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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 21544 | . . . 4 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
2 | 1 | oveq1i 7436 | . . 3 ⊢ (ℝfld ↾s ℕ0) = ((ℂfld ↾s ℝ) ↾s ℕ0) |
3 | reex 11237 | . . . 4 ⊢ ℝ ∈ V | |
4 | nn0ssre 12514 | . . . 4 ⊢ ℕ0 ⊆ ℝ | |
5 | ressabs 17237 | . . . 4 ⊢ ((ℝ ∈ V ∧ ℕ0 ⊆ ℝ) → ((ℂfld ↾s ℝ) ↾s ℕ0) = (ℂfld ↾s ℕ0)) | |
6 | 3, 4, 5 | mp2an 690 | . . 3 ⊢ ((ℂfld ↾s ℝ) ↾s ℕ0) = (ℂfld ↾s ℕ0) |
7 | 2, 6 | eqtri 2756 | . 2 ⊢ (ℝfld ↾s ℕ0) = (ℂfld ↾s ℕ0) |
8 | reofld 33080 | . . . 4 ⊢ ℝfld ∈ oField | |
9 | isofld 33041 | . . . . . 6 ⊢ (ℝfld ∈ oField ↔ (ℝfld ∈ Field ∧ ℝfld ∈ oRing)) | |
10 | 9 | simprbi 495 | . . . . 5 ⊢ (ℝfld ∈ oField → ℝfld ∈ oRing) |
11 | orngogrp 33040 | . . . . 5 ⊢ (ℝfld ∈ oRing → ℝfld ∈ oGrp) | |
12 | isogrp 32803 | . . . . . 6 ⊢ (ℝfld ∈ oGrp ↔ (ℝfld ∈ Grp ∧ ℝfld ∈ oMnd)) | |
13 | 12 | simprbi 495 | . . . . 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 21362 | . . . . 5 ⊢ ℕ0 ∈ (SubMnd‘ℂfld) | |
17 | eqid 2728 | . . . . . 6 ⊢ (ℂfld ↾s ℕ0) = (ℂfld ↾s ℕ0) | |
18 | 17 | submmnd 18772 | . . . . 5 ⊢ (ℕ0 ∈ (SubMnd‘ℂfld) → (ℂfld ↾s ℕ0) ∈ Mnd) |
19 | 16, 18 | ax-mp 5 | . . . 4 ⊢ (ℂfld ↾s ℕ0) ∈ Mnd |
20 | 7, 19 | eqeltri 2825 | . . 3 ⊢ (ℝfld ↾s ℕ0) ∈ Mnd |
21 | submomnd 32811 | . . 3 ⊢ ((ℝfld ∈ oMnd ∧ (ℝfld ↾s ℕ0) ∈ Mnd) → (ℝfld ↾s ℕ0) ∈ oMnd) | |
22 | 15, 20, 21 | mp2an 690 | . 2 ⊢ (ℝfld ↾s ℕ0) ∈ oMnd |
23 | 7, 22 | eqeltrri 2826 | 1 ⊢ (ℂfld ↾s ℕ0) ∈ oMnd |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 Vcvv 3473 ⊆ wss 3949 ‘cfv 6553 (class class class)co 7426 ℝcr 11145 ℕ0cn0 12510 ↾s cress 17216 Mndcmnd 18701 SubMndcsubmnd 18746 Grpcgrp 18897 Fieldcfield 20632 ℂfldccnfld 21286 ℝfldcrefld 21543 oMndcomnd 32798 oGrpcogrp 32799 oRingcorng 33034 oFieldcofld 33035 |
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 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-addf 11225 ax-mulf 11226 |
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 7877 df-1st 7999 df-2nd 8000 df-tpos 8238 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-0g 17430 df-proset 18294 df-poset 18312 df-plt 18329 df-toset 18416 df-ps 18565 df-tsr 18566 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 df-grp 18900 df-minusg 18901 df-subg 19085 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-cring 20183 df-oppr 20280 df-dvdsr 20303 df-unit 20304 df-invr 20334 df-dvr 20347 df-subrng 20490 df-subrg 20515 df-drng 20633 df-field 20634 df-cnfld 21287 df-refld 21544 df-omnd 32800 df-ogrp 32801 df-orng 33036 df-ofld 33037 |
This theorem is referenced by: (None) |
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