qndenserrn - Mathbox for Glauco Siliprandi

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Theorem qndenserrn 45750

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
qndenserrn.i	⊢ (𝜑 → 𝐼 ∈ Fin)
qndenserrn.j	⊢ 𝐽 = (TopOpen‘(ℝ^‘𝐼))

**Assertion**
Ref	Expression
qndenserrn	⊢ (𝜑 → ((cls‘𝐽)‘(ℚ ↑_m 𝐼)) = (ℝ ↑_m 𝐼))

Proof of Theorem qndenserrn Dummy variables 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		qndenserrn.i	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ Fin)
2		qndenserrn.j	. . . . . 6 ⊢ 𝐽 = (TopOpen‘(ℝ^‘𝐼))
3	2	rrxtop 45740	. . . . 5 ⊢ (𝐼 ∈ Fin → 𝐽 ∈ Top)
4	1, 3	syl 17	. . . 4 ⊢ (𝜑 → 𝐽 ∈ Top)
5		reex 11229	. . . . . . 7 ⊢ ℝ ∈ V
6		qssre 12973	. . . . . . 7 ⊢ ℚ ⊆ ℝ
7		mapss 8906	. . . . . . 7 ⊢ ((ℝ ∈ V ∧ ℚ ⊆ ℝ) → (ℚ ↑_m 𝐼) ⊆ (ℝ ↑_m 𝐼))
8	5, 6, 7	mp2an 690	. . . . . 6 ⊢ (ℚ ↑_m 𝐼) ⊆ (ℝ ↑_m 𝐼)
9	8	a1i 11	. . . . 5 ⊢ (𝜑 → (ℚ ↑_m 𝐼) ⊆ (ℝ ↑_m 𝐼))
10		eqid 2725	. . . . . . . 8 ⊢ (ℝ^‘𝐼) = (ℝ^‘𝐼)
11		eqid 2725	. . . . . . . 8 ⊢ (Base‘(ℝ^‘𝐼)) = (Base‘(ℝ^‘𝐼))
12	1, 10, 11	rrxbasefi 25356	. . . . . . 7 ⊢ (𝜑 → (Base‘(ℝ^‘𝐼)) = (ℝ ↑_m 𝐼))
13	12	eqcomd 2731	. . . . . 6 ⊢ (𝜑 → (ℝ ↑_m 𝐼) = (Base‘(ℝ^‘𝐼)))
14		rrxtps 45737	. . . . . . 7 ⊢ (𝐼 ∈ Fin → (ℝ^‘𝐼) ∈ TopSp)
15		eqid 2725	. . . . . . . 8 ⊢ (TopOpen‘(ℝ^‘𝐼)) = (TopOpen‘(ℝ^‘𝐼))
16	11, 15	tpsuni 22856	. . . . . . 7 ⊢ ((ℝ^‘𝐼) ∈ TopSp → (Base‘(ℝ^‘𝐼)) = ∪ (TopOpen‘(ℝ^‘𝐼)))
17	1, 14, 16	3syl 18	. . . . . 6 ⊢ (𝜑 → (Base‘(ℝ^‘𝐼)) = ∪ (TopOpen‘(ℝ^‘𝐼)))
18	2	unieqi 4915	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ (TopOpen‘(ℝ^‘𝐼))
19	18	eqcomi 2734	. . . . . . 7 ⊢ ∪ (TopOpen‘(ℝ^‘𝐼)) = ∪ 𝐽
20	19	a1i 11	. . . . . 6 ⊢ (𝜑 → ∪ (TopOpen‘(ℝ^‘𝐼)) = ∪ 𝐽)
21	13, 17, 20	3eqtrd 2769	. . . . 5 ⊢ (𝜑 → (ℝ ↑_m 𝐼) = ∪ 𝐽)
22	9, 21	sseqtrd 4013	. . . 4 ⊢ (𝜑 → (ℚ ↑_m 𝐼) ⊆ ∪ 𝐽)
23		eqid 2725	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
24	23	clsss3 22981	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (ℚ ↑_m 𝐼) ⊆ ∪ 𝐽) → ((cls‘𝐽)‘(ℚ ↑_m 𝐼)) ⊆ ∪ 𝐽)
25	4, 22, 24	syl2anc 582	. . 3 ⊢ (𝜑 → ((cls‘𝐽)‘(ℚ ↑_m 𝐼)) ⊆ ∪ 𝐽)
26	21	eqcomd 2731	. . 3 ⊢ (𝜑 → ∪ 𝐽 = (ℝ ↑_m 𝐼))
27	25, 26	sseqtrd 4013	. 2 ⊢ (𝜑 → ((cls‘𝐽)‘(ℚ ↑_m 𝐼)) ⊆ (ℝ ↑_m 𝐼))
28	1	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → 𝐼 ∈ Fin)
29		id 22	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ 𝐽 → 𝑣 ∈ 𝐽)
30	29, 2	eleqtrdi 2835	. . . . . . . . . . 11 ⊢ (𝑣 ∈ 𝐽 → 𝑣 ∈ (TopOpen‘(ℝ^‘𝐼)))
31	30	ad2antlr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → 𝑣 ∈ (TopOpen‘(ℝ^‘𝐼)))
32		ne0i 4330	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑣 → 𝑣 ≠ ∅)
33	32	adantl 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → 𝑣 ≠ ∅)
34	28, 15, 31, 33	qndenserrnopn 45749	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → ∃𝑦 ∈ (ℚ ↑_m 𝐼)𝑦 ∈ 𝑣)
35		df-rex 3061	. . . . . . . . 9 ⊢ (∃𝑦 ∈ (ℚ ↑_m 𝐼)𝑦 ∈ 𝑣 ↔ ∃𝑦(𝑦 ∈ (ℚ ↑_m 𝐼) ∧ 𝑦 ∈ 𝑣))
36	34, 35	sylib 217	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → ∃𝑦(𝑦 ∈ (ℚ ↑_m 𝐼) ∧ 𝑦 ∈ 𝑣))
37		simpr 483	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (ℚ ↑_m 𝐼) ∧ 𝑦 ∈ 𝑣) → 𝑦 ∈ 𝑣)
38		simpl 481	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (ℚ ↑_m 𝐼) ∧ 𝑦 ∈ 𝑣) → 𝑦 ∈ (ℚ ↑_m 𝐼))
39	37, 38	elind 4188	. . . . . . . . . 10 ⊢ ((𝑦 ∈ (ℚ ↑_m 𝐼) ∧ 𝑦 ∈ 𝑣) → 𝑦 ∈ (𝑣 ∩ (ℚ ↑_m 𝐼)))
40	39	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → ((𝑦 ∈ (ℚ ↑_m 𝐼) ∧ 𝑦 ∈ 𝑣) → 𝑦 ∈ (𝑣 ∩ (ℚ ↑_m 𝐼))))
41	40	eximdv 1912	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → (∃𝑦(𝑦 ∈ (ℚ ↑_m 𝐼) ∧ 𝑦 ∈ 𝑣) → ∃𝑦 𝑦 ∈ (𝑣 ∩ (ℚ ↑_m 𝐼))))
42	36, 41	mpd 15	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → ∃𝑦 𝑦 ∈ (𝑣 ∩ (ℚ ↑_m 𝐼)))
43		n0 4342	. . . . . . 7 ⊢ ((𝑣 ∩ (ℚ ↑_m 𝐼)) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝑣 ∩ (ℚ ↑_m 𝐼)))
44	42, 43	sylibr 233	. . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑥 ∈ 𝑣) → (𝑣 ∩ (ℚ ↑_m 𝐼)) ≠ ∅)
45	44	ex 411	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐽) → (𝑥 ∈ 𝑣 → (𝑣 ∩ (ℚ ↑_m 𝐼)) ≠ ∅))
46	45	adantlr 713	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑣 ∈ 𝐽) → (𝑥 ∈ 𝑣 → (𝑣 ∩ (ℚ ↑_m 𝐼)) ≠ ∅))
47	46	ralrimiva 3136	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑_m 𝐼)) → ∀𝑣 ∈ 𝐽 (𝑥 ∈ 𝑣 → (𝑣 ∩ (ℚ ↑_m 𝐼)) ≠ ∅))
48	4	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑_m 𝐼)) → 𝐽 ∈ Top)
49	22	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑_m 𝐼)) → (ℚ ↑_m 𝐼) ⊆ ∪ 𝐽)
50		simpr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑_m 𝐼)) → 𝑥 ∈ (ℝ ↑_m 𝐼))
51	21	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑_m 𝐼)) → (ℝ ↑_m 𝐼) = ∪ 𝐽)
52	50, 51	eleqtrd 2827	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑_m 𝐼)) → 𝑥 ∈ ∪ 𝐽)
53	23	elcls 22995	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (ℚ ↑_m 𝐼) ⊆ ∪ 𝐽 ∧ 𝑥 ∈ ∪ 𝐽) → (𝑥 ∈ ((cls‘𝐽)‘(ℚ ↑_m 𝐼)) ↔ ∀𝑣 ∈ 𝐽 (𝑥 ∈ 𝑣 → (𝑣 ∩ (ℚ ↑_m 𝐼)) ≠ ∅)))
54	48, 49, 52, 53	syl3anc 1368	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑_m 𝐼)) → (𝑥 ∈ ((cls‘𝐽)‘(ℚ ↑_m 𝐼)) ↔ ∀𝑣 ∈ 𝐽 (𝑥 ∈ 𝑣 → (𝑣 ∩ (ℚ ↑_m 𝐼)) ≠ ∅)))
55	47, 54	mpbird 256	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ↑_m 𝐼)) → 𝑥 ∈ ((cls‘𝐽)‘(ℚ ↑_m 𝐼)))
56	27, 55	eqelssd 3994	1 ⊢ (𝜑 → ((cls‘𝐽)‘(ℚ ↑_m 𝐼)) = (ℝ ↑_m 𝐼))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ≠ wne 2930 ∀wral 3051 ∃wrex 3060 Vcvv 3463 ∩ cin 3938 ⊆ wss 3939 ∅c0 4318 ∪ cuni 4903 ‘cfv 6543 (class class class)co 7416 ↑_m cmap 8843 Fincfn 8962 ℝcr 11137 ℚcq 12962 Basecbs 17179 TopOpenctopn 17402 Topctop 22813 TopSpctps 22852 clsccl 22940 ℝ^crrx 25329

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 ax-addf 11217 ax-mulf 11218

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7991 df-2nd 7992 df-supp 8164 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-sup 9465 df-inf 9466 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-ioo 13360 df-ico 13362 df-fz 13517 df-fzo 13660 df-seq 13999 df-exp 14059 df-hash 14322 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-clim 15464 df-sum 15665 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-hom 17256 df-cco 17257 df-rest 17403 df-topn 17404 df-0g 17422 df-gsum 17423 df-topgen 17424 df-prds 17428 df-pws 17430 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18739 df-submnd 18740 df-grp 18897 df-minusg 18898 df-sbg 18899 df-subg 19082 df-ghm 19172 df-cntz 19272 df-cmn 19741 df-abl 19742 df-mgp 20079 df-rng 20097 df-ur 20126 df-ring 20179 df-cring 20180 df-oppr 20277 df-dvdsr 20300 df-unit 20301 df-invr 20331 df-dvr 20344 df-rhm 20415 df-subrng 20487 df-subrg 20512 df-drng 20630 df-field 20631 df-abv 20701 df-staf 20729 df-srng 20730 df-lmod 20749 df-lss 20820 df-lmhm 20911 df-lvec 20992 df-sra 21062 df-rgmod 21063 df-psmet 21275 df-xmet 21276 df-met 21277 df-bl 21278 df-mopn 21279 df-cnfld 21284 df-refld 21541 df-phl 21562 df-dsmm 21670 df-frlm 21685 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22867 df-cld 22941 df-ntr 22942 df-cls 22943 df-xms 24244 df-ms 24245 df-nm 24509 df-ngp 24510 df-tng 24511 df-nrg 24512 df-nlm 24513 df-clm 25008 df-cph 25114 df-tcph 25115 df-rrx 25331

This theorem is referenced by: (None)

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