MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nn0srg Structured version   Unicode version

Theorem nn0srg 18251
Description: The nonnegative integers form a semiring (commutative by subcmn 16635). (Contributed by Thierry Arnoux, 1-May-2018.)
Assertion
Ref Expression
nn0srg  |-  (flds  NN0 )  e. SRing

Proof of Theorem nn0srg
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnrng 18208 . . . 4  |-fld  e.  Ring
2 rngcmn 17013 . . . 4  |-  (fld  e.  Ring  ->fld  e. CMnd )
31, 2ax-mp 5 . . 3  |-fld  e. CMnd
4 nn0subm 18238 . . 3  |-  NN0  e.  (SubMnd ` fld )
5 eqid 2467 . . . 4  |-  (flds  NN0 )  =  (flds  NN0 )
65submcmn 16636 . . 3  |-  ( (fld  e. CMnd  /\  NN0  e.  (SubMnd ` fld )
)  ->  (flds  NN0 )  e. CMnd )
73, 4, 6mp2an 672 . 2  |-  (flds  NN0 )  e. CMnd
8 nn0ex 10797 . . . 4  |-  NN0  e.  _V
9 eqid 2467 . . . . 5  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
105, 9mgpress 16939 . . . 4  |-  ( (fld  e. CMnd  /\  NN0  e.  _V )  ->  ( (mulGrp ` fld )s  NN0 )  =  (mulGrp `  (flds  NN0 ) ) )
113, 8, 10mp2an 672 . . 3  |-  ( (mulGrp ` fld )s  NN0 )  =  (mulGrp `  (flds  NN0 ) )
12 nn0sscn 10796 . . . . . 6  |-  NN0  C_  CC
13 1nn0 10807 . . . . . 6  |-  1  e.  NN0
14 nn0mulcl 10828 . . . . . . 7  |-  ( ( x  e.  NN0  /\  y  e.  NN0 )  -> 
( x  x.  y
)  e.  NN0 )
1514rgen2 2889 . . . . . 6  |-  A. x  e.  NN0  A. y  e. 
NN0  ( x  x.  y )  e.  NN0
1612, 13, 153pm3.2i 1174 . . . . 5  |-  ( NN0  C_  CC  /\  1  e. 
NN0  /\  A. x  e.  NN0  A. y  e. 
NN0  ( x  x.  y )  e.  NN0 )
179rngmgp 16989 . . . . . . 7  |-  (fld  e.  Ring  -> 
(mulGrp ` fld )  e.  Mnd )
181, 17ax-mp 5 . . . . . 6  |-  (mulGrp ` fld )  e.  Mnd
19 cnfldbas 18192 . . . . . . . 8  |-  CC  =  ( Base ` fld )
209, 19mgpbas 16934 . . . . . . 7  |-  CC  =  ( Base `  (mulGrp ` fld ) )
21 cnfld1 18211 . . . . . . . 8  |-  1  =  ( 1r ` fld )
229, 21rngidval 16942 . . . . . . 7  |-  1  =  ( 0g `  (mulGrp ` fld ) )
23 cnfldmul 18194 . . . . . . . 8  |-  x.  =  ( .r ` fld )
249, 23mgpplusg 16932 . . . . . . 7  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
2520, 22, 24issubm 15785 . . . . . 6  |-  ( (mulGrp ` fld )  e.  Mnd  ->  ( NN0  e.  (SubMnd `  (mulGrp ` fld ) )  <->  ( NN0  C_  CC  /\  1  e.  NN0  /\  A. x  e.  NN0  A. y  e.  NN0  ( x  x.  y )  e.  NN0 ) ) )
2618, 25ax-mp 5 . . . . 5  |-  ( NN0 
e.  (SubMnd `  (mulGrp ` fld ) )  <->  ( NN0  C_  CC  /\  1  e. 
NN0  /\  A. x  e.  NN0  A. y  e. 
NN0  ( x  x.  y )  e.  NN0 ) )
2716, 26mpbir 209 . . . 4  |-  NN0  e.  (SubMnd `  (mulGrp ` fld ) )
28 eqid 2467 . . . . 5  |-  ( (mulGrp ` fld )s  NN0 )  =  (
(mulGrp ` fld )s  NN0 )
2928submmnd 15792 . . . 4  |-  ( NN0 
e.  (SubMnd `  (mulGrp ` fld ) )  ->  (
(mulGrp ` fld )s  NN0 )  e.  Mnd )
3027, 29ax-mp 5 . . 3  |-  ( (mulGrp ` fld )s  NN0 )  e.  Mnd
3111, 30eqeltrri 2552 . 2  |-  (mulGrp `  (flds  NN0 )
)  e.  Mnd
32 simpl 457 . . . . . . . 8  |-  ( ( x  e.  NN0  /\  ( y  e.  NN0  /\  z  e.  NN0 )
)  ->  x  e.  NN0 )
33 nn0cn 10801 . . . . . . . 8  |-  ( x  e.  NN0  ->  x  e.  CC )
3432, 33syl 16 . . . . . . 7  |-  ( ( x  e.  NN0  /\  ( y  e.  NN0  /\  z  e.  NN0 )
)  ->  x  e.  CC )
35 simprl 755 . . . . . . . 8  |-  ( ( x  e.  NN0  /\  ( y  e.  NN0  /\  z  e.  NN0 )
)  ->  y  e.  NN0 )
3612, 35sseldi 3502 . . . . . . 7  |-  ( ( x  e.  NN0  /\  ( y  e.  NN0  /\  z  e.  NN0 )
)  ->  y  e.  CC )
37 simprr 756 . . . . . . . 8  |-  ( ( x  e.  NN0  /\  ( y  e.  NN0  /\  z  e.  NN0 )
)  ->  z  e.  NN0 )
3812, 37sseldi 3502 . . . . . . 7  |-  ( ( x  e.  NN0  /\  ( y  e.  NN0  /\  z  e.  NN0 )
)  ->  z  e.  CC )
3934, 36, 38adddid 9616 . . . . . 6  |-  ( ( x  e.  NN0  /\  ( y  e.  NN0  /\  z  e.  NN0 )
)  ->  ( x  x.  ( y  +  z ) )  =  ( ( x  x.  y
)  +  ( x  x.  z ) ) )
4034, 36, 38adddird 9617 . . . . . 6  |-  ( ( x  e.  NN0  /\  ( y  e.  NN0  /\  z  e.  NN0 )
)  ->  ( (
x  +  y )  x.  z )  =  ( ( x  x.  z )  +  ( y  x.  z ) ) )
4139, 40jca 532 . . . . 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
) ) ) )
4241ralrimivva 2885 . . . 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 ) ) ) )
4333mul02d 9773 . . . . 5  |-  ( x  e.  NN0  ->  ( 0  x.  x )  =  0 )
4433mul01d 9774 . . . . 5  |-  ( x  e.  NN0  ->  ( x  x.  0 )  =  0 )
4543, 44jca 532 . . . 4  |-  ( x  e.  NN0  ->  ( ( 0  x.  x )  =  0  /\  (
x  x.  0 )  =  0 ) )
4642, 45jca 532 . . 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 ) ) )
4746rgen 2824 . 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 ) )
485, 19ressbas2 14539 . . . 4  |-  ( NN0  C_  CC  ->  NN0  =  (
Base `  (flds  NN0 ) ) )
4912, 48ax-mp 5 . . 3  |-  NN0  =  ( Base `  (flds  NN0 ) )
50 eqid 2467 . . 3  |-  (mulGrp `  (flds  NN0 )
)  =  (mulGrp `  (flds  NN0 )
)
51 cnfldadd 18193 . . . . 5  |-  +  =  ( +g  ` fld )
525, 51ressplusg 14590 . . . 4  |-  ( NN0 
e.  _V  ->  +  =  ( +g  `  (flds  NN0 ) ) )
538, 52ax-mp 5 . . 3  |-  +  =  ( +g  `  (flds  NN0 ) )
545, 23ressmulr 14601 . . . 4  |-  ( NN0 
e.  _V  ->  x.  =  ( .r `  (flds  NN0 ) ) )
558, 54ax-mp 5 . . 3  |-  x.  =  ( .r `  (flds  NN0 ) )
56 rngmnd 16992 . . . . 5  |-  (fld  e.  Ring  ->fld  e.  Mnd )
571, 56ax-mp 5 . . . 4  |-fld  e.  Mnd
58 0nn0 10806 . . . 4  |-  0  e.  NN0
59 cnfld0 18210 . . . . 5  |-  0  =  ( 0g ` fld )
605, 19, 59ress0g 15760 . . . 4  |-  ( (fld  e. 
Mnd  /\  0  e.  NN0 
/\  NN0  C_  CC )  ->  0  =  ( 0g `  (flds  NN0 ) ) )
6157, 58, 12, 60mp3an 1324 . . 3  |-  0  =  ( 0g `  (flds  NN0 )
)
6249, 50, 53, 55, 61issrg 16946 . 2  |-  ( (flds  NN0 )  e. SRing  <-> 
( (flds  NN0 )  e. CMnd  /\  (mulGrp `  (flds  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 ) ) ) )
637, 31, 47, 62mpbir3an 1178 1  |-  (flds  NN0 )  e. SRing
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113    C_ wss 3476   ` cfv 5586  (class class class)co 6282   CCcc 9486   0cc0 9488   1c1 9489    + caddc 9491    x. cmul 9493   NN0cn0 10791   Basecbs 14483   ↾s cress 14484   +g cplusg 14548   .rcmulr 14549   0gc0g 14688   Mndcmnd 15719  SubMndcsubmnd 15773  CMndccmn 16591  mulGrpcmgp 16928  SRingcsrg 16944   Ringcrg 16983  ℂfldccnfld 18188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-addf 9567  ax-mulf 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-fz 11669  df-struct 14485  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-ress 14490  df-plusg 14561  df-mulr 14562  df-starv 14563  df-tset 14567  df-ple 14568  df-ds 14570  df-unif 14571  df-0g 14690  df-mnd 15725  df-submnd 15775  df-grp 15855  df-minusg 15856  df-cmn 16593  df-abl 16594  df-mgp 16929  df-ur 16941  df-srg 16945  df-rng 16985  df-cring 16986  df-cnfld 18189
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator