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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > qndenserrnbl | Structured version Visualization version GIF version |
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.) |
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+) |
Ref | Expression |
---|---|
qndenserrnbl | ⊢ (𝜑 → ∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5264 | . . . . . 6 ⊢ ∅ ∈ V | |
2 | 1 | snid 4622 | . . . . 5 ⊢ ∅ ∈ {∅} |
3 | 2 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∅ ∈ {∅}) |
4 | oveq2 7365 | . . . . . 6 ⊢ (𝐼 = ∅ → (ℚ ↑m 𝐼) = (ℚ ↑m ∅)) | |
5 | qex 12886 | . . . . . . . 8 ⊢ ℚ ∈ V | |
6 | mapdm0 8780 | . . . . . . . 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 2777 | . . . . 5 ⊢ (𝐼 = ∅ → {∅} = (ℚ ↑m 𝐼)) |
10 | 9 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝐼 = ∅) → {∅} = (ℚ ↑m 𝐼)) |
11 | 3, 10 | eleqtrd 2840 | . . 3 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∅ ∈ (ℚ ↑m 𝐼)) |
12 | qndenserrnbl.i | . . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ Fin) | |
13 | qndenserrnbl.d | . . . . . . . . 9 ⊢ 𝐷 = (dist‘(ℝ^‘𝐼)) | |
14 | 13 | rrxmetfi 24776 | . . . . . . . 8 ⊢ (𝐼 ∈ Fin → 𝐷 ∈ (Met‘(ℝ ↑m 𝐼))) |
15 | 12, 14 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (Met‘(ℝ ↑m 𝐼))) |
16 | metxmet 23687 | . . . . . . 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 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝐼 = ∅) → 𝐷 ∈ (∞Met‘(ℝ ↑m 𝐼))) |
19 | qndenserrnbl.x | . . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (ℝ ↑m 𝐼)) | |
20 | 19 | adantr 481 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐼 = ∅) → 𝑋 ∈ (ℝ ↑m 𝐼)) |
21 | oveq2 7365 | . . . . . . . . . . 11 ⊢ (𝐼 = ∅ → (ℝ ↑m 𝐼) = (ℝ ↑m ∅)) | |
22 | reex 11142 | . . . . . . . . . . . . 13 ⊢ ℝ ∈ V | |
23 | mapdm0 8780 | . . . . . . . . . . . . 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 2776 | . . . . . . . . . 10 ⊢ (𝐼 = ∅ → (ℝ ↑m 𝐼) = {∅}) |
27 | 26 | adantl 482 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐼 = ∅) → (ℝ ↑m 𝐼) = {∅}) |
28 | 20, 27 | eleqtrd 2840 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐼 = ∅) → 𝑋 ∈ {∅}) |
29 | elsng 4600 | . . . . . . . . . 10 ⊢ (𝑋 ∈ (ℝ ↑m 𝐼) → (𝑋 ∈ {∅} ↔ 𝑋 = ∅)) | |
30 | 19, 29 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → (𝑋 ∈ {∅} ↔ 𝑋 = ∅)) |
31 | 30 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐼 = ∅) → (𝑋 ∈ {∅} ↔ 𝑋 = ∅)) |
32 | 28, 31 | mpbid 231 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐼 = ∅) → 𝑋 = ∅) |
33 | 32 | eqcomd 2742 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∅ = 𝑋) |
34 | 33, 20 | eqeltrd 2838 | . . . . 5 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∅ ∈ (ℝ ↑m 𝐼)) |
35 | qndenserrnbl.e | . . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ℝ+) | |
36 | 35 | rpxrd 12958 | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ*) |
37 | 35 | rpgt0d 12960 | . . . . . . 7 ⊢ (𝜑 → 0 < 𝐸) |
38 | 36, 37 | jca 512 | . . . . . 6 ⊢ (𝜑 → (𝐸 ∈ ℝ* ∧ 0 < 𝐸)) |
39 | 38 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝐼 = ∅) → (𝐸 ∈ ℝ* ∧ 0 < 𝐸)) |
40 | xblcntr 23764 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘(ℝ ↑m 𝐼)) ∧ ∅ ∈ (ℝ ↑m 𝐼) ∧ (𝐸 ∈ ℝ* ∧ 0 < 𝐸)) → ∅ ∈ (∅(ball‘𝐷)𝐸)) | |
41 | 18, 34, 39, 40 | syl3anc 1371 | . . . 4 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∅ ∈ (∅(ball‘𝐷)𝐸)) |
42 | 33 | oveq1d 7372 | . . . 4 ⊢ ((𝜑 ∧ 𝐼 = ∅) → (∅(ball‘𝐷)𝐸) = (𝑋(ball‘𝐷)𝐸)) |
43 | 41, 42 | eleqtrd 2840 | . . 3 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∅ ∈ (𝑋(ball‘𝐷)𝐸)) |
44 | eleq1 2825 | . . . 4 ⊢ (𝑦 = ∅ → (𝑦 ∈ (𝑋(ball‘𝐷)𝐸) ↔ ∅ ∈ (𝑋(ball‘𝐷)𝐸))) | |
45 | 44 | rspcev 3581 | . . 3 ⊢ ((∅ ∈ (ℚ ↑m 𝐼) ∧ ∅ ∈ (𝑋(ball‘𝐷)𝐸)) → ∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸)) |
46 | 11, 43, 45 | syl2anc 584 | . 2 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸)) |
47 | 12 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐼 = ∅) → 𝐼 ∈ Fin) |
48 | neqne 2951 | . . . 4 ⊢ (¬ 𝐼 = ∅ → 𝐼 ≠ ∅) | |
49 | 48 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐼 = ∅) → 𝐼 ≠ ∅) |
50 | 19 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐼 = ∅) → 𝑋 ∈ (ℝ ↑m 𝐼)) |
51 | 35 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐼 = ∅) → 𝐸 ∈ ℝ+) |
52 | 47, 49, 50, 13, 51 | qndenserrnbllem 44525 | . 2 ⊢ ((𝜑 ∧ ¬ 𝐼 = ∅) → ∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸)) |
53 | 46, 52 | pm2.61dan 811 | 1 ⊢ (𝜑 → ∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3073 Vcvv 3445 ∅c0 4282 {csn 4586 class class class wbr 5105 ‘cfv 6496 (class class class)co 7357 ↑m cmap 8765 Fincfn 8883 ℝcr 11050 0cc0 11051 ℝ*cxr 11188 < clt 11189 ℚcq 12873 ℝ+crp 12915 distcds 17142 ∞Metcxmet 20781 Metcmet 20782 ballcbl 20783 ℝ^crrx 24747 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-tpos 8157 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xadd 13034 df-ioo 13268 df-ico 13270 df-fz 13425 df-fzo 13568 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-sum 15571 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-0g 17323 df-gsum 17324 df-prds 17329 df-pws 17331 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-mhm 18601 df-grp 18751 df-minusg 18752 df-sbg 18753 df-subg 18925 df-ghm 19006 df-cntz 19097 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-ring 19966 df-cring 19967 df-oppr 20049 df-dvdsr 20070 df-unit 20071 df-invr 20101 df-dvr 20112 df-rnghom 20146 df-drng 20187 df-field 20188 df-subrg 20220 df-staf 20304 df-srng 20305 df-lmod 20324 df-lss 20393 df-sra 20633 df-rgmod 20634 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-cnfld 20797 df-refld 21009 df-dsmm 21138 df-frlm 21153 df-nm 23938 df-tng 23940 df-tcph 24533 df-rrx 24749 |
This theorem is referenced by: qndenserrnopnlem 44528 |
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