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Theorem rge0srg 18466
Description: The nonnegative real numbers form a semiring (commutative by subcmn 16824). (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 18419 . . . 4  |-fld  e.  Ring
2 ringcmn 17208 . . . 4  |-  (fld  e.  Ring  ->fld  e. CMnd )
31, 2ax-mp 5 . . 3  |-fld  e. CMnd
4 rege0subm 18453 . . 3  |-  ( 0 [,) +oo )  e.  (SubMnd ` fld )
5 eqid 2443 . . . 4  |-  (flds  ( 0 [,) +oo ) )  =  (flds  ( 0 [,) +oo ) )
65submcmn 16825 . . 3  |-  ( (fld  e. CMnd  /\  ( 0 [,) +oo )  e.  (SubMnd ` fld ) )  ->  (flds  ( 0 [,) +oo ) )  e. CMnd )
73, 4, 6mp2an 672 . 2  |-  (flds  ( 0 [,) +oo ) )  e. CMnd
8 rge0ssre 11639 . . . . 5  |-  ( 0 [,) +oo )  C_  RR
9 ax-resscn 9552 . . . . 5  |-  RR  C_  CC
108, 9sstri 3498 . . . 4  |-  ( 0 [,) +oo )  C_  CC
11 1re 9598 . . . . 5  |-  1  e.  RR
12 0le1 10083 . . . . 5  |-  0  <_  1
13 ltpnf 11342 . . . . . 6  |-  ( 1  e.  RR  ->  1  < +oo )
1411, 13ax-mp 5 . . . . 5  |-  1  < +oo
15 0re 9599 . . . . . 6  |-  0  e.  RR
16 pnfxr 11332 . . . . . 6  |- +oo  e.  RR*
17 elico2 11599 . . . . . 6  |-  ( ( 0  e.  RR  /\ +oo  e.  RR* )  ->  (
1  e.  ( 0 [,) +oo )  <->  ( 1  e.  RR  /\  0  <_  1  /\  1  < +oo ) ) )
1815, 16, 17mp2an 672 . . . . 5  |-  ( 1  e.  ( 0 [,) +oo )  <->  ( 1  e.  RR  /\  0  <_ 
1  /\  1  < +oo ) )
1911, 12, 14, 18mpbir3an 1179 . . . 4  |-  1  e.  ( 0 [,) +oo )
20 ge0mulcl 11644 . . . . 5  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  ( x  x.  y )  e.  ( 0 [,) +oo )
)
2120rgen2a 2870 . . . 4  |-  A. x  e.  ( 0 [,) +oo ) A. y  e.  ( 0 [,) +oo )
( x  x.  y
)  e.  ( 0 [,) +oo )
22 eqid 2443 . . . . . 6  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
2322ringmgp 17183 . . . . 5  |-  (fld  e.  Ring  -> 
(mulGrp ` fld )  e.  Mnd )
24 cnfldbas 18403 . . . . . . 7  |-  CC  =  ( Base ` fld )
2522, 24mgpbas 17126 . . . . . 6  |-  CC  =  ( Base `  (mulGrp ` fld ) )
26 cnfld1 18422 . . . . . . 7  |-  1  =  ( 1r ` fld )
2722, 26ringidval 17134 . . . . . 6  |-  1  =  ( 0g `  (mulGrp ` fld ) )
28 cnfldmul 18405 . . . . . . 7  |-  x.  =  ( .r ` fld )
2922, 28mgpplusg 17124 . . . . . 6  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
3025, 27, 29issubm 15957 . . . . 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 1179 . . 3  |-  ( 0 [,) +oo )  e.  (SubMnd `  (mulGrp ` fld ) )
33 eqid 2443 . . . 4  |-  ( (mulGrp ` fld )s  ( 0 [,) +oo ) )  =  ( (mulGrp ` fld )s  ( 0 [,) +oo ) )
3433submmnd 15964 . . 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 753 . . . . . . . . 9  |-  ( ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  (
0 [,) +oo )
)  /\  z  e.  ( 0 [,) +oo ) )  ->  x  e.  ( 0 [,) +oo ) )
3710, 36sseldi 3487 . . . . . . . 8  |-  ( ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  (
0 [,) +oo )
)  /\  z  e.  ( 0 [,) +oo ) )  ->  x  e.  CC )
38 simplr 755 . . . . . . . . 9  |-  ( ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  (
0 [,) +oo )
)  /\  z  e.  ( 0 [,) +oo ) )  ->  y  e.  ( 0 [,) +oo ) )
3910, 38sseldi 3487 . . . . . . . 8  |-  ( ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  (
0 [,) +oo )
)  /\  z  e.  ( 0 [,) +oo ) )  ->  y  e.  CC )
40 simpr 461 . . . . . . . . 9  |-  ( ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  (
0 [,) +oo )
)  /\  z  e.  ( 0 [,) +oo ) )  ->  z  e.  ( 0 [,) +oo ) )
4110, 40sseldi 3487 . . . . . . . 8  |-  ( ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  (
0 [,) +oo )
)  /\  z  e.  ( 0 [,) +oo ) )  ->  z  e.  CC )
4237, 39, 41adddid 9623 . . . . . . 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 9624 . . . . . . 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 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 ) ) ) )
4544ralrimiva 2857 . . . . 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 2857 . . . 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 3485 . . . . 5  |-  ( x  e.  ( 0 [,) +oo )  ->  x  e.  CC )
4847mul02d 9781 . . . 4  |-  ( x  e.  ( 0 [,) +oo )  ->  ( 0  x.  x )  =  0 )
4947mul01d 9782 . . . 4  |-  ( x  e.  ( 0 [,) +oo )  ->  ( x  x.  0 )  =  0 )
5046, 48, 49jca32 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 ) ) )
5150rgen 2803 . 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 14670 . . . 4  |-  ( ( 0 [,) +oo )  C_  CC  ->  ( 0 [,) +oo )  =  ( Base `  (flds  ( 0 [,) +oo ) ) ) )
5310, 52ax-mp 5 . . 3  |-  ( 0 [,) +oo )  =  ( Base `  (flds  ( 0 [,) +oo ) ) )
54 cnfldex 18402 . . . 4  |-fld  e.  _V
55 ovex 6309 . . . 4  |-  ( 0 [,) +oo )  e. 
_V
565, 22mgpress 17131 . . . 4  |-  ( (fld  e. 
_V  /\  ( 0 [,) +oo )  e. 
_V )  ->  (
(mulGrp ` fld )s  ( 0 [,) +oo ) )  =  (mulGrp `  (flds  ( 0 [,) +oo )
) ) )
5754, 55, 56mp2an 672 . . 3  |-  ( (mulGrp ` fld )s  ( 0 [,) +oo ) )  =  (mulGrp `  (flds  ( 0 [,) +oo )
) )
58 cnfldadd 18404 . . . . 5  |-  +  =  ( +g  ` fld )
595, 58ressplusg 14721 . . . 4  |-  ( ( 0 [,) +oo )  e.  _V  ->  +  =  ( +g  `  (flds  ( 0 [,) +oo ) ) ) )
6055, 59ax-mp 5 . . 3  |-  +  =  ( +g  `  (flds  ( 0 [,) +oo ) ) )
615, 28ressmulr 14732 . . . 4  |-  ( ( 0 [,) +oo )  e.  _V  ->  x.  =  ( .r `  (flds  ( 0 [,) +oo ) ) ) )
6255, 61ax-mp 5 . . 3  |-  x.  =  ( .r `  (flds  ( 0 [,) +oo ) ) )
63 ringmnd 17186 . . . . 5  |-  (fld  e.  Ring  ->fld  e.  Mnd )
641, 63ax-mp 5 . . . 4  |-fld  e.  Mnd
65 0e0icopnf 11641 . . . 4  |-  0  e.  ( 0 [,) +oo )
66 cnfld0 18421 . . . . 5  |-  0  =  ( 0g ` fld )
675, 24, 66ress0g 15928 . . . 4  |-  ( (fld  e. 
Mnd  /\  0  e.  ( 0 [,) +oo )  /\  ( 0 [,) +oo )  C_  CC )  ->  0  =  ( 0g `  (flds  ( 0 [,) +oo ) ) ) )
6864, 65, 10, 67mp3an 1325 . . 3  |-  0  =  ( 0g `  (flds  (
0 [,) +oo )
) )
6953, 57, 60, 62, 68issrg 17138 . 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 1179 1  |-  (flds  ( 0 [,) +oo ) )  e. SRing
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   A.wral 2793   _Vcvv 3095    C_ wss 3461   class class class wbr 4437   ` cfv 5578  (class class class)co 6281   CCcc 9493   RRcr 9494   0cc0 9495   1c1 9496    + caddc 9498    x. cmul 9500   +oocpnf 9628   RR*cxr 9630    < clt 9631    <_ cle 9632   [,)cico 11542   Basecbs 14614   ↾s cress 14615   +g cplusg 14679   .rcmulr 14680   0gc0g 14819   Mndcmnd 15898  SubMndcsubmnd 15944  CMndccmn 16777  mulGrpcmgp 17120  SRingcsrg 17136   Ringcrg 17177  ℂfldccnfld 18399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10987  df-uz 11093  df-ico 11546  df-fz 11684  df-struct 14616  df-ndx 14617  df-slot 14618  df-base 14619  df-sets 14620  df-ress 14621  df-plusg 14692  df-mulr 14693  df-starv 14694  df-tset 14698  df-ple 14699  df-ds 14701  df-unif 14702  df-0g 14821  df-mgm 15851  df-sgrp 15890  df-mnd 15900  df-submnd 15946  df-grp 16036  df-minusg 16037  df-cmn 16779  df-abl 16780  df-mgp 17121  df-ur 17133  df-srg 17137  df-ring 17179  df-cring 17180  df-cnfld 18400
This theorem is referenced by:  xrge0slmod  27812
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