ {∅} ↔ 𝑋 = ∅))
31	30	adantr 474	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐼 = ∅) → (𝑋 ∈ {∅} ↔ 𝑋 = ∅))
32	28, 31	mpbid 224	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐼 = ∅) → 𝑋 = ∅)
33	32	eqcomd 2783	. . . . . 6 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∅ = 𝑋)
34	33, 20	eqeltrd 2858	. . . . 5 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∅ ∈ (ℝ ↑_𝑚 𝐼))
35		qndenserrnbl.e	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
36	35	rpxrd 12182	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ^*)
37	35	rpgt0d 12184	. . . . . . 7 ⊢ (𝜑 → 0 < 𝐸)
38	36, 37	jca 507	. . . . . 6 ⊢ (𝜑 → (𝐸 ∈ ℝ^* ∧ 0 < 𝐸))
39	38	adantr 474	. . . . 5 ⊢ ((𝜑 ∧ 𝐼 = ∅) → (𝐸 ∈ ℝ^* ∧ 0 < 𝐸))
40		xblcntr 22624	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘(ℝ ↑_𝑚 𝐼)) ∧ ∅ ∈ (ℝ ↑_𝑚 𝐼) ∧ (𝐸 ∈ ℝ^* ∧ 0 < 𝐸)) → ∅ ∈ (∅(ball‘𝐷)𝐸))
41	18, 34, 39, 40	syl3anc 1439	. . . 4 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∅ ∈ (∅(ball‘𝐷)𝐸))
42	33	oveq1d 6937	. . . 4 ⊢ ((𝜑 ∧ 𝐼 = ∅) → (∅(ball‘𝐷)𝐸) = (𝑋(ball‘𝐷)𝐸))
43	41, 42	eleqtrd 2860	. . 3 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∅ ∈ (𝑋(ball‘𝐷)𝐸))
44		eleq1 2846	. . . 4 ⊢ (𝑦 = ∅ → (𝑦 ∈ (𝑋(ball‘𝐷)𝐸) ↔ ∅ ∈ (𝑋(ball‘𝐷)𝐸)))
45	44	rspcev 3510	. . 3 ⊢ ((∅ ∈ (ℚ ↑_𝑚 𝐼) ∧ ∅ ∈ (𝑋(ball‘𝐷)𝐸)) → ∃𝑦 ∈ (ℚ ↑_𝑚 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸))
46	11, 43, 45	syl2anc 579	. 2 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∃𝑦 ∈ (ℚ ↑_𝑚 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸))
47	12	adantr 474	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐼 = ∅) → 𝐼 ∈ Fin)
48		neqne 2976	. . . 4 ⊢ (¬ 𝐼 = ∅ → 𝐼 ≠ ∅)
49	48	adantl 475	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐼 = ∅) → 𝐼 ≠ ∅)
50	19	adantr 474	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐼 = ∅) → 𝑋 ∈ (ℝ ↑_𝑚 𝐼))
51	35	adantr 474	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐼 = ∅) → 𝐸 ∈ ℝ⁺)
52	47, 49, 50, 13, 51	qndenserrnbllem 41431	. 2 ⊢ ((𝜑 ∧ ¬ 𝐼 = ∅) → ∃𝑦 ∈ (ℚ ↑_𝑚 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸))
53	46, 52	pm2.61dan 803	1 ⊢ (𝜑 → ∃𝑦 ∈ (ℚ ↑_𝑚 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ≠ wne 2968 ∃wrex 3090 Vcvv 3397 ∅c0 4140 {csn 4397 class class class wbr 4886 ‘cfv 6135 (class class class)co 6922 ↑_𝑚 cmap 8140 Fincfn 8241 ℝcr 10271 0cc0 10272 ℝ^*cxr 10410 < clt 10411 ℚcq 12095 ℝ⁺crp 12137 distcds 16347 ∞Metcxmet 20127 Metcmet 20128 ballcbl 20129 ℝ^crrx 23589

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-addf 10351 ax-mulf 10352

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-tpos 7634 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-xadd 12258 df-ioo 12491 df-ico 12493 df-fz 12644 df-fzo 12785 df-seq 13120 df-exp 13179 df-hash 13436 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-clim 14627 df-sum 14825 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-hom 16362 df-cco 16363 df-0g 16488 df-gsum 16489 df-prds 16494 df-pws 16496 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-mhm 17721 df-grp 17812 df-minusg 17813 df-sbg 17814 df-subg 17975 df-ghm 18042 df-cntz 18133 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-cring 18937 df-oppr 19010 df-dvdsr 19028 df-unit 19029 df-invr 19059 df-dvr 19070 df-rnghom 19104 df-drng 19141 df-field 19142 df-subrg 19170 df-staf 19237 df-srng 19238 df-lmod 19257 df-lss 19325 df-sra 19569 df-rgmod 19570 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-cnfld 20143 df-refld 20348 df-dsmm 20475 df-frlm 20490 df-nm 22795 df-tng 22797 df-tcph 23376 df-rrx 23591

This theorem is referenced by: qndenserrnopnlem 41434

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