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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 20174 | . . . 4 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
2 | 1 | oveq1i 6825 | . . 3 ⊢ (ℝfld ↾s ℕ0) = ((ℂfld ↾s ℝ) ↾s ℕ0) |
3 | reex 10240 | . . . 4 ⊢ ℝ ∈ V | |
4 | nn0ssre 11509 | . . . 4 ⊢ ℕ0 ⊆ ℝ | |
5 | ressabs 16162 | . . . 4 ⊢ ((ℝ ∈ V ∧ ℕ0 ⊆ ℝ) → ((ℂfld ↾s ℝ) ↾s ℕ0) = (ℂfld ↾s ℕ0)) | |
6 | 3, 4, 5 | mp2an 710 | . . 3 ⊢ ((ℂfld ↾s ℝ) ↾s ℕ0) = (ℂfld ↾s ℕ0) |
7 | 2, 6 | eqtri 2783 | . 2 ⊢ (ℝfld ↾s ℕ0) = (ℂfld ↾s ℕ0) |
8 | reofld 30171 | . . . 4 ⊢ ℝfld ∈ oField | |
9 | isofld 30133 | . . . . . 6 ⊢ (ℝfld ∈ oField ↔ (ℝfld ∈ Field ∧ ℝfld ∈ oRing)) | |
10 | 9 | simprbi 483 | . . . . 5 ⊢ (ℝfld ∈ oField → ℝfld ∈ oRing) |
11 | orngogrp 30132 | . . . . 5 ⊢ (ℝfld ∈ oRing → ℝfld ∈ oGrp) | |
12 | isogrp 30033 | . . . . . 6 ⊢ (ℝfld ∈ oGrp ↔ (ℝfld ∈ Grp ∧ ℝfld ∈ oMnd)) | |
13 | 12 | simprbi 483 | . . . . 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 20024 | . . . . 5 ⊢ ℕ0 ∈ (SubMnd‘ℂfld) | |
17 | eqid 2761 | . . . . . 6 ⊢ (ℂfld ↾s ℕ0) = (ℂfld ↾s ℕ0) | |
18 | 17 | submmnd 17576 | . . . . 5 ⊢ (ℕ0 ∈ (SubMnd‘ℂfld) → (ℂfld ↾s ℕ0) ∈ Mnd) |
19 | 16, 18 | ax-mp 5 | . . . 4 ⊢ (ℂfld ↾s ℕ0) ∈ Mnd |
20 | 7, 19 | eqeltri 2836 | . . 3 ⊢ (ℝfld ↾s ℕ0) ∈ Mnd |
21 | submomnd 30041 | . . 3 ⊢ ((ℝfld ∈ oMnd ∧ (ℝfld ↾s ℕ0) ∈ Mnd) → (ℝfld ↾s ℕ0) ∈ oMnd) | |
22 | 15, 20, 21 | mp2an 710 | . 2 ⊢ (ℝfld ↾s ℕ0) ∈ oMnd |
23 | 7, 22 | eqeltrri 2837 | 1 ⊢ (ℂfld ↾s ℕ0) ∈ oMnd |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1632 ∈ wcel 2140 Vcvv 3341 ⊆ wss 3716 ‘cfv 6050 (class class class)co 6815 ℝcr 10148 ℕ0cn0 11505 ↾s cress 16081 Mndcmnd 17516 SubMndcsubmnd 17556 Grpcgrp 17644 Fieldcfield 18971 ℂfldccnfld 19969 ℝfldcrefld 20173 oMndcomnd 30028 oGrpcogrp 30029 oRingcorng 30126 oFieldcofld 30127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1989 ax-6 2055 ax-7 2091 ax-8 2142 ax-9 2149 ax-10 2169 ax-11 2184 ax-12 2197 ax-13 2392 ax-ext 2741 ax-rep 4924 ax-sep 4934 ax-nul 4942 ax-pow 4993 ax-pr 5056 ax-un 7116 ax-cnex 10205 ax-resscn 10206 ax-1cn 10207 ax-icn 10208 ax-addcl 10209 ax-addrcl 10210 ax-mulcl 10211 ax-mulrcl 10212 ax-mulcom 10213 ax-addass 10214 ax-mulass 10215 ax-distr 10216 ax-i2m1 10217 ax-1ne0 10218 ax-1rid 10219 ax-rnegex 10220 ax-rrecex 10221 ax-cnre 10222 ax-pre-lttri 10223 ax-pre-lttrn 10224 ax-pre-ltadd 10225 ax-pre-mulgt0 10226 ax-addf 10228 ax-mulf 10229 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2048 df-eu 2612 df-mo 2613 df-clab 2748 df-cleq 2754 df-clel 2757 df-nfc 2892 df-ne 2934 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3343 df-sbc 3578 df-csb 3676 df-dif 3719 df-un 3721 df-in 3723 df-ss 3730 df-pss 3732 df-nul 4060 df-if 4232 df-pw 4305 df-sn 4323 df-pr 4325 df-tp 4327 df-op 4329 df-uni 4590 df-int 4629 df-iun 4675 df-br 4806 df-opab 4866 df-mpt 4883 df-tr 4906 df-id 5175 df-eprel 5180 df-po 5188 df-so 5189 df-fr 5226 df-we 5228 df-xp 5273 df-rel 5274 df-cnv 5275 df-co 5276 df-dm 5277 df-rn 5278 df-res 5279 df-ima 5280 df-pred 5842 df-ord 5888 df-on 5889 df-lim 5890 df-suc 5891 df-iota 6013 df-fun 6052 df-fn 6053 df-f 6054 df-f1 6055 df-fo 6056 df-f1o 6057 df-fv 6058 df-riota 6776 df-ov 6818 df-oprab 6819 df-mpt2 6820 df-om 7233 df-1st 7335 df-2nd 7336 df-tpos 7523 df-wrecs 7578 df-recs 7639 df-rdg 7677 df-1o 7731 df-oadd 7735 df-er 7914 df-en 8125 df-dom 8126 df-sdom 8127 df-fin 8128 df-pnf 10289 df-mnf 10290 df-xr 10291 df-ltxr 10292 df-le 10293 df-sub 10481 df-neg 10482 df-div 10898 df-nn 11234 df-2 11292 df-3 11293 df-4 11294 df-5 11295 df-6 11296 df-7 11297 df-8 11298 df-9 11299 df-n0 11506 df-z 11591 df-dec 11707 df-uz 11901 df-fz 12541 df-struct 16082 df-ndx 16083 df-slot 16084 df-base 16086 df-sets 16087 df-ress 16088 df-plusg 16177 df-mulr 16178 df-starv 16179 df-tset 16183 df-ple 16184 df-ds 16187 df-unif 16188 df-0g 16325 df-preset 17150 df-poset 17168 df-plt 17180 df-toset 17256 df-ps 17422 df-tsr 17423 df-mgm 17464 df-sgrp 17506 df-mnd 17517 df-submnd 17558 df-grp 17647 df-minusg 17648 df-subg 17813 df-cmn 18416 df-mgp 18711 df-ur 18723 df-ring 18770 df-cring 18771 df-oppr 18844 df-dvdsr 18862 df-unit 18863 df-invr 18893 df-dvr 18904 df-drng 18972 df-field 18973 df-subrg 19001 df-cnfld 19970 df-refld 20174 df-omnd 30030 df-ogrp 30031 df-orng 30128 df-ofld 30129 |
This theorem is referenced by: (None) |
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