Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > nn0omnd | Structured version Visualization version Unicode version |
Description: The nonnegative integers form an ordered monoid. (Contributed by Thierry Arnoux, 23-Mar-2018.) |
Ref | Expression |
---|---|
nn0omnd | ℂfld ↾s oMnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-refld 19951 | . . . 4 RRfld ℂfld ↾s | |
2 | 1 | oveq1i 6660 | . . 3 RRfld ↾s ℂfld ↾s ↾s |
3 | reex 10027 | . . . 4 | |
4 | nn0ssre 11296 | . . . 4 | |
5 | ressabs 15939 | . . . 4 ℂfld ↾s ↾s ℂfld ↾s | |
6 | 3, 4, 5 | mp2an 708 | . . 3 ℂfld ↾s ↾s ℂfld ↾s |
7 | 2, 6 | eqtri 2644 | . 2 RRfld ↾s ℂfld ↾s |
8 | reofld 29840 | . . . 4 RRfld oField | |
9 | isofld 29802 | . . . . . 6 RRfld oField RRfld Field RRfld oRing | |
10 | 9 | simprbi 480 | . . . . 5 RRfld oField RRfld oRing |
11 | orngogrp 29801 | . . . . 5 RRfld oRing RRfld oGrp | |
12 | isogrp 29702 | . . . . . 6 RRfld oGrp RRfld RRfld oMnd | |
13 | 12 | simprbi 480 | . . . . 5 RRfld oGrp RRfld oMnd |
14 | 10, 11, 13 | 3syl 18 | . . . 4 RRfld oField RRfld oMnd |
15 | 8, 14 | ax-mp 5 | . . 3 RRfld oMnd |
16 | nn0subm 19801 | . . . . 5 SubMndℂfld | |
17 | eqid 2622 | . . . . . 6 ℂfld ↾s ℂfld ↾s | |
18 | 17 | submmnd 17354 | . . . . 5 SubMndℂfld ℂfld ↾s |
19 | 16, 18 | ax-mp 5 | . . . 4 ℂfld ↾s |
20 | 7, 19 | eqeltri 2697 | . . 3 RRfld ↾s |
21 | submomnd 29710 | . . 3 RRfld oMnd RRfld ↾s RRfld ↾s oMnd | |
22 | 15, 20, 21 | mp2an 708 | . 2 RRfld ↾s oMnd |
23 | 7, 22 | eqeltrri 2698 | 1 ℂfld ↾s oMnd |
Colors of variables: wff setvar class |
Syntax hints: wceq 1483 wcel 1990 cvv 3200 wss 3574 cfv 5888 (class class class)co 6650 cr 9935 cn0 11292 ↾s cress 15858 cmnd 17294 SubMndcsubmnd 17334 cgrp 17422 Fieldcfield 18748 ℂfldccnfld 19746 RRfldcrefld 19950 oMndcomnd 29697 oGrpcogrp 29698 oRingcorng 29795 oFieldcofld 29796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-0g 16102 df-preset 16928 df-poset 16946 df-plt 16958 df-toset 17034 df-ps 17200 df-tsr 17201 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-subg 17591 df-cmn 18195 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 df-field 18750 df-subrg 18778 df-cnfld 19747 df-refld 19951 df-omnd 29699 df-ogrp 29700 df-orng 29797 df-ofld 29798 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |