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 20752 | . . . 4 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
2 | 1 | oveq1i 7169 | . . 3 ⊢ (ℝfld ↾s ℕ0) = ((ℂfld ↾s ℝ) ↾s ℕ0) |
3 | resubdrg 20755 | . . . . 5 ⊢ (ℝ ∈ (SubRing‘ℂfld) ∧ ℝfld ∈ DivRing) | |
4 | 3 | simpli 486 | . . . 4 ⊢ ℝ ∈ (SubRing‘ℂfld) |
5 | nn0ssre 11904 | . . . 4 ⊢ ℕ0 ⊆ ℝ | |
6 | ressabs 16566 | . . . 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 2847 | . 2 ⊢ (ℝfld ↾s ℕ0) = (ℂfld ↾s ℕ0) |
9 | retos 20765 | . . . 4 ⊢ ℝfld ∈ Toset | |
10 | rearchi 30919 | . . . 4 ⊢ ℝfld ∈ Archi | |
11 | 9, 10 | pm3.2i 473 | . . 3 ⊢ (ℝfld ∈ Toset ∧ ℝfld ∈ Archi) |
12 | nn0subm 20603 | . . . 4 ⊢ ℕ0 ∈ (SubMnd‘ℂfld) | |
13 | subrgsubg 19544 | . . . . . 6 ⊢ (ℝ ∈ (SubRing‘ℂfld) → ℝ ∈ (SubGrp‘ℂfld)) | |
14 | subgsubm 18304 | . . . . . 6 ⊢ (ℝ ∈ (SubGrp‘ℂfld) → ℝ ∈ (SubMnd‘ℂfld)) | |
15 | 4, 13, 14 | mp2b 10 | . . . . 5 ⊢ ℝ ∈ (SubMnd‘ℂfld) |
16 | 1 | subsubm 17984 | . . . . 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 30819 | . . 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 2913 | 1 ⊢ (ℂfld ↾s ℕ0) ∈ Archi |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ⊆ wss 3939 ‘cfv 6358 (class class class)co 7159 ℝcr 10539 ℕ0cn0 11900 ↾s cress 16487 Tosetctos 17646 SubMndcsubmnd 17958 SubGrpcsubg 18276 DivRingcdr 19505 SubRingcsubrg 19534 ℂfldccnfld 20548 ℝfldcrefld 20751 Archicarchi 30810 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 ax-addf 10619 ax-mulf 10620 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-tpos 7895 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-seq 13373 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-starv 16583 df-tset 16587 df-ple 16588 df-ds 16590 df-unif 16591 df-0g 16718 df-proset 17541 df-poset 17559 df-plt 17571 df-toset 17647 df-ps 17813 df-tsr 17814 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-mhm 17959 df-submnd 17960 df-grp 18109 df-minusg 18110 df-sbg 18111 df-mulg 18228 df-subg 18279 df-ghm 18359 df-cmn 18911 df-mgp 19243 df-ur 19255 df-ring 19302 df-cring 19303 df-oppr 19376 df-dvdsr 19394 df-unit 19395 df-invr 19425 df-dvr 19436 df-rnghom 19470 df-drng 19507 df-field 19508 df-subrg 19536 df-cnfld 20549 df-zring 20621 df-zrh 20654 df-refld 20752 df-omnd 30704 df-ogrp 30705 df-inftm 30811 df-archi 30812 df-orng 30874 df-ofld 30875 |
This theorem is referenced by: (None) |
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