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

Theorem nn0srg 17879
Description: The nonnegative integers form a semiring (commutative by subcmn 16319). (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 17836 . . . 4  |-fld  e.  Ring
2 rngcmn 16673 . . . 4  |-  (fld  e.  Ring  ->fld  e. CMnd )
31, 2ax-mp 5 . . 3  |-fld  e. CMnd
4 nn0subm 17866 . . 3  |-  NN0  e.  (SubMnd ` fld )
5 eqid 2441 . . . 4  |-  (flds  NN0 )  =  (flds  NN0 )
65submcmn 16320 . . 3  |-  ( (fld  e. CMnd  /\  NN0  e.  (SubMnd ` fld )
)  ->  (flds  NN0 )  e. CMnd )
73, 4, 6mp2an 672 . 2  |-  (flds  NN0 )  e. CMnd
8 nn0ex 10583 . . . 4  |-  NN0  e.  _V
9 eqid 2441 . . . . 5  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
105, 9mgpress 16600 . . . 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 10582 . . . . . 6  |-  NN0  C_  CC
13 1nn0 10593 . . . . . 6  |-  1  e.  NN0
14 nn0mulcl 10614 . . . . . . 7  |-  ( ( x  e.  NN0  /\  y  e.  NN0 )  -> 
( x  x.  y
)  e.  NN0 )
1514rgen2 2810 . . . . . 6  |-  A. x  e.  NN0  A. y  e. 
NN0  ( x  x.  y )  e.  NN0
1612, 13, 153pm3.2i 1166 . . . . 5  |-  ( NN0  C_  CC  /\  1  e. 
NN0  /\  A. x  e.  NN0  A. y  e. 
NN0  ( x  x.  y )  e.  NN0 )
179rngmgp 16649 . . . . . . 7  |-  (fld  e.  Ring  -> 
(mulGrp ` fld )  e.  Mnd )
181, 17ax-mp 5 . . . . . 6  |-  (mulGrp ` fld )  e.  Mnd
19 cnfldbas 17820 . . . . . . . 8  |-  CC  =  ( Base ` fld )
209, 19mgpbas 16595 . . . . . . 7  |-  CC  =  ( Base `  (mulGrp ` fld ) )
21 cnfld1 17839 . . . . . . . 8  |-  1  =  ( 1r ` fld )
229, 21rngidval 16603 . . . . . . 7  |-  1  =  ( 0g `  (mulGrp ` fld ) )
23 cnfldmul 17822 . . . . . . . 8  |-  x.  =  ( .r ` fld )
249, 23mgpplusg 16593 . . . . . . 7  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
2520, 22, 24issubm 15473 . . . . . 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 2441 . . . . 5  |-  ( (mulGrp ` fld )s  NN0 )  =  (
(mulGrp ` fld )s  NN0 )
2928submmnd 15480 . . . 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 2512 . 2  |-  (mulGrp `  (flds  NN0 )
)  e.  Mnd
32 simpl 457 . . . . . . . 8  |-  ( ( x  e.  NN0  /\  ( y  e.  NN0  /\  z  e.  NN0 )
)  ->  x  e.  NN0 )
33 nn0cn 10587 . . . . . . . 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 3352 . . . . . . 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 3352 . . . . . . 7  |-  ( ( x  e.  NN0  /\  ( y  e.  NN0  /\  z  e.  NN0 )
)  ->  z  e.  CC )
3934, 36, 38adddid 9408 . . . . . 6  |-  ( ( x  e.  NN0  /\  ( y  e.  NN0  /\  z  e.  NN0 )
)  ->  ( x  x.  ( y  +  z ) )  =  ( ( x  x.  y
)  +  ( x  x.  z ) ) )
4034, 36, 38adddird 9409 . . . . . 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 2806 . . . 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 9565 . . . . 5  |-  ( x  e.  NN0  ->  ( 0  x.  x )  =  0 )
4433mul01d 9566 . . . . 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 2779 . 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 14227 . . . 4  |-  ( NN0  C_  CC  ->  NN0  =  (
Base `  (flds  NN0 ) ) )
4912, 48ax-mp 5 . . 3  |-  NN0  =  ( Base `  (flds  NN0 ) )
50 eqid 2441 . . 3  |-  (mulGrp `  (flds  NN0 )
)  =  (mulGrp `  (flds  NN0 )
)
51 cnfldadd 17821 . . . . 5  |-  +  =  ( +g  ` fld )
525, 51ressplusg 14278 . . . 4  |-  ( NN0 
e.  _V  ->  +  =  ( +g  `  (flds  NN0 ) ) )
538, 52ax-mp 5 . . 3  |-  +  =  ( +g  `  (flds  NN0 ) )
545, 23ressmulr 14289 . . . 4  |-  ( NN0 
e.  _V  ->  x.  =  ( .r `  (flds  NN0 ) ) )
558, 54ax-mp 5 . . 3  |-  x.  =  ( .r `  (flds  NN0 ) )
56 rngmnd 16652 . . . . 5  |-  (fld  e.  Ring  ->fld  e.  Mnd )
571, 56ax-mp 5 . . . 4  |-fld  e.  Mnd
58 0nn0 10592 . . . 4  |-  0  e.  NN0
59 cnfld0 17838 . . . . 5  |-  0  =  ( 0g ` fld )
605, 19, 59ress0g 15448 . . . 4  |-  ( (fld  e. 
Mnd  /\  0  e.  NN0 
/\  NN0  C_  CC )  ->  0  =  ( 0g `  (flds  NN0 ) ) )
6157, 58, 12, 60mp3an 1314 . . 3  |-  0  =  ( 0g `  (flds  NN0 )
)
6249, 50, 53, 55, 61issrg 16607 . 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 1170 1  |-  (flds  NN0 )  e. SRing
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2713   _Vcvv 2970    C_ wss 3326   ` cfv 5416  (class class class)co 6089   CCcc 9278   0cc0 9280   1c1 9281    + caddc 9283    x. cmul 9285   NN0cn0 10577   Basecbs 14172   ↾s cress 14173   +g cplusg 14236   .rcmulr 14237   0gc0g 14376   Mndcmnd 15407  SubMndcsubmnd 15461  CMndccmn 16275  mulGrpcmgp 16589  SRingcsrg 16605   Ringcrg 16643  ℂfldccnfld 17816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357  ax-addf 9359  ax-mulf 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-2 10378  df-3 10379  df-4 10380  df-5 10381  df-6 10382  df-7 10383  df-8 10384  df-9 10385  df-10 10386  df-n0 10578  df-z 10645  df-dec 10754  df-uz 10860  df-fz 11436  df-struct 14174  df-ndx 14175  df-slot 14176  df-base 14177  df-sets 14178  df-ress 14179  df-plusg 14249  df-mulr 14250  df-starv 14251  df-tset 14255  df-ple 14256  df-ds 14258  df-unif 14259  df-0g 14378  df-mnd 15413  df-submnd 15463  df-grp 15543  df-minusg 15544  df-cmn 16277  df-abl 16278  df-mgp 16590  df-ur 16602  df-srg 16606  df-rng 16645  df-cring 16646  df-cnfld 17817
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator