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Theorem rge0srg 18600
Description: The nonnegative real numbers form a semiring (commutative by subcmn 16962). (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 cnring 18553 . . . 4  |-fld  e.  Ring
2 ringcmn 17342 . . . 4  |-  (fld  e.  Ring  ->fld  e. CMnd )
31, 2ax-mp 5 . . 3  |-fld  e. CMnd
4 rege0subm 18587 . . 3  |-  ( 0 [,) +oo )  e.  (SubMnd ` fld )
5 eqid 2382 . . . 4  |-  (flds  ( 0 [,) +oo ) )  =  (flds  ( 0 [,) +oo ) )
65submcmn 16963 . . 3  |-  ( (fld  e. CMnd  /\  ( 0 [,) +oo )  e.  (SubMnd ` fld ) )  ->  (flds  ( 0 [,) +oo ) )  e. CMnd )
73, 4, 6mp2an 670 . 2  |-  (flds  ( 0 [,) +oo ) )  e. CMnd
8 rge0ssre 11549 . . . . 5  |-  ( 0 [,) +oo )  C_  RR
9 ax-resscn 9460 . . . . 5  |-  RR  C_  CC
108, 9sstri 3426 . . . 4  |-  ( 0 [,) +oo )  C_  CC
11 1re 9506 . . . . 5  |-  1  e.  RR
12 0le1 9993 . . . . 5  |-  0  <_  1
13 ltpnf 11252 . . . . . 6  |-  ( 1  e.  RR  ->  1  < +oo )
1411, 13ax-mp 5 . . . . 5  |-  1  < +oo
15 0re 9507 . . . . . 6  |-  0  e.  RR
16 pnfxr 11242 . . . . . 6  |- +oo  e.  RR*
17 elico2 11509 . . . . . 6  |-  ( ( 0  e.  RR  /\ +oo  e.  RR* )  ->  (
1  e.  ( 0 [,) +oo )  <->  ( 1  e.  RR  /\  0  <_  1  /\  1  < +oo ) ) )
1815, 16, 17mp2an 670 . . . . 5  |-  ( 1  e.  ( 0 [,) +oo )  <->  ( 1  e.  RR  /\  0  <_ 
1  /\  1  < +oo ) )
1911, 12, 14, 18mpbir3an 1176 . . . 4  |-  1  e.  ( 0 [,) +oo )
20 ge0mulcl 11554 . . . . 5  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  ( x  x.  y )  e.  ( 0 [,) +oo )
)
2120rgen2a 2809 . . . 4  |-  A. x  e.  ( 0 [,) +oo ) A. y  e.  ( 0 [,) +oo )
( x  x.  y
)  e.  ( 0 [,) +oo )
22 eqid 2382 . . . . . 6  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
2322ringmgp 17317 . . . . 5  |-  (fld  e.  Ring  -> 
(mulGrp ` fld )  e.  Mnd )
24 cnfldbas 18537 . . . . . . 7  |-  CC  =  ( Base ` fld )
2522, 24mgpbas 17260 . . . . . 6  |-  CC  =  ( Base `  (mulGrp ` fld ) )
26 cnfld1 18556 . . . . . . 7  |-  1  =  ( 1r ` fld )
2722, 26ringidval 17268 . . . . . 6  |-  1  =  ( 0g `  (mulGrp ` fld ) )
28 cnfldmul 18539 . . . . . . 7  |-  x.  =  ( .r ` fld )
2922, 28mgpplusg 17258 . . . . . 6  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
3025, 27, 29issubm 16095 . . . . 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 ) ) ) )
311, 23, 30mp2b 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 ) ) )
3210, 19, 21, 31mpbir3an 1176 . . 3  |-  ( 0 [,) +oo )  e.  (SubMnd `  (mulGrp ` fld ) )
33 eqid 2382 . . . 4  |-  ( (mulGrp ` fld )s  ( 0 [,) +oo ) )  =  ( (mulGrp ` fld )s  ( 0 [,) +oo ) )
3433submmnd 16102 . . 3  |-  ( ( 0 [,) +oo )  e.  (SubMnd `  (mulGrp ` fld ) )  ->  (
(mulGrp ` fld )s  ( 0 [,) +oo ) )  e.  Mnd )
3532, 34ax-mp 5 . 2  |-  ( (mulGrp ` fld )s  ( 0 [,) +oo ) )  e.  Mnd
36 simpll 751 . . . . . . . . 9  |-  ( ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  (
0 [,) +oo )
)  /\  z  e.  ( 0 [,) +oo ) )  ->  x  e.  ( 0 [,) +oo ) )
3710, 36sseldi 3415 . . . . . . . 8  |-  ( ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  (
0 [,) +oo )
)  /\  z  e.  ( 0 [,) +oo ) )  ->  x  e.  CC )
38 simplr 753 . . . . . . . . 9  |-  ( ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  (
0 [,) +oo )
)  /\  z  e.  ( 0 [,) +oo ) )  ->  y  e.  ( 0 [,) +oo ) )
3910, 38sseldi 3415 . . . . . . . 8  |-  ( ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  (
0 [,) +oo )
)  /\  z  e.  ( 0 [,) +oo ) )  ->  y  e.  CC )
40 simpr 459 . . . . . . . . 9  |-  ( ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  (
0 [,) +oo )
)  /\  z  e.  ( 0 [,) +oo ) )  ->  z  e.  ( 0 [,) +oo ) )
4110, 40sseldi 3415 . . . . . . . 8  |-  ( ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  (
0 [,) +oo )
)  /\  z  e.  ( 0 [,) +oo ) )  ->  z  e.  CC )
4237, 39, 41adddid 9531 . . . . . . 7  |-  ( ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  (
0 [,) +oo )
)  /\  z  e.  ( 0 [,) +oo ) )  ->  (
x  x.  ( y  +  z ) )  =  ( ( x  x.  y )  +  ( x  x.  z
) ) )
4337, 39, 41adddird 9532 . . . . . . 7  |-  ( ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  (
0 [,) +oo )
)  /\  z  e.  ( 0 [,) +oo ) )  ->  (
( x  +  y )  x.  z )  =  ( ( x  x.  z )  +  ( y  x.  z
) ) )
4442, 43jca 530 . . . . . 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 ) ) ) )
4544ralrimiva 2796 . . . . 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 ) ) ) )
4645ralrimiva 2796 . . . 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 ) ) ) )
4710sseli 3413 . . . . 5  |-  ( x  e.  ( 0 [,) +oo )  ->  x  e.  CC )
4847mul02d 9689 . . . 4  |-  ( x  e.  ( 0 [,) +oo )  ->  ( 0  x.  x )  =  0 )
4947mul01d 9690 . . . 4  |-  ( x  e.  ( 0 [,) +oo )  ->  ( x  x.  0 )  =  0 )
5046, 48, 49jca32 533 . . 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 ) ) )
5150rgen 2742 . 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 ) )
525, 24ressbas2 14692 . . . 4  |-  ( ( 0 [,) +oo )  C_  CC  ->  ( 0 [,) +oo )  =  ( Base `  (flds  ( 0 [,) +oo ) ) ) )
5310, 52ax-mp 5 . . 3  |-  ( 0 [,) +oo )  =  ( Base `  (flds  ( 0 [,) +oo ) ) )
54 cnfldex 18536 . . . 4  |-fld  e.  _V
55 ovex 6224 . . . 4  |-  ( 0 [,) +oo )  e. 
_V
565, 22mgpress 17265 . . . 4  |-  ( (fld  e. 
_V  /\  ( 0 [,) +oo )  e. 
_V )  ->  (
(mulGrp ` fld )s  ( 0 [,) +oo ) )  =  (mulGrp `  (flds  ( 0 [,) +oo )
) ) )
5754, 55, 56mp2an 670 . . 3  |-  ( (mulGrp ` fld )s  ( 0 [,) +oo ) )  =  (mulGrp `  (flds  ( 0 [,) +oo )
) )
58 cnfldadd 18538 . . . . 5  |-  +  =  ( +g  ` fld )
595, 58ressplusg 14748 . . . 4  |-  ( ( 0 [,) +oo )  e.  _V  ->  +  =  ( +g  `  (flds  ( 0 [,) +oo ) ) ) )
6055, 59ax-mp 5 . . 3  |-  +  =  ( +g  `  (flds  ( 0 [,) +oo ) ) )
615, 28ressmulr 14759 . . . 4  |-  ( ( 0 [,) +oo )  e.  _V  ->  x.  =  ( .r `  (flds  ( 0 [,) +oo ) ) ) )
6255, 61ax-mp 5 . . 3  |-  x.  =  ( .r `  (flds  ( 0 [,) +oo ) ) )
63 ringmnd 17320 . . . . 5  |-  (fld  e.  Ring  ->fld  e.  Mnd )
641, 63ax-mp 5 . . . 4  |-fld  e.  Mnd
65 0e0icopnf 11551 . . . 4  |-  0  e.  ( 0 [,) +oo )
66 cnfld0 18555 . . . . 5  |-  0  =  ( 0g ` fld )
675, 24, 66ress0g 16066 . . . 4  |-  ( (fld  e. 
Mnd  /\  0  e.  ( 0 [,) +oo )  /\  ( 0 [,) +oo )  C_  CC )  ->  0  =  ( 0g `  (flds  ( 0 [,) +oo ) ) ) )
6864, 65, 10, 67mp3an 1322 . . 3  |-  0  =  ( 0g `  (flds  (
0 [,) +oo )
) )
6953, 57, 60, 62, 68issrg 17272 . 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 ) ) ) )
707, 35, 51, 69mpbir3an 1176 1  |-  (flds  ( 0 [,) +oo ) )  e. SRing
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826   A.wral 2732   _Vcvv 3034    C_ wss 3389   class class class wbr 4367   ` cfv 5496  (class class class)co 6196   CCcc 9401   RRcr 9402   0cc0 9403   1c1 9404    + caddc 9406    x. cmul 9408   +oocpnf 9536   RR*cxr 9538    < clt 9539    <_ cle 9540   [,)cico 11452   Basecbs 14634   ↾s cress 14635   +g cplusg 14702   .rcmulr 14703   0gc0g 14847   Mndcmnd 16036  SubMndcsubmnd 16082  CMndccmn 16915  mulGrpcmgp 17254  SRingcsrg 17270   Ringcrg 17311  ℂfldccnfld 18533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-addf 9482  ax-mulf 9483
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-7 10516  df-8 10517  df-9 10518  df-10 10519  df-n0 10713  df-z 10782  df-dec 10896  df-uz 11002  df-ico 11456  df-fz 11594  df-struct 14636  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-mulr 14716  df-starv 14717  df-tset 14721  df-ple 14722  df-ds 14724  df-unif 14725  df-0g 14849  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-submnd 16084  df-grp 16174  df-minusg 16175  df-cmn 16917  df-abl 16918  df-mgp 17255  df-ur 17267  df-srg 17271  df-ring 17313  df-cring 17314  df-cnfld 18534
This theorem is referenced by:  xrge0slmod  27988
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