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Theorem rge0srg 18255
Description: The nonnegative real numbers form a semiring (commutative by subcmn 16638). (Contributed by Thierry Arnoux, 6-Sep-2018.)
Assertion
Ref Expression
rge0srg  |-  (flds  ( 0 [,) +oo ) )  e. SRing

Proof of Theorem rge0srg
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnrng 18211 . . . 4  |-fld  e.  Ring
2 rngcmn 17016 . . . 4  |-  (fld  e.  Ring  ->fld  e. CMnd )
31, 2ax-mp 5 . . 3  |-fld  e. CMnd
4 rge0ssre 11624 . . . . . 6  |-  ( 0 [,) +oo )  C_  RR
5 ax-resscn 9545 . . . . . 6  |-  RR  C_  CC
64, 5sstri 3513 . . . . 5  |-  ( 0 [,) +oo )  C_  CC
7 0e0icopnf 11626 . . . . 5  |-  0  e.  ( 0 [,) +oo )
8 ge0addcl 11628 . . . . . 6  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  ( x  +  y )  e.  ( 0 [,) +oo )
)
98rgen2 2889 . . . . 5  |-  A. x  e.  ( 0 [,) +oo ) A. y  e.  ( 0 [,) +oo )
( x  +  y )  e.  ( 0 [,) +oo )
106, 7, 93pm3.2i 1174 . . . 4  |-  ( ( 0 [,) +oo )  C_  CC  /\  0  e.  ( 0 [,) +oo )  /\  A. x  e.  ( 0 [,) +oo ) A. y  e.  ( 0 [,) +oo )
( x  +  y )  e.  ( 0 [,) +oo ) )
11 rngmnd 16995 . . . . . 6  |-  (fld  e.  Ring  ->fld  e.  Mnd )
121, 11ax-mp 5 . . . . 5  |-fld  e.  Mnd
13 cnfldbas 18195 . . . . . 6  |-  CC  =  ( Base ` fld )
14 cnfld0 18213 . . . . . 6  |-  0  =  ( 0g ` fld )
15 cnfldadd 18196 . . . . . 6  |-  +  =  ( +g  ` fld )
1613, 14, 15issubm 15788 . . . . 5  |-  (fld  e.  Mnd  ->  ( ( 0 [,) +oo )  e.  (SubMnd ` fld )  <-> 
( ( 0 [,) +oo )  C_  CC  /\  0  e.  ( 0 [,) +oo )  /\  A. x  e.  ( 0 [,) +oo ) A. y  e.  ( 0 [,) +oo ) ( x  +  y )  e.  ( 0 [,) +oo ) ) ) )
1712, 16ax-mp 5 . . . 4  |-  ( ( 0 [,) +oo )  e.  (SubMnd ` fld )  <->  ( ( 0 [,) +oo )  C_  CC  /\  0  e.  ( 0 [,) +oo )  /\  A. x  e.  ( 0 [,) +oo ) A. y  e.  (
0 [,) +oo )
( x  +  y )  e.  ( 0 [,) +oo ) ) )
1810, 17mpbir 209 . . 3  |-  ( 0 [,) +oo )  e.  (SubMnd ` fld )
19 eqid 2467 . . . 4  |-  (flds  ( 0 [,) +oo ) )  =  (flds  ( 0 [,) +oo ) )
2019submcmn 16639 . . 3  |-  ( (fld  e. CMnd  /\  ( 0 [,) +oo )  e.  (SubMnd ` fld ) )  ->  (flds  ( 0 [,) +oo ) )  e. CMnd )
213, 18, 20mp2an 672 . 2  |-  (flds  ( 0 [,) +oo ) )  e. CMnd
22 1re 9591 . . . . . 6  |-  1  e.  RR
23 0le1 10072 . . . . . 6  |-  0  <_  1
24 ltpnf 11327 . . . . . . 7  |-  ( 1  e.  RR  ->  1  < +oo )
2522, 24ax-mp 5 . . . . . 6  |-  1  < +oo
26 0re 9592 . . . . . . 7  |-  0  e.  RR
27 pnfxr 11317 . . . . . . 7  |- +oo  e.  RR*
28 elico2 11584 . . . . . . 7  |-  ( ( 0  e.  RR  /\ +oo  e.  RR* )  ->  (
1  e.  ( 0 [,) +oo )  <->  ( 1  e.  RR  /\  0  <_  1  /\  1  < +oo ) ) )
2926, 27, 28mp2an 672 . . . . . 6  |-  ( 1  e.  ( 0 [,) +oo )  <->  ( 1  e.  RR  /\  0  <_ 
1  /\  1  < +oo ) )
3022, 23, 25, 29mpbir3an 1178 . . . . 5  |-  1  e.  ( 0 [,) +oo )
31 ge0mulcl 11629 . . . . . 6  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  ( x  x.  y )  e.  ( 0 [,) +oo )
)
3231rgen2 2889 . . . . 5  |-  A. x  e.  ( 0 [,) +oo ) A. y  e.  ( 0 [,) +oo )
( x  x.  y
)  e.  ( 0 [,) +oo )
336, 30, 323pm3.2i 1174 . . . 4  |-  ( ( 0 [,) +oo )  C_  CC  /\  1  e.  ( 0 [,) +oo )  /\  A. x  e.  ( 0 [,) +oo ) A. y  e.  ( 0 [,) +oo )
( x  x.  y
)  e.  ( 0 [,) +oo ) )
34 eqid 2467 . . . . . 6  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
3534rngmgp 16992 . . . . 5  |-  (fld  e.  Ring  -> 
(mulGrp ` fld )  e.  Mnd )
3634, 13mgpbas 16937 . . . . . 6  |-  CC  =  ( Base `  (mulGrp ` fld ) )
37 cnfld1 18214 . . . . . . 7  |-  1  =  ( 1r ` fld )
3834, 37rngidval 16945 . . . . . 6  |-  1  =  ( 0g `  (mulGrp ` fld ) )
39 cnfldmul 18197 . . . . . . 7  |-  x.  =  ( .r ` fld )
4034, 39mgpplusg 16935 . . . . . 6  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
4136, 38, 40issubm 15788 . . . . 5  |-  ( (mulGrp ` fld )  e.  Mnd  ->  (
( 0 [,) +oo )  e.  (SubMnd `  (mulGrp ` fld ) )  <->  ( ( 0 [,) +oo )  C_  CC  /\  1  e.  ( 0 [,) +oo )  /\  A. x  e.  ( 0 [,) +oo ) A. y  e.  (
0 [,) +oo )
( x  x.  y
)  e.  ( 0 [,) +oo ) ) ) )
421, 35, 41mp2b 10 . . . 4  |-  ( ( 0 [,) +oo )  e.  (SubMnd `  (mulGrp ` fld ) )  <->  ( (
0 [,) +oo )  C_  CC  /\  1  e.  ( 0 [,) +oo )  /\  A. x  e.  ( 0 [,) +oo ) A. y  e.  ( 0 [,) +oo )
( x  x.  y
)  e.  ( 0 [,) +oo ) ) )
4333, 42mpbir 209 . . 3  |-  ( 0 [,) +oo )  e.  (SubMnd `  (mulGrp ` fld ) )
44 eqid 2467 . . . 4  |-  ( (mulGrp ` fld )s  ( 0 [,) +oo ) )  =  ( (mulGrp ` fld )s  ( 0 [,) +oo ) )
4544submmnd 15795 . . 3  |-  ( ( 0 [,) +oo )  e.  (SubMnd `  (mulGrp ` fld ) )  ->  (
(mulGrp ` fld )s  ( 0 [,) +oo ) )  e.  Mnd )
4643, 45ax-mp 5 . 2  |-  ( (mulGrp ` fld )s  ( 0 [,) +oo ) )  e.  Mnd
47 simpll 753 . . . . . . . . 9  |-  ( ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  (
0 [,) +oo )
)  /\  z  e.  ( 0 [,) +oo ) )  ->  x  e.  ( 0 [,) +oo ) )
486, 47sseldi 3502 . . . . . . . 8  |-  ( ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  (
0 [,) +oo )
)  /\  z  e.  ( 0 [,) +oo ) )  ->  x  e.  CC )
49 simplr 754 . . . . . . . . 9  |-  ( ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  (
0 [,) +oo )
)  /\  z  e.  ( 0 [,) +oo ) )  ->  y  e.  ( 0 [,) +oo ) )
506, 49sseldi 3502 . . . . . . . 8  |-  ( ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  (
0 [,) +oo )
)  /\  z  e.  ( 0 [,) +oo ) )  ->  y  e.  CC )
51 simpr 461 . . . . . . . . 9  |-  ( ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  (
0 [,) +oo )
)  /\  z  e.  ( 0 [,) +oo ) )  ->  z  e.  ( 0 [,) +oo ) )
526, 51sseldi 3502 . . . . . . . 8  |-  ( ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  (
0 [,) +oo )
)  /\  z  e.  ( 0 [,) +oo ) )  ->  z  e.  CC )
5348, 50, 52adddid 9616 . . . . . . 7  |-  ( ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  (
0 [,) +oo )
)  /\  z  e.  ( 0 [,) +oo ) )  ->  (
x  x.  ( y  +  z ) )  =  ( ( x  x.  y )  +  ( x  x.  z
) ) )
5448, 50, 52adddird 9617 . . . . . . 7  |-  ( ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  (
0 [,) +oo )
)  /\  z  e.  ( 0 [,) +oo ) )  ->  (
( x  +  y )  x.  z )  =  ( ( x  x.  z )  +  ( y  x.  z
) ) )
5553, 54jca 532 . . . . . 6  |-  ( ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  (
0 [,) +oo )
)  /\  z  e.  ( 0 [,) +oo ) )  ->  (
( x  x.  (
y  +  z ) )  =  ( ( x  x.  y )  +  ( x  x.  z ) )  /\  ( ( x  +  y )  x.  z
)  =  ( ( x  x.  z )  +  ( y  x.  z ) ) ) )
5655ralrimiva 2878 . . . . 5  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  A. z  e.  ( 0 [,) +oo )
( ( x  x.  ( y  +  z ) )  =  ( ( x  x.  y
)  +  ( x  x.  z ) )  /\  ( ( x  +  y )  x.  z )  =  ( ( x  x.  z
)  +  ( y  x.  z ) ) ) )
5756ralrimiva 2878 . . . 4  |-  ( x  e.  ( 0 [,) +oo )  ->  A. y  e.  ( 0 [,) +oo ) A. z  e.  ( 0 [,) +oo )
( ( x  x.  ( y  +  z ) )  =  ( ( x  x.  y
)  +  ( x  x.  z ) )  /\  ( ( x  +  y )  x.  z )  =  ( ( x  x.  z
)  +  ( y  x.  z ) ) ) )
586sseli 3500 . . . . 5  |-  ( x  e.  ( 0 [,) +oo )  ->  x  e.  CC )
5958mul02d 9773 . . . 4  |-  ( x  e.  ( 0 [,) +oo )  ->  ( 0  x.  x )  =  0 )
6058mul01d 9774 . . . 4  |-  ( x  e.  ( 0 [,) +oo )  ->  ( x  x.  0 )  =  0 )
6157, 59, 60jca32 535 . . 3  |-  ( x  e.  ( 0 [,) +oo )  ->  ( A. y  e.  ( 0 [,) +oo ) A. z  e.  ( 0 [,) +oo ) ( ( 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 ) ) )
6261rgen 2824 . 2  |-  A. x  e.  ( 0 [,) +oo ) ( A. y  e.  ( 0 [,) +oo ) A. z  e.  ( 0 [,) +oo )
( ( 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 ) )
6319, 13ressbas2 14542 . . . 4  |-  ( ( 0 [,) +oo )  C_  CC  ->  ( 0 [,) +oo )  =  ( Base `  (flds  ( 0 [,) +oo ) ) ) )
646, 63ax-mp 5 . . 3  |-  ( 0 [,) +oo )  =  ( Base `  (flds  ( 0 [,) +oo ) ) )
65 cnfldex 18194 . . . 4  |-fld  e.  _V
66 ovex 6307 . . . 4  |-  ( 0 [,) +oo )  e. 
_V
6719, 34mgpress 16942 . . . 4  |-  ( (fld  e. 
_V  /\  ( 0 [,) +oo )  e. 
_V )  ->  (
(mulGrp ` fld )s  ( 0 [,) +oo ) )  =  (mulGrp `  (flds  ( 0 [,) +oo )
) ) )
6865, 66, 67mp2an 672 . . 3  |-  ( (mulGrp ` fld )s  ( 0 [,) +oo ) )  =  (mulGrp `  (flds  ( 0 [,) +oo )
) )
6919, 15ressplusg 14593 . . . 4  |-  ( ( 0 [,) +oo )  e.  _V  ->  +  =  ( +g  `  (flds  ( 0 [,) +oo ) ) ) )
7066, 69ax-mp 5 . . 3  |-  +  =  ( +g  `  (flds  ( 0 [,) +oo ) ) )
7119, 39ressmulr 14604 . . . 4  |-  ( ( 0 [,) +oo )  e.  _V  ->  x.  =  ( .r `  (flds  ( 0 [,) +oo ) ) ) )
7266, 71ax-mp 5 . . 3  |-  x.  =  ( .r `  (flds  ( 0 [,) +oo ) ) )
7319, 13, 14ress0g 15763 . . . 4  |-  ( (fld  e. 
Mnd  /\  0  e.  ( 0 [,) +oo )  /\  ( 0 [,) +oo )  C_  CC )  ->  0  =  ( 0g `  (flds  ( 0 [,) +oo ) ) ) )
7412, 7, 6, 73mp3an 1324 . . 3  |-  0  =  ( 0g `  (flds  (
0 [,) +oo )
) )
7564, 68, 70, 72, 74issrg 16949 . 2  |-  ( (flds  ( 0 [,) +oo ) )  e. SRing 
<->  ( (flds  ( 0 [,) +oo )
)  e. CMnd  /\  (
(mulGrp ` fld )s  ( 0 [,) +oo ) )  e.  Mnd  /\ 
A. x  e.  ( 0 [,) +oo )
( A. y  e.  ( 0 [,) +oo ) A. z  e.  ( 0 [,) +oo )
( ( 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 ) ) ) )
7621, 46, 62, 75mpbir3an 1178 1  |-  (flds  ( 0 [,) +oo ) )  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   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489    + caddc 9491    x. cmul 9493   +oocpnf 9621   RR*cxr 9623    < clt 9624    <_ cle 9625   [,)cico 11527   Basecbs 14486   ↾s cress 14487   +g cplusg 14551   .rcmulr 14552   0gc0g 14691   Mndcmnd 15722  SubMndcsubmnd 15776  CMndccmn 16594  mulGrpcmgp 16931  SRingcsrg 16947   Ringcrg 16986  ℂfldccnfld 18191
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-ico 11531  df-fz 11669  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-starv 14566  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-0g 14693  df-mnd 15728  df-submnd 15778  df-grp 15858  df-minusg 15859  df-cmn 16596  df-abl 16597  df-mgp 16932  df-ur 16944  df-srg 16948  df-rng 16988  df-cring 16989  df-cnfld 18192
This theorem is referenced by:  xrge0slmod  27497
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