Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > nn0omnd | Structured version Visualization version GIF version |
Description: The nonnegative integers form an ordered monoid. (Contributed by Thierry Arnoux, 23-Mar-2018.) |
Ref | Expression |
---|---|
nn0omnd | ⊢ (ℂfld ↾s ℕ0) ∈ oMnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-refld 20521 | . . . 4 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
2 | 1 | oveq1i 7201 | . . 3 ⊢ (ℝfld ↾s ℕ0) = ((ℂfld ↾s ℝ) ↾s ℕ0) |
3 | reex 10785 | . . . 4 ⊢ ℝ ∈ V | |
4 | nn0ssre 12059 | . . . 4 ⊢ ℕ0 ⊆ ℝ | |
5 | ressabs 16747 | . . . 4 ⊢ ((ℝ ∈ V ∧ ℕ0 ⊆ ℝ) → ((ℂfld ↾s ℝ) ↾s ℕ0) = (ℂfld ↾s ℕ0)) | |
6 | 3, 4, 5 | mp2an 692 | . . 3 ⊢ ((ℂfld ↾s ℝ) ↾s ℕ0) = (ℂfld ↾s ℕ0) |
7 | 2, 6 | eqtri 2759 | . 2 ⊢ (ℝfld ↾s ℕ0) = (ℂfld ↾s ℕ0) |
8 | reofld 31212 | . . . 4 ⊢ ℝfld ∈ oField | |
9 | isofld 31174 | . . . . . 6 ⊢ (ℝfld ∈ oField ↔ (ℝfld ∈ Field ∧ ℝfld ∈ oRing)) | |
10 | 9 | simprbi 500 | . . . . 5 ⊢ (ℝfld ∈ oField → ℝfld ∈ oRing) |
11 | orngogrp 31173 | . . . . 5 ⊢ (ℝfld ∈ oRing → ℝfld ∈ oGrp) | |
12 | isogrp 31001 | . . . . . 6 ⊢ (ℝfld ∈ oGrp ↔ (ℝfld ∈ Grp ∧ ℝfld ∈ oMnd)) | |
13 | 12 | simprbi 500 | . . . . 5 ⊢ (ℝfld ∈ oGrp → ℝfld ∈ oMnd) |
14 | 10, 11, 13 | 3syl 18 | . . . 4 ⊢ (ℝfld ∈ oField → ℝfld ∈ oMnd) |
15 | 8, 14 | ax-mp 5 | . . 3 ⊢ ℝfld ∈ oMnd |
16 | nn0subm 20372 | . . . . 5 ⊢ ℕ0 ∈ (SubMnd‘ℂfld) | |
17 | eqid 2736 | . . . . . 6 ⊢ (ℂfld ↾s ℕ0) = (ℂfld ↾s ℕ0) | |
18 | 17 | submmnd 18194 | . . . . 5 ⊢ (ℕ0 ∈ (SubMnd‘ℂfld) → (ℂfld ↾s ℕ0) ∈ Mnd) |
19 | 16, 18 | ax-mp 5 | . . . 4 ⊢ (ℂfld ↾s ℕ0) ∈ Mnd |
20 | 7, 19 | eqeltri 2827 | . . 3 ⊢ (ℝfld ↾s ℕ0) ∈ Mnd |
21 | submomnd 31009 | . . 3 ⊢ ((ℝfld ∈ oMnd ∧ (ℝfld ↾s ℕ0) ∈ Mnd) → (ℝfld ↾s ℕ0) ∈ oMnd) | |
22 | 15, 20, 21 | mp2an 692 | . 2 ⊢ (ℝfld ↾s ℕ0) ∈ oMnd |
23 | 7, 22 | eqeltrri 2828 | 1 ⊢ (ℂfld ↾s ℕ0) ∈ oMnd |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2112 Vcvv 3398 ⊆ wss 3853 ‘cfv 6358 (class class class)co 7191 ℝcr 10693 ℕ0cn0 12055 ↾s cress 16667 Mndcmnd 18127 SubMndcsubmnd 18171 Grpcgrp 18319 Fieldcfield 19722 ℂfldccnfld 20317 ℝfldcrefld 20520 oMndcomnd 30996 oGrpcogrp 30997 oRingcorng 31167 oFieldcofld 31168 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-addf 10773 ax-mulf 10774 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-tpos 7946 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-fz 13061 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-starv 16764 df-tset 16768 df-ple 16769 df-ds 16771 df-unif 16772 df-0g 16900 df-proset 17756 df-poset 17774 df-plt 17790 df-toset 17877 df-ps 18026 df-tsr 18027 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-submnd 18173 df-grp 18322 df-minusg 18323 df-subg 18494 df-cmn 19126 df-mgp 19459 df-ur 19471 df-ring 19518 df-cring 19519 df-oppr 19595 df-dvdsr 19613 df-unit 19614 df-invr 19644 df-dvr 19655 df-drng 19723 df-field 19724 df-subrg 19752 df-cnfld 20318 df-refld 20521 df-omnd 30998 df-ogrp 30999 df-orng 31169 df-ofld 31170 |
This theorem is referenced by: (None) |
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