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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 20431 | . . . 4 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
2 | 1 | oveq1i 7026 | . . 3 ⊢ (ℝfld ↾s ℕ0) = ((ℂfld ↾s ℝ) ↾s ℕ0) |
3 | reex 10474 | . . . 4 ⊢ ℝ ∈ V | |
4 | nn0ssre 11749 | . . . 4 ⊢ ℕ0 ⊆ ℝ | |
5 | ressabs 16392 | . . . 4 ⊢ ((ℝ ∈ V ∧ ℕ0 ⊆ ℝ) → ((ℂfld ↾s ℝ) ↾s ℕ0) = (ℂfld ↾s ℕ0)) | |
6 | 3, 4, 5 | mp2an 688 | . . 3 ⊢ ((ℂfld ↾s ℝ) ↾s ℕ0) = (ℂfld ↾s ℕ0) |
7 | 2, 6 | eqtri 2819 | . 2 ⊢ (ℝfld ↾s ℕ0) = (ℂfld ↾s ℕ0) |
8 | reofld 30567 | . . . 4 ⊢ ℝfld ∈ oField | |
9 | isofld 30529 | . . . . . 6 ⊢ (ℝfld ∈ oField ↔ (ℝfld ∈ Field ∧ ℝfld ∈ oRing)) | |
10 | 9 | simprbi 497 | . . . . 5 ⊢ (ℝfld ∈ oField → ℝfld ∈ oRing) |
11 | orngogrp 30528 | . . . . 5 ⊢ (ℝfld ∈ oRing → ℝfld ∈ oGrp) | |
12 | isogrp 30363 | . . . . . 6 ⊢ (ℝfld ∈ oGrp ↔ (ℝfld ∈ Grp ∧ ℝfld ∈ oMnd)) | |
13 | 12 | simprbi 497 | . . . . 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 20282 | . . . . 5 ⊢ ℕ0 ∈ (SubMnd‘ℂfld) | |
17 | eqid 2795 | . . . . . 6 ⊢ (ℂfld ↾s ℕ0) = (ℂfld ↾s ℕ0) | |
18 | 17 | submmnd 17793 | . . . . 5 ⊢ (ℕ0 ∈ (SubMnd‘ℂfld) → (ℂfld ↾s ℕ0) ∈ Mnd) |
19 | 16, 18 | ax-mp 5 | . . . 4 ⊢ (ℂfld ↾s ℕ0) ∈ Mnd |
20 | 7, 19 | eqeltri 2879 | . . 3 ⊢ (ℝfld ↾s ℕ0) ∈ Mnd |
21 | submomnd 30371 | . . 3 ⊢ ((ℝfld ∈ oMnd ∧ (ℝfld ↾s ℕ0) ∈ Mnd) → (ℝfld ↾s ℕ0) ∈ oMnd) | |
22 | 15, 20, 21 | mp2an 688 | . 2 ⊢ (ℝfld ↾s ℕ0) ∈ oMnd |
23 | 7, 22 | eqeltrri 2880 | 1 ⊢ (ℂfld ↾s ℕ0) ∈ oMnd |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1522 ∈ wcel 2081 Vcvv 3437 ⊆ wss 3859 ‘cfv 6225 (class class class)co 7016 ℝcr 10382 ℕ0cn0 11745 ↾s cress 16313 Mndcmnd 17733 SubMndcsubmnd 17773 Grpcgrp 17861 Fieldcfield 19193 ℂfldccnfld 20227 ℝfldcrefld 20430 oMndcomnd 30358 oGrpcogrp 30359 oRingcorng 30522 oFieldcofld 30523 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-addf 10462 ax-mulf 10463 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-tpos 7743 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-oadd 7957 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-5 11551 df-6 11552 df-7 11553 df-8 11554 df-9 11555 df-n0 11746 df-z 11830 df-dec 11948 df-uz 12094 df-fz 12743 df-struct 16314 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-ress 16320 df-plusg 16407 df-mulr 16408 df-starv 16409 df-tset 16413 df-ple 16414 df-ds 16416 df-unif 16417 df-0g 16544 df-proset 17367 df-poset 17385 df-plt 17397 df-toset 17473 df-ps 17639 df-tsr 17640 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-submnd 17775 df-grp 17864 df-minusg 17865 df-subg 18030 df-cmn 18635 df-mgp 18930 df-ur 18942 df-ring 18989 df-cring 18990 df-oppr 19063 df-dvdsr 19081 df-unit 19082 df-invr 19112 df-dvr 19123 df-drng 19194 df-field 19195 df-subrg 19223 df-cnfld 20228 df-refld 20431 df-omnd 30360 df-ogrp 30361 df-orng 30524 df-ofld 30525 |
This theorem is referenced by: (None) |
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