Theorem nn0srg 19816

Description: The nonnegative integers form a semiring (commutative by subcmn 18242). (Contributed by Thierry Arnoux, 1-May-2018.)

Assertion

Ref	Expression
nn0srg	|- (ℂfld ↾s NN0 ) e. SRing

Proof of Theorem nn0srg

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnring 19768	. . . 4 |- ℂfld e. Ring
2		ringcmn 18581	. . . 4 |- (ℂfld e. Ring -> ℂfld e. CMnd )
3	1, 2	ax-mp 5	. . 3 |- ℂfld e. CMnd
4		nn0subm 19801	. . 3 |- NN0 e. (SubMnd ` ℂfld )
5		eqid 2622	. . . 4 |- (ℂfld ↾s NN0 ) = (ℂfld ↾s NN0 )
6	5	submcmn 18243	. . 3 |- ( (ℂfld e. CMnd /\ NN0 e. (SubMnd ` ℂfld ) ) -> (ℂfld ↾s NN0 ) e. CMnd )
7	3, 4, 6	mp2an 708	. 2 |- (ℂfld ↾s NN0 ) e. CMnd
8		nn0ex 11298	. . . 4 |- NN0 e. _V
9		eqid 2622	. . . . 5 |- (mulGrp ` ℂfld ) = (mulGrp ` ℂfld )
10	5, 9	mgpress 18500	. . . 4 |- ( (ℂfld e. CMnd /\ NN0 e. _V ) -> ( (mulGrp ` ℂfld ) ↾s NN0 ) = (mulGrp ` (ℂfld ↾s NN0 ) ) )
11	3, 8, 10	mp2an 708	. . 3 |- ( (mulGrp ` ℂfld ) ↾s NN0 ) = (mulGrp ` (ℂfld ↾s NN0 ) )
12		nn0sscn 11297	. . . . 5 |- NN0 C_ CC
13		1nn0 11308	. . . . 5 |- 1 e. NN0
14		nn0mulcl 11329	. . . . . 6 |- ( ( x e. NN0 /\ y e. NN0 ) -> ( x x. y ) e. NN0 )
15	14	rgen2a 2977	. . . . 5 |- A. x e. NN0 A. y e. NN0 ( x x. y ) e. NN0
16	9	ringmgp 18553	. . . . . . 7 |- (ℂfld e. Ring -> (mulGrp ` ℂfld ) e. Mnd )
17	1, 16	ax-mp 5	. . . . . 6 |- (mulGrp ` ℂfld ) e. Mnd
18		cnfldbas 19750	. . . . . . . 8 |- CC = ( Base ` ℂfld )
19	9, 18	mgpbas 18495	. . . . . . 7 |- CC = ( Base ` (mulGrp ` ℂfld ) )
20		cnfld1 19771	. . . . . . . 8 |- 1 = ( 1r ` ℂfld )
21	9, 20	ringidval 18503	. . . . . . 7 |- 1 = ( 0g ` (mulGrp ` ℂfld ) )
22		cnfldmul 19752	. . . . . . . 8 |- x. = ( .r ` ℂfld )
23	9, 22	mgpplusg 18493	. . . . . . 7 |- x. = ( +g ` (mulGrp ` ℂfld ) )
24	19, 21, 23	issubm 17347	. . . . . 6 |- ( (mulGrp ` ℂfld ) e. Mnd -> ( NN0 e. (SubMnd ` (mulGrp ` ℂfld ) ) <-> ( NN0 C_ CC /\ 1 e. NN0 /\ A. x e. NN0 A. y e. NN0 ( x x. y ) e. NN0 ) ) )
25	17, 24	ax-mp 5	. . . . 5 |- ( NN0 e. (SubMnd ` (mulGrp ` ℂfld ) ) <-> ( NN0 C_ CC /\ 1 e. NN0 /\ A. x e. NN0 A. y e. NN0 ( x x. y ) e. NN0 ) )
26	12, 13, 15, 25	mpbir3an 1244	. . . 4 |- NN0 e. (SubMnd ` (mulGrp ` ℂfld ) )
27		eqid 2622	. . . . 5 |- ( (mulGrp ` ℂfld ) ↾s NN0 ) = ( (mulGrp ` ℂfld ) ↾s NN0 )
28	27	submmnd 17354	. . . 4 |- ( NN0 e. (SubMnd ` (mulGrp ` ℂfld ) ) -> ( (mulGrp ` ℂfld ) ↾s NN0 ) e. Mnd )
29	26, 28	ax-mp 5	. . 3 |- ( (mulGrp ` ℂfld ) ↾s NN0 ) e. Mnd
30	11, 29	eqeltrri 2698	. 2 |- (mulGrp ` (ℂfld ↾s NN0 ) ) e. Mnd
31		simpl 473	. . . . . . . 8 |- ( ( x e. NN0 /\ ( y e. NN0 /\ z e. NN0 ) ) -> x e. NN0 )
32	31	nn0cnd 11353	. . . . . . 7 |- ( ( x e. NN0 /\ ( y e. NN0 /\ z e. NN0 ) ) -> x e. CC )
33		simprl 794	. . . . . . . 8 |- ( ( x e. NN0 /\ ( y e. NN0 /\ z e. NN0 ) ) -> y e. NN0 )
34	33	nn0cnd 11353	. . . . . . 7 |- ( ( x e. NN0 /\ ( y e. NN0 /\ z e. NN0 ) ) -> y e. CC )
35		simprr 796	. . . . . . . 8 |- ( ( x e. NN0 /\ ( y e. NN0 /\ z e. NN0 ) ) -> z e. NN0 )
36	35	nn0cnd 11353	. . . . . . 7 |- ( ( x e. NN0 /\ ( y e. NN0 /\ z e. NN0 ) ) -> z e. CC )
37	32, 34, 36	adddid 10064	. . . . . 6 |- ( ( x e. NN0 /\ ( y e. NN0 /\ z e. NN0 ) ) -> ( x x. ( y + z ) ) = ( ( x x. y ) + ( x x. z ) ) )
38	32, 34, 36	adddird 10065	. . . . . 6 |- ( ( x e. NN0 /\ ( y e. NN0 /\ z e. NN0 ) ) -> ( ( x + y ) x. z ) = ( ( x x. z ) + ( y x. z ) ) )
39	37, 38	jca 554	. . . . 5 |- ( ( x e. NN0 /\ ( y e. NN0 /\ z e. NN0 ) ) -> ( ( x x. ( y + z ) ) = ( ( x x. y ) + ( x x. z ) ) /\ ( ( x + y ) x. z ) = ( ( x x. z ) + ( y x. z ) ) ) )
40	39	ralrimivva 2971	. . . 4 |- ( x e. NN0 -> A. y e. NN0 A. z e. NN0 ( ( x x. ( y + z ) ) = ( ( x x. y ) + ( x x. z ) ) /\ ( ( x + y ) x. z ) = ( ( x x. z ) + ( y x. z ) ) ) )
41		nn0cn 11302	. . . . 5 |- ( x e. NN0 -> x e. CC )
42	41	mul02d 10234	. . . 4 |- ( x e. NN0 -> ( 0 x. x ) = 0 )
43	41	mul01d 10235	. . . 4 |- ( x e. NN0 -> ( x x. 0 ) = 0 )
44	40, 42, 43	jca32 558	. . 3 |- ( x e. NN0 -> ( A. y e. NN0 A. z e. NN0 ( ( x x. ( y + z ) ) = ( ( x x. y ) + ( x x. z ) ) /\ ( ( x + y ) x. z ) = ( ( x x. z ) + ( y x. z ) ) ) /\ ( ( 0 x. x ) = 0 /\ ( x x. 0 ) = 0 ) ) )
45	44	rgen 2922	. 2 |- A. x e. NN0 ( A. y e. NN0 A. z e. NN0 ( ( x x. ( y + z ) ) = ( ( x x. y ) + ( x x. z ) ) /\ ( ( x + y ) x. z ) = ( ( x x. z ) + ( y x. z ) ) ) /\ ( ( 0 x. x ) = 0 /\ ( x x. 0 ) = 0 ) )
46	5, 18	ressbas2 15931	. . . 4 |- ( NN0 C_ CC -> NN0 = ( Base ` (ℂfld ↾s NN0 ) ) )
47	12, 46	ax-mp 5	. . 3 |- NN0 = ( Base ` (ℂfld ↾s NN0 ) )
48		eqid 2622	. . 3 |- (mulGrp ` (ℂfld ↾s NN0 ) ) = (mulGrp ` (ℂfld ↾s NN0 ) )
49		cnfldadd 19751	. . . . 5 |- + = ( +g ` ℂfld )
50	5, 49	ressplusg 15993	. . . 4 |- ( NN0 e. _V -> + = ( +g ` (ℂfld ↾s NN0 ) ) )
51	8, 50	ax-mp 5	. . 3 |- + = ( +g ` (ℂfld ↾s NN0 ) )
52	5, 22	ressmulr 16006	. . . 4 |- ( NN0 e. _V -> x. = ( .r ` (ℂfld ↾s NN0 ) ) )
53	8, 52	ax-mp 5	. . 3 |- x. = ( .r ` (ℂfld ↾s NN0 ) )
54		ringmnd 18556	. . . . 5 |- (ℂfld e. Ring -> ℂfld e. Mnd )
55	1, 54	ax-mp 5	. . . 4 |- ℂfld e. Mnd
56		0nn0 11307	. . . 4 |- 0 e. NN0
57		cnfld0 19770	. . . . 5 |- 0 = ( 0g ` ℂfld )
58	5, 18, 57	ress0g 17319	. . . 4 |- ( (ℂfld e. Mnd /\ 0 e. NN0 /\ NN0 C_ CC ) -> 0 = ( 0g ` (ℂfld ↾s NN0 ) ) )
59	55, 56, 12, 58	mp3an 1424	. . 3 |- 0 = ( 0g ` (ℂfld ↾s NN0 ) )
60	47, 48, 51, 53, 59	issrg 18507	. 2 |- ( (ℂfld ↾s NN0 ) e. SRing <-> ( (ℂfld ↾s NN0 ) e. CMnd /\ (mulGrp ` (ℂfld ↾s NN0 ) ) e. Mnd /\ A. x e. NN0 ( A. y e. NN0 A. z e. NN0 ( ( x x. ( y + z ) ) = ( ( x x. y ) + ( x x. z ) ) /\ ( ( x + y ) x. z ) = ( ( x x. z ) + ( y x. z ) ) ) /\ ( ( 0 x. x ) = 0 /\ ( x x. 0 ) = 0 ) ) ) )
61	7, 30, 45, 60	mpbir3an 1244	1 |- (ℂfld ↾s NN0 ) e. SRing

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   NN0cn0 11292   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941   .rcmulr 15942   0gc0g 16100   Mndcmnd 17294  SubMndcsubmnd 17334  CMndccmn 18193  mulGrpcmgp 18489  SRingcsrg 18505   Ringcrg 18547  ℂ_fldccnfld 19746

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-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-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-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-srg 18506  df-ring 18549  df-cring 18550  df-cnfld 19747

This theorem is referenced by: (None)