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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 5313 | . . . . . 6 ⊢ ∅ ∈ V | |
2 | 1 | snid 4667 | . . . . 5 ⊢ ∅ ∈ {∅} |
3 | 2 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∅ ∈ {∅}) |
4 | oveq2 7439 | . . . . . 6 ⊢ (𝐼 = ∅ → (ℚ ↑m 𝐼) = (ℚ ↑m ∅)) | |
5 | qex 13001 | . . . . . . . 8 ⊢ ℚ ∈ V | |
6 | mapdm0 8881 | . . . . . . . 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 2776 | . . . . 5 ⊢ (𝐼 = ∅ → {∅} = (ℚ ↑m 𝐼)) |
10 | 9 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐼 = ∅) → {∅} = (ℚ ↑m 𝐼)) |
11 | 3, 10 | eleqtrd 2841 | . . 3 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∅ ∈ (ℚ ↑m 𝐼)) |
12 | qndenserrnbl.i | . . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ Fin) | |
13 | qndenserrnbl.d | . . . . . . . . 9 ⊢ 𝐷 = (dist‘(ℝ^‘𝐼)) | |
14 | 13 | rrxmetfi 25460 | . . . . . . . 8 ⊢ (𝐼 ∈ Fin → 𝐷 ∈ (Met‘(ℝ ↑m 𝐼))) |
15 | 12, 14 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (Met‘(ℝ ↑m 𝐼))) |
16 | metxmet 24360 | . . . . . . 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 7439 | . . . . . . . . . . 11 ⊢ (𝐼 = ∅ → (ℝ ↑m 𝐼) = (ℝ ↑m ∅)) | |
22 | reex 11244 | . . . . . . . . . . . . 13 ⊢ ℝ ∈ V | |
23 | mapdm0 8881 | . . . . . . . . . . . . 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 2775 | . . . . . . . . . 10 ⊢ (𝐼 = ∅ → (ℝ ↑m 𝐼) = {∅}) |
27 | 26 | adantl 481 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐼 = ∅) → (ℝ ↑m 𝐼) = {∅}) |
28 | 20, 27 | eleqtrd 2841 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐼 = ∅) → 𝑋 ∈ {∅}) |
29 | elsng 4645 | . . . . . . . . . 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 232 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐼 = ∅) → 𝑋 = ∅) |
33 | 32 | eqcomd 2741 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∅ = 𝑋) |
34 | 33, 20 | eqeltrd 2839 | . . . . 5 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∅ ∈ (ℝ ↑m 𝐼)) |
35 | qndenserrnbl.e | . . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ℝ+) | |
36 | 35 | rpxrd 13076 | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ*) |
37 | 35 | rpgt0d 13078 | . . . . . . 7 ⊢ (𝜑 → 0 < 𝐸) |
38 | 36, 37 | jca 511 | . . . . . 6 ⊢ (𝜑 → (𝐸 ∈ ℝ* ∧ 0 < 𝐸)) |
39 | 38 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝐼 = ∅) → (𝐸 ∈ ℝ* ∧ 0 < 𝐸)) |
40 | xblcntr 24437 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘(ℝ ↑m 𝐼)) ∧ ∅ ∈ (ℝ ↑m 𝐼) ∧ (𝐸 ∈ ℝ* ∧ 0 < 𝐸)) → ∅ ∈ (∅(ball‘𝐷)𝐸)) | |
41 | 18, 34, 39, 40 | syl3anc 1370 | . . . 4 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∅ ∈ (∅(ball‘𝐷)𝐸)) |
42 | 33 | oveq1d 7446 | . . . 4 ⊢ ((𝜑 ∧ 𝐼 = ∅) → (∅(ball‘𝐷)𝐸) = (𝑋(ball‘𝐷)𝐸)) |
43 | 41, 42 | eleqtrd 2841 | . . 3 ⊢ ((𝜑 ∧ 𝐼 = ∅) → ∅ ∈ (𝑋(ball‘𝐷)𝐸)) |
44 | eleq1 2827 | . . . 4 ⊢ (𝑦 = ∅ → (𝑦 ∈ (𝑋(ball‘𝐷)𝐸) ↔ ∅ ∈ (𝑋(ball‘𝐷)𝐸))) | |
45 | 44 | rspcev 3622 | . . 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 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐼 = ∅) → 𝐼 ∈ Fin) |
48 | neqne 2946 | . . . 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 46250 | . 2 ⊢ ((𝜑 ∧ ¬ 𝐼 = ∅) → ∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸)) |
53 | 46, 52 | pm2.61dan 813 | 1 ⊢ (𝜑 → ∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∃wrex 3068 Vcvv 3478 ∅c0 4339 {csn 4631 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 Fincfn 8984 ℝcr 11152 0cc0 11153 ℝ*cxr 11292 < clt 11293 ℚcq 12988 ℝ+crp 13032 distcds 17307 ∞Metcxmet 21367 Metcmet 21368 ballcbl 21369 ℝ^crrx 25431 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xadd 13153 df-ioo 13388 df-ico 13390 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-0g 17488 df-gsum 17489 df-prds 17494 df-pws 17496 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-ghm 19244 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-rhm 20489 df-subrng 20563 df-subrg 20587 df-drng 20748 df-field 20749 df-staf 20857 df-srng 20858 df-lmod 20877 df-lss 20948 df-sra 21190 df-rgmod 21191 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-cnfld 21383 df-refld 21641 df-dsmm 21770 df-frlm 21785 df-nm 24611 df-tng 24613 df-tcph 25217 df-rrx 25433 |
This theorem is referenced by: qndenserrnopnlem 46253 |
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