qndenserrnbl - Mathbox for Glauco Siliprandi

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Theorem qndenserrnbl 39191

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 4718	. . . . . 6 ⊢ ∅ ∈ V
2	1	snid 4155	. . . . 5 ⊢ ∅ ∈ {∅}
3	2	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∅ ∈ {∅})
4		oveq2 6557	. . . . . 6 ⊢ (𝐼 = ∅ → (ℚ ↑_𝑚 𝐼) = (ℚ ↑_𝑚 ∅))
5		qex 11676	. . . . . . . 8 ⊢ ℚ ∈ V
6		mapdm0 38378	. . . . . . . 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 2645	. . . . 5 ⊢ (𝐼 = ∅ → {∅} = (ℚ ↑_𝑚 𝐼))
10	9	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝐼 = ∅) → {∅} = (ℚ ↑_𝑚 𝐼))
11	3, 10	eleqtrd 2690	. . 3 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∅ ∈ (ℚ ↑_𝑚 𝐼))
12		qndenserrnbl.i	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ Fin)
13		qndenserrnbl.d	. . . . . . . . 9 ⊢ 𝐷 = (dist‘(ℝ^‘𝐼))
14	13	rrxmetfi 39183	. . . . . . . 8 ⊢ (𝐼 ∈ Fin → 𝐷 ∈ (Met‘(ℝ ↑_𝑚 𝐼)))
15	12, 14	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (Met‘(ℝ ↑_𝑚 𝐼)))
16		metxmet 21949	. . . . . . 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 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐼 = ∅) → 𝐷 ∈ (∞Met‘(ℝ ↑_𝑚 𝐼)))
19		qndenserrnbl.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (ℝ ↑_𝑚 𝐼))
20	19	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐼 = ∅) → 𝑋 ∈ (ℝ ↑_𝑚 𝐼))
21		oveq2 6557	. . . . . . . . . . 11 ⊢ (𝐼 = ∅ → (ℝ ↑_𝑚 𝐼) = (ℝ ↑_𝑚 ∅))
22		reex 9906	. . . . . . . . . . . . 13 ⊢ ℝ ∈ V
23		mapdm0 38378	. . . . . . . . . . . . 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 2644	. . . . . . . . . 10 ⊢ (𝐼 = ∅ → (ℝ ↑_𝑚 𝐼) = {∅})
27	26	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐼 = ∅) → (ℝ ↑_𝑚 𝐼) = {∅})
28	20, 27	eleqtrd 2690	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐼 = ∅) → 𝑋 ∈ {∅})
29		elsng 4139	. . . . . . . . . 10 ⊢ (𝑋 ∈ (ℝ ↑_𝑚 𝐼) → (𝑋 ∈ {∅} ↔ 𝑋 = ∅))
30	19, 29	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 ∈ {∅} ↔ 𝑋 = ∅))
31	30	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐼 = ∅) → (𝑋 ∈ {∅} ↔ 𝑋 = ∅))
32	28, 31	mpbid 221	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐼 = ∅) → 𝑋 = ∅)
33	32	eqcomd 2616	. . . . . 6 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∅ = 𝑋)
34	33, 20	eqeltrd 2688	. . . . 5 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∅ ∈ (ℝ ↑_𝑚 𝐼))
35		qndenserrnbl.e	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
36	35	rpxrd 11749	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ^*)
37	35	rpgt0d 11751	. . . . . . 7 ⊢ (𝜑 → 0 < 𝐸)
38	36, 37	jca 553	. . . . . 6 ⊢ (𝜑 → (𝐸 ∈ ℝ^* ∧ 0 < 𝐸))
39	38	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐼 = ∅) → (𝐸 ∈ ℝ^* ∧ 0 < 𝐸))
40		xblcntr 22026	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘(ℝ ↑_𝑚 𝐼)) ∧ ∅ ∈ (ℝ ↑_𝑚 𝐼) ∧ (𝐸 ∈ ℝ^* ∧ 0 < 𝐸)) → ∅ ∈ (∅(ball‘𝐷)𝐸))
41	18, 34, 39, 40	syl3anc 1318	. . . 4 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∅ ∈ (∅(ball‘𝐷)𝐸))
42	33	oveq1d 6564	. . . 4 ⊢ ((𝜑 ∧ 𝐼 = ∅) → (∅(ball‘𝐷)𝐸) = (𝑋(ball‘𝐷)𝐸))
43	41, 42	eleqtrd 2690	. . 3 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∅ ∈ (𝑋(ball‘𝐷)𝐸))
44		eleq1 2676	. . . 4 ⊢ (𝑦 = ∅ → (𝑦 ∈ (𝑋(ball‘𝐷)𝐸) ↔ ∅ ∈ (𝑋(ball‘𝐷)𝐸)))
45	44	rspcev 3282	. . 3 ⊢ ((∅ ∈ (ℚ ↑_𝑚 𝐼) ∧ ∅ ∈ (𝑋(ball‘𝐷)𝐸)) → ∃𝑦 ∈ (ℚ ↑_𝑚 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸))
46	11, 43, 45	syl2anc 691	. 2 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∃𝑦 ∈ (ℚ ↑_𝑚 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸))
47	12	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐼 = ∅) → 𝐼 ∈ Fin)
48		neqne 2790	. . . 4 ⊢ (¬ 𝐼 = ∅ → 𝐼 ≠ ∅)
49	48	adantl 481	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐼 = ∅) → 𝐼 ≠ ∅)
50	19	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐼 = ∅) → 𝑋 ∈ (ℝ ↑_𝑚 𝐼))
51	35	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐼 = ∅) → 𝐸 ∈ ℝ⁺)
52	47, 49, 50, 13, 51	qndenserrnbllem 39190	. 2 ⊢ ((𝜑 ∧ ¬ 𝐼 = ∅) → ∃𝑦 ∈ (ℚ ↑_𝑚 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸))
53	46, 52	pm2.61dan 828	1 ⊢ (𝜑 → ∃𝑦 ∈ (ℚ ↑_𝑚 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∃wrex 2897 Vcvv 3173 ∅c0 3874 {csn 4125 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ↑_𝑚 cmap 7744 Fincfn 7841 ℝcr 9814 0cc0 9815 ℝ^*cxr 9952 < clt 9953 ℚcq 11664 ℝ⁺crp 11708 distcds 15777 ∞Metcxmt 19552 Metcme 19553 ballcbl 19554 ℝ^crrx 22979

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xadd 11823 df-ioo 12050 df-ico 12052 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-0g 15925 df-gsum 15926 df-prds 15931 df-pws 15933 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-grp 17248 df-minusg 17249 df-sbg 17250 df-subg 17414 df-ghm 17481 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-dvr 18506 df-rnghom 18538 df-drng 18572 df-field 18573 df-subrg 18601 df-staf 18668 df-srng 18669 df-lmod 18688 df-lss 18754 df-sra 18993 df-rgmod 18994 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-cnfld 19568 df-refld 19770 df-dsmm 19895 df-frlm 19910 df-nm 22197 df-tng 22199 df-tch 22777 df-rrx 22981

This theorem is referenced by: qndenserrnopnlem 39193

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