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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 21641 | . . . 4 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
2 | 1 | oveq1i 7441 | . . 3 ⊢ (ℝfld ↾s ℕ0) = ((ℂfld ↾s ℝ) ↾s ℕ0) |
3 | reex 11244 | . . . 4 ⊢ ℝ ∈ V | |
4 | nn0ssre 12528 | . . . 4 ⊢ ℕ0 ⊆ ℝ | |
5 | ressabs 17295 | . . . 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 2763 | . 2 ⊢ (ℝfld ↾s ℕ0) = (ℂfld ↾s ℕ0) |
8 | reofld 33352 | . . . 4 ⊢ ℝfld ∈ oField | |
9 | isofld 33312 | . . . . . 6 ⊢ (ℝfld ∈ oField ↔ (ℝfld ∈ Field ∧ ℝfld ∈ oRing)) | |
10 | 9 | simprbi 496 | . . . . 5 ⊢ (ℝfld ∈ oField → ℝfld ∈ oRing) |
11 | orngogrp 33311 | . . . . 5 ⊢ (ℝfld ∈ oRing → ℝfld ∈ oGrp) | |
12 | isogrp 33062 | . . . . . 6 ⊢ (ℝfld ∈ oGrp ↔ (ℝfld ∈ Grp ∧ ℝfld ∈ oMnd)) | |
13 | 12 | simprbi 496 | . . . . 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 21458 | . . . . 5 ⊢ ℕ0 ∈ (SubMnd‘ℂfld) | |
17 | eqid 2735 | . . . . . 6 ⊢ (ℂfld ↾s ℕ0) = (ℂfld ↾s ℕ0) | |
18 | 17 | submmnd 18839 | . . . . 5 ⊢ (ℕ0 ∈ (SubMnd‘ℂfld) → (ℂfld ↾s ℕ0) ∈ Mnd) |
19 | 16, 18 | ax-mp 5 | . . . 4 ⊢ (ℂfld ↾s ℕ0) ∈ Mnd |
20 | 7, 19 | eqeltri 2835 | . . 3 ⊢ (ℝfld ↾s ℕ0) ∈ Mnd |
21 | submomnd 33070 | . . 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 2836 | 1 ⊢ (ℂfld ↾s ℕ0) ∈ oMnd |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 Vcvv 3478 ⊆ wss 3963 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 ℕ0cn0 12524 ↾s cress 17274 Mndcmnd 18760 SubMndcsubmnd 18808 Grpcgrp 18964 Fieldcfield 20747 ℂfldccnfld 21382 ℝfldcrefld 21640 oMndcomnd 33057 oGrpcogrp 33058 oRingcorng 33305 oFieldcofld 33306 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-0g 17488 df-proset 18352 df-poset 18371 df-plt 18388 df-toset 18475 df-ps 18624 df-tsr 18625 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-minusg 18968 df-subg 19154 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-subrng 20563 df-subrg 20587 df-drng 20748 df-field 20749 df-cnfld 21383 df-refld 21641 df-omnd 33059 df-ogrp 33060 df-orng 33307 df-ofld 33308 |
This theorem is referenced by: (None) |
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