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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > nn0archi | Structured version Visualization version GIF version |
Description: The monoid of the nonnegative integers is Archimedean. (Contributed by Thierry Arnoux, 16-Sep-2018.) |
Ref | Expression |
---|---|
nn0archi | ⊢ (ℂfld ↾s ℕ0) ∈ Archi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-refld 20319 | . . . 4 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
2 | 1 | oveq1i 6920 | . . 3 ⊢ (ℝfld ↾s ℕ0) = ((ℂfld ↾s ℝ) ↾s ℕ0) |
3 | resubdrg 20322 | . . . . 5 ⊢ (ℝ ∈ (SubRing‘ℂfld) ∧ ℝfld ∈ DivRing) | |
4 | 3 | simpli 478 | . . . 4 ⊢ ℝ ∈ (SubRing‘ℂfld) |
5 | nn0ssre 11629 | . . . 4 ⊢ ℕ0 ⊆ ℝ | |
6 | ressabs 16310 | . . . 4 ⊢ ((ℝ ∈ (SubRing‘ℂfld) ∧ ℕ0 ⊆ ℝ) → ((ℂfld ↾s ℝ) ↾s ℕ0) = (ℂfld ↾s ℕ0)) | |
7 | 4, 5, 6 | mp2an 683 | . . 3 ⊢ ((ℂfld ↾s ℝ) ↾s ℕ0) = (ℂfld ↾s ℕ0) |
8 | 2, 7 | eqtri 2849 | . 2 ⊢ (ℝfld ↾s ℕ0) = (ℂfld ↾s ℕ0) |
9 | retos 20332 | . . . 4 ⊢ ℝfld ∈ Toset | |
10 | rearchi 30383 | . . . 4 ⊢ ℝfld ∈ Archi | |
11 | 9, 10 | pm3.2i 464 | . . 3 ⊢ (ℝfld ∈ Toset ∧ ℝfld ∈ Archi) |
12 | nn0subm 20168 | . . . 4 ⊢ ℕ0 ∈ (SubMnd‘ℂfld) | |
13 | subrgsubg 19149 | . . . . . 6 ⊢ (ℝ ∈ (SubRing‘ℂfld) → ℝ ∈ (SubGrp‘ℂfld)) | |
14 | subgsubm 17974 | . . . . . 6 ⊢ (ℝ ∈ (SubGrp‘ℂfld) → ℝ ∈ (SubMnd‘ℂfld)) | |
15 | 4, 13, 14 | mp2b 10 | . . . . 5 ⊢ ℝ ∈ (SubMnd‘ℂfld) |
16 | 1 | subsubm 17717 | . . . . 5 ⊢ (ℝ ∈ (SubMnd‘ℂfld) → (ℕ0 ∈ (SubMnd‘ℝfld) ↔ (ℕ0 ∈ (SubMnd‘ℂfld) ∧ ℕ0 ⊆ ℝ))) |
17 | 15, 16 | ax-mp 5 | . . . 4 ⊢ (ℕ0 ∈ (SubMnd‘ℝfld) ↔ (ℕ0 ∈ (SubMnd‘ℂfld) ∧ ℕ0 ⊆ ℝ)) |
18 | 12, 5, 17 | mpbir2an 702 | . . 3 ⊢ ℕ0 ∈ (SubMnd‘ℝfld) |
19 | submarchi 30281 | . . 3 ⊢ (((ℝfld ∈ Toset ∧ ℝfld ∈ Archi) ∧ ℕ0 ∈ (SubMnd‘ℝfld)) → (ℝfld ↾s ℕ0) ∈ Archi) | |
20 | 11, 18, 19 | mp2an 683 | . 2 ⊢ (ℝfld ↾s ℕ0) ∈ Archi |
21 | 8, 20 | eqeltrri 2903 | 1 ⊢ (ℂfld ↾s ℕ0) ∈ Archi |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ⊆ wss 3798 ‘cfv 6127 (class class class)co 6910 ℝcr 10258 ℕ0cn0 11625 ↾s cress 16230 Tosetctos 17393 SubMndcsubmnd 17694 SubGrpcsubg 17946 DivRingcdr 19110 SubRingcsubrg 19139 ℂfldccnfld 20113 ℝfldcrefld 20318 Archicarchi 30272 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-pre-sup 10337 ax-addf 10338 ax-mulf 10339 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-tpos 7622 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-map 8129 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-dec 11829 df-uz 11976 df-fz 12627 df-seq 13103 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-starv 16327 df-tset 16331 df-ple 16332 df-ds 16334 df-unif 16335 df-0g 16462 df-proset 17288 df-poset 17306 df-plt 17318 df-toset 17394 df-ps 17560 df-tsr 17561 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-mhm 17695 df-submnd 17696 df-grp 17786 df-minusg 17787 df-sbg 17788 df-mulg 17902 df-subg 17949 df-ghm 18016 df-cmn 18555 df-mgp 18851 df-ur 18863 df-ring 18910 df-cring 18911 df-oppr 18984 df-dvdsr 19002 df-unit 19003 df-invr 19033 df-dvr 19044 df-rnghom 19078 df-drng 19112 df-field 19113 df-subrg 19141 df-cnfld 20114 df-zring 20186 df-zrh 20219 df-refld 20319 df-omnd 30240 df-ogrp 30241 df-inftm 30273 df-archi 30274 df-orng 30338 df-ofld 30339 |
This theorem is referenced by: (None) |
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