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Theorem rge0srg 17897
Description: The nonnegative real numbers form a semiring (commutative by subcmn 16336). (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 17853 . . . 4  |-fld  e.  Ring
2 rngcmn 16690 . . . 4  |-  (fld  e.  Ring  ->fld  e. CMnd )
31, 2ax-mp 5 . . 3  |-fld  e. CMnd
4 rge0ssre 11408 . . . . . 6  |-  ( 0 [,) +oo )  C_  RR
5 ax-resscn 9354 . . . . . 6  |-  RR  C_  CC
64, 5sstri 3380 . . . . 5  |-  ( 0 [,) +oo )  C_  CC
7 0e0icopnf 11410 . . . . 5  |-  0  e.  ( 0 [,) +oo )
8 ge0addcl 11412 . . . . . 6  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  ( x  +  y )  e.  ( 0 [,) +oo )
)
98rgen2 2827 . . . . 5  |-  A. x  e.  ( 0 [,) +oo ) A. y  e.  ( 0 [,) +oo )
( x  +  y )  e.  ( 0 [,) +oo )
106, 7, 93pm3.2i 1166 . . . 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 16669 . . . . . 6  |-  (fld  e.  Ring  ->fld  e.  Mnd )
121, 11ax-mp 5 . . . . 5  |-fld  e.  Mnd
13 cnfldbas 17837 . . . . . 6  |-  CC  =  ( Base ` fld )
14 cnfld0 17855 . . . . . 6  |-  0  =  ( 0g ` fld )
15 cnfldadd 17838 . . . . . 6  |-  +  =  ( +g  ` fld )
1613, 14, 15issubm 15490 . . . . 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 2443 . . . 4  |-  (flds  ( 0 [,) +oo ) )  =  (flds  ( 0 [,) +oo ) )
2019submcmn 16337 . . 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 9400 . . . . . 6  |-  1  e.  RR
23 0le1 9878 . . . . . 6  |-  0  <_  1
24 ltpnf 11117 . . . . . . 7  |-  ( 1  e.  RR  ->  1  < +oo )
2522, 24ax-mp 5 . . . . . 6  |-  1  < +oo
26 0re 9401 . . . . . . 7  |-  0  e.  RR
27 pnfxr 11107 . . . . . . 7  |- +oo  e.  RR*
28 elico2 11374 . . . . . . 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 1170 . . . . 5  |-  1  e.  ( 0 [,) +oo )
31 ge0mulcl 11413 . . . . . 6  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  ( x  x.  y )  e.  ( 0 [,) +oo )
)
3231rgen2 2827 . . . . 5  |-  A. x  e.  ( 0 [,) +oo ) A. y  e.  ( 0 [,) +oo )
( x  x.  y
)  e.  ( 0 [,) +oo )
336, 30, 323pm3.2i 1166 . . . 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 2443 . . . . . 6  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
3534rngmgp 16666 . . . . 5  |-  (fld  e.  Ring  -> 
(mulGrp ` fld )  e.  Mnd )
3634, 13mgpbas 16612 . . . . . 6  |-  CC  =  ( Base `  (mulGrp ` fld ) )
37 cnfld1 17856 . . . . . . 7  |-  1  =  ( 1r ` fld )
3834, 37rngidval 16620 . . . . . 6  |-  1  =  ( 0g `  (mulGrp ` fld ) )
39 cnfldmul 17839 . . . . . . 7  |-  x.  =  ( .r ` fld )
4034, 39mgpplusg 16610 . . . . . 6  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
4136, 38, 40issubm 15490 . . . . 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 2443 . . . 4  |-  ( (mulGrp ` fld )s  ( 0 [,) +oo ) )  =  ( (mulGrp ` fld )s  ( 0 [,) +oo ) )
4544submmnd 15497 . . 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 3369 . . . . . . . 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 3369 . . . . . . . 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 3369 . . . . . . . 8  |-  ( ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  (
0 [,) +oo )
)  /\  z  e.  ( 0 [,) +oo ) )  ->  z  e.  CC )
5348, 50, 52adddid 9425 . . . . . . 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 9426 . . . . . . 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 2814 . . . . 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 2814 . . . 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 3367 . . . . 5  |-  ( x  e.  ( 0 [,) +oo )  ->  x  e.  CC )
5958mul02d 9582 . . . 4  |-  ( x  e.  ( 0 [,) +oo )  ->  ( 0  x.  x )  =  0 )
6058mul01d 9583 . . . 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 2796 . 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 14244 . . . 4  |-  ( ( 0 [,) +oo )  C_  CC  ->  ( 0 [,) +oo )  =  ( Base `  (flds  ( 0 [,) +oo ) ) ) )
646, 63ax-mp 5 . . 3  |-  ( 0 [,) +oo )  =  ( Base `  (flds  ( 0 [,) +oo ) ) )
65 cnfldex 17836 . . . 4  |-fld  e.  _V
66 ovex 6131 . . . 4  |-  ( 0 [,) +oo )  e. 
_V
6719, 34mgpress 16617 . . . 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 14295 . . . 4  |-  ( ( 0 [,) +oo )  e.  _V  ->  +  =  ( +g  `  (flds  ( 0 [,) +oo ) ) ) )
7066, 69ax-mp 5 . . 3  |-  +  =  ( +g  `  (flds  ( 0 [,) +oo ) ) )
7119, 39ressmulr 14306 . . . 4  |-  ( ( 0 [,) +oo )  e.  _V  ->  x.  =  ( .r `  (flds  ( 0 [,) +oo ) ) ) )
7266, 71ax-mp 5 . . 3  |-  x.  =  ( .r `  (flds  ( 0 [,) +oo ) ) )
7319, 13, 14ress0g 15465 . . . 4  |-  ( (fld  e. 
Mnd  /\  0  e.  ( 0 [,) +oo )  /\  ( 0 [,) +oo )  C_  CC )  ->  0  =  ( 0g `  (flds  ( 0 [,) +oo ) ) ) )
7412, 7, 6, 73mp3an 1314 . . 3  |-  0  =  ( 0g `  (flds  (
0 [,) +oo )
) )
7564, 68, 70, 72, 74issrg 16624 . 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 1170 1  |-  (flds  ( 0 [,) +oo ) )  e. SRing
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2730   _Vcvv 2987    C_ wss 3343   class class class wbr 4307   ` cfv 5433  (class class class)co 6106   CCcc 9295   RRcr 9296   0cc0 9297   1c1 9298    + caddc 9300    x. cmul 9302   +oocpnf 9430   RR*cxr 9432    < clt 9433    <_ cle 9434   [,)cico 11317   Basecbs 14189   ↾s cress 14190   +g cplusg 14253   .rcmulr 14254   0gc0g 14393   Mndcmnd 15424  SubMndcsubmnd 15478  CMndccmn 16292  mulGrpcmgp 16606  SRingcsrg 16622   Ringcrg 16660  ℂfldccnfld 17833
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 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374  ax-addf 9376  ax-mulf 9377
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-int 4144  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-om 6492  df-1st 6592  df-2nd 6593  df-recs 6847  df-rdg 6881  df-1o 6935  df-oadd 6939  df-er 7116  df-en 7326  df-dom 7327  df-sdom 7328  df-fin 7329  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-nn 10338  df-2 10395  df-3 10396  df-4 10397  df-5 10398  df-6 10399  df-7 10400  df-8 10401  df-9 10402  df-10 10403  df-n0 10595  df-z 10662  df-dec 10771  df-uz 10877  df-ico 11321  df-fz 11453  df-struct 14191  df-ndx 14192  df-slot 14193  df-base 14194  df-sets 14195  df-ress 14196  df-plusg 14266  df-mulr 14267  df-starv 14268  df-tset 14272  df-ple 14273  df-ds 14275  df-unif 14276  df-0g 14395  df-mnd 15430  df-submnd 15480  df-grp 15560  df-minusg 15561  df-cmn 16294  df-abl 16295  df-mgp 16607  df-ur 16619  df-srg 16623  df-rng 16662  df-cring 16663  df-cnfld 17834
This theorem is referenced by:  xrge0slmod  26327
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