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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 21551 | . . . 4 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
2 | 1 | oveq1i 7436 | . . 3 ⊢ (ℝfld ↾s ℕ0) = ((ℂfld ↾s ℝ) ↾s ℕ0) |
3 | resubdrg 21554 | . . . . 5 ⊢ (ℝ ∈ (SubRing‘ℂfld) ∧ ℝfld ∈ DivRing) | |
4 | 3 | simpli 482 | . . . 4 ⊢ ℝ ∈ (SubRing‘ℂfld) |
5 | nn0ssre 12516 | . . . 4 ⊢ ℕ0 ⊆ ℝ | |
6 | ressabs 17239 | . . . 4 ⊢ ((ℝ ∈ (SubRing‘ℂfld) ∧ ℕ0 ⊆ ℝ) → ((ℂfld ↾s ℝ) ↾s ℕ0) = (ℂfld ↾s ℕ0)) | |
7 | 4, 5, 6 | mp2an 690 | . . 3 ⊢ ((ℂfld ↾s ℝ) ↾s ℕ0) = (ℂfld ↾s ℕ0) |
8 | 2, 7 | eqtri 2756 | . 2 ⊢ (ℝfld ↾s ℕ0) = (ℂfld ↾s ℕ0) |
9 | retos 21564 | . . . 4 ⊢ ℝfld ∈ Toset | |
10 | rearchi 33090 | . . . 4 ⊢ ℝfld ∈ Archi | |
11 | 9, 10 | pm3.2i 469 | . . 3 ⊢ (ℝfld ∈ Toset ∧ ℝfld ∈ Archi) |
12 | nn0subm 21369 | . . . 4 ⊢ ℕ0 ∈ (SubMnd‘ℂfld) | |
13 | subrgsubg 20530 | . . . . . 6 ⊢ (ℝ ∈ (SubRing‘ℂfld) → ℝ ∈ (SubGrp‘ℂfld)) | |
14 | subgsubm 19117 | . . . . . 6 ⊢ (ℝ ∈ (SubGrp‘ℂfld) → ℝ ∈ (SubMnd‘ℂfld)) | |
15 | 4, 13, 14 | mp2b 10 | . . . . 5 ⊢ ℝ ∈ (SubMnd‘ℂfld) |
16 | 1 | subsubm 18782 | . . . . 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 709 | . . 3 ⊢ ℕ0 ∈ (SubMnd‘ℝfld) |
19 | submarchi 32923 | . . 3 ⊢ (((ℝfld ∈ Toset ∧ ℝfld ∈ Archi) ∧ ℕ0 ∈ (SubMnd‘ℝfld)) → (ℝfld ↾s ℕ0) ∈ Archi) | |
20 | 11, 18, 19 | mp2an 690 | . 2 ⊢ (ℝfld ↾s ℕ0) ∈ Archi |
21 | 8, 20 | eqeltrri 2826 | 1 ⊢ (ℂfld ↾s ℕ0) ∈ Archi |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ⊆ wss 3949 ‘cfv 6553 (class class class)co 7426 ℝcr 11147 ℕ0cn0 12512 ↾s cress 17218 Tosetctos 18417 SubMndcsubmnd 18748 SubGrpcsubg 19089 SubRingcsubrg 20520 DivRingcdr 20638 ℂfldccnfld 21293 ℝfldcrefld 21550 Archicarchi 32914 |
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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-pre-sup 11226 ax-addf 11227 ax-mulf 11228 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-tpos 8240 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-map 8855 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-7 12320 df-8 12321 df-9 12322 df-n0 12513 df-z 12599 df-dec 12718 df-uz 12863 df-fz 13527 df-seq 14009 df-struct 17125 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-mulr 17256 df-starv 17257 df-tset 17261 df-ple 17262 df-ds 17264 df-unif 17265 df-0g 17432 df-proset 18296 df-poset 18314 df-plt 18331 df-toset 18418 df-ps 18567 df-tsr 18568 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-mhm 18749 df-submnd 18750 df-grp 18907 df-minusg 18908 df-sbg 18909 df-mulg 19038 df-subg 19092 df-ghm 19182 df-cmn 19751 df-abl 19752 df-mgp 20089 df-rng 20107 df-ur 20136 df-ring 20189 df-cring 20190 df-oppr 20287 df-dvdsr 20310 df-unit 20311 df-invr 20341 df-dvr 20354 df-rhm 20425 df-subrng 20497 df-subrg 20522 df-drng 20640 df-field 20641 df-cnfld 21294 df-zring 21387 df-zrh 21443 df-refld 21551 df-omnd 32808 df-ogrp 32809 df-inftm 32915 df-archi 32916 df-orng 33044 df-ofld 33045 |
This theorem is referenced by: (None) |
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