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Theorem nn0srg 18613
Description: The nonnegative integers form a semiring (commutative by subcmn 16972). (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 cnring 18567 . . . 4  |-fld  e.  Ring
2 ringcmn 17356 . . . 4  |-  (fld  e.  Ring  ->fld  e. CMnd )
31, 2ax-mp 5 . . 3  |-fld  e. CMnd
4 nn0subm 18600 . . 3  |-  NN0  e.  (SubMnd ` fld )
5 eqid 2457 . . . 4  |-  (flds  NN0 )  =  (flds  NN0 )
65submcmn 16973 . . 3  |-  ( (fld  e. CMnd  /\  NN0  e.  (SubMnd ` fld )
)  ->  (flds  NN0 )  e. CMnd )
73, 4, 6mp2an 672 . 2  |-  (flds  NN0 )  e. CMnd
8 nn0ex 10822 . . . 4  |-  NN0  e.  _V
9 eqid 2457 . . . . 5  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
105, 9mgpress 17279 . . . 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 10821 . . . . 5  |-  NN0  C_  CC
13 1nn0 10832 . . . . 5  |-  1  e.  NN0
14 nn0mulcl 10853 . . . . . 6  |-  ( ( x  e.  NN0  /\  y  e.  NN0 )  -> 
( x  x.  y
)  e.  NN0 )
1514rgen2a 2884 . . . . 5  |-  A. x  e.  NN0  A. y  e. 
NN0  ( x  x.  y )  e.  NN0
169ringmgp 17331 . . . . . . 7  |-  (fld  e.  Ring  -> 
(mulGrp ` fld )  e.  Mnd )
171, 16ax-mp 5 . . . . . 6  |-  (mulGrp ` fld )  e.  Mnd
18 cnfldbas 18551 . . . . . . . 8  |-  CC  =  ( Base ` fld )
199, 18mgpbas 17274 . . . . . . 7  |-  CC  =  ( Base `  (mulGrp ` fld ) )
20 cnfld1 18570 . . . . . . . 8  |-  1  =  ( 1r ` fld )
219, 20ringidval 17282 . . . . . . 7  |-  1  =  ( 0g `  (mulGrp ` fld ) )
22 cnfldmul 18553 . . . . . . . 8  |-  x.  =  ( .r ` fld )
239, 22mgpplusg 17272 . . . . . . 7  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
2419, 21, 23issubm 16105 . . . . . 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 ) ) )
2517, 24ax-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 ) )
2612, 13, 15, 25mpbir3an 1178 . . . 4  |-  NN0  e.  (SubMnd `  (mulGrp ` fld ) )
27 eqid 2457 . . . . 5  |-  ( (mulGrp ` fld )s  NN0 )  =  (
(mulGrp ` fld )s  NN0 )
2827submmnd 16112 . . . 4  |-  ( NN0 
e.  (SubMnd `  (mulGrp ` fld ) )  ->  (
(mulGrp ` fld )s  NN0 )  e.  Mnd )
2926, 28ax-mp 5 . . 3  |-  ( (mulGrp ` fld )s  NN0 )  e.  Mnd
3011, 29eqeltrri 2542 . 2  |-  (mulGrp `  (flds  NN0 )
)  e.  Mnd
31 simpl 457 . . . . . . . 8  |-  ( ( x  e.  NN0  /\  ( y  e.  NN0  /\  z  e.  NN0 )
)  ->  x  e.  NN0 )
3231nn0cnd 10875 . . . . . . 7  |-  ( ( x  e.  NN0  /\  ( y  e.  NN0  /\  z  e.  NN0 )
)  ->  x  e.  CC )
33 simprl 756 . . . . . . . 8  |-  ( ( x  e.  NN0  /\  ( y  e.  NN0  /\  z  e.  NN0 )
)  ->  y  e.  NN0 )
3433nn0cnd 10875 . . . . . . 7  |-  ( ( x  e.  NN0  /\  ( y  e.  NN0  /\  z  e.  NN0 )
)  ->  y  e.  CC )
35 simprr 757 . . . . . . . 8  |-  ( ( x  e.  NN0  /\  ( y  e.  NN0  /\  z  e.  NN0 )
)  ->  z  e.  NN0 )
3635nn0cnd 10875 . . . . . . 7  |-  ( ( x  e.  NN0  /\  ( y  e.  NN0  /\  z  e.  NN0 )
)  ->  z  e.  CC )
3732, 34, 36adddid 9637 . . . . . 6  |-  ( ( x  e.  NN0  /\  ( y  e.  NN0  /\  z  e.  NN0 )
)  ->  ( x  x.  ( y  +  z ) )  =  ( ( x  x.  y
)  +  ( x  x.  z ) ) )
3832, 34, 36adddird 9638 . . . . . 6  |-  ( ( x  e.  NN0  /\  ( y  e.  NN0  /\  z  e.  NN0 )
)  ->  ( (
x  +  y )  x.  z )  =  ( ( x  x.  z )  +  ( y  x.  z ) ) )
3937, 38jca 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
) ) ) )
4039ralrimivva 2878 . . . 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 10826 . . . . 5  |-  ( x  e.  NN0  ->  x  e.  CC )
4241mul02d 9795 . . . 4  |-  ( x  e.  NN0  ->  ( 0  x.  x )  =  0 )
4341mul01d 9796 . . . 4  |-  ( x  e.  NN0  ->  ( x  x.  0 )  =  0 )
4440, 42, 43jca32 535 . . 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 ) ) )
4544rgen 2817 . 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 ) )
465, 18ressbas2 14702 . . . 4  |-  ( NN0  C_  CC  ->  NN0  =  (
Base `  (flds  NN0 ) ) )
4712, 46ax-mp 5 . . 3  |-  NN0  =  ( Base `  (flds  NN0 ) )
48 eqid 2457 . . 3  |-  (mulGrp `  (flds  NN0 )
)  =  (mulGrp `  (flds  NN0 )
)
49 cnfldadd 18552 . . . . 5  |-  +  =  ( +g  ` fld )
505, 49ressplusg 14758 . . . 4  |-  ( NN0 
e.  _V  ->  +  =  ( +g  `  (flds  NN0 ) ) )
518, 50ax-mp 5 . . 3  |-  +  =  ( +g  `  (flds  NN0 ) )
525, 22ressmulr 14769 . . . 4  |-  ( NN0 
e.  _V  ->  x.  =  ( .r `  (flds  NN0 ) ) )
538, 52ax-mp 5 . . 3  |-  x.  =  ( .r `  (flds  NN0 ) )
54 ringmnd 17334 . . . . 5  |-  (fld  e.  Ring  ->fld  e.  Mnd )
551, 54ax-mp 5 . . . 4  |-fld  e.  Mnd
56 0nn0 10831 . . . 4  |-  0  e.  NN0
57 cnfld0 18569 . . . . 5  |-  0  =  ( 0g ` fld )
585, 18, 57ress0g 16076 . . . 4  |-  ( (fld  e. 
Mnd  /\  0  e.  NN0 
/\  NN0  C_  CC )  ->  0  =  ( 0g `  (flds  NN0 ) ) )
5955, 56, 12, 58mp3an 1324 . . 3  |-  0  =  ( 0g `  (flds  NN0 )
)
6047, 48, 51, 53, 59issrg 17286 . 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 ) ) ) )
617, 30, 45, 60mpbir3an 1178 1  |-  (flds  NN0 )  e. SRing
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807   _Vcvv 3109    C_ wss 3471   ` cfv 5594  (class class class)co 6296   CCcc 9507   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514   NN0cn0 10816   Basecbs 14644   ↾s cress 14645   +g cplusg 14712   .rcmulr 14713   0gc0g 14857   Mndcmnd 16046  SubMndcsubmnd 16092  CMndccmn 16925  mulGrpcmgp 17268  SRingcsrg 17284   Ringcrg 17325  ℂfldccnfld 18547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-fz 11698  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-starv 14727  df-tset 14731  df-ple 14732  df-ds 14734  df-unif 14735  df-0g 14859  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-submnd 16094  df-grp 16184  df-minusg 16185  df-cmn 16927  df-abl 16928  df-mgp 17269  df-ur 17281  df-srg 17285  df-ring 17327  df-cring 17328  df-cnfld 18548
This theorem is referenced by: (None)
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