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Theorem xrge00 26146
Description: The zero of the extended nonnegative real numbers monoid. (Contributed by Thierry Arnoux, 30-Jan-2017.)
Assertion
Ref Expression
xrge00  |-  0  =  ( 0g `  ( RR*ss  ( 0 [,] +oo ) ) )

Proof of Theorem xrge00
StepHypRef Expression
1 eqid 2442 . . 3  |-  ( RR*ss  ( RR*  \  { -oo } ) )  =  (
RR*ss  ( RR*  \  { -oo } ) )
21xrs1mnd 17850 . 2  |-  ( RR*ss  ( RR*  \  { -oo } ) )  e.  Mnd
3 xrge0cmn 17854 . . 3  |-  ( RR*ss  ( 0 [,] +oo ) )  e. CMnd
4 cmnmnd 16291 . . 3  |-  ( (
RR*ss  ( 0 [,] +oo ) )  e. CMnd  ->  (
RR*ss  ( 0 [,] +oo ) )  e.  Mnd )
53, 4ax-mp 5 . 2  |-  ( RR*ss  ( 0 [,] +oo ) )  e.  Mnd
6 mnflt0 11104 . . . . . . 7  |- -oo  <  0
7 mnfxr 11093 . . . . . . . 8  |- -oo  e.  RR*
8 0xr 9429 . . . . . . . 8  |-  0  e.  RR*
9 xrltnle 9442 . . . . . . . 8  |-  ( ( -oo  e.  RR*  /\  0  e.  RR* )  ->  ( -oo  <  0  <->  -.  0  <_ -oo ) )
107, 8, 9mp2an 672 . . . . . . 7  |-  ( -oo  <  0  <->  -.  0  <_ -oo )
116, 10mpbi 208 . . . . . 6  |-  -.  0  <_ -oo
1211intnan 905 . . . . 5  |-  -.  ( -oo  e.  RR*  /\  0  <_ -oo )
13 elxrge0 11393 . . . . 5  |-  ( -oo  e.  ( 0 [,] +oo ) 
<->  ( -oo  e.  RR*  /\  0  <_ -oo )
)
1412, 13mtbir 299 . . . 4  |-  -. -oo  e.  ( 0 [,] +oo )
15 difsn 4007 . . . 4  |-  ( -. -oo  e.  ( 0 [,] +oo )  ->  ( ( 0 [,] +oo )  \  { -oo } )  =  ( 0 [,] +oo ) )
1614, 15ax-mp 5 . . 3  |-  ( ( 0 [,] +oo )  \  { -oo } )  =  ( 0 [,] +oo )
17 iccssxr 11377 . . . 4  |-  ( 0 [,] +oo )  C_  RR*
18 ssdif 3490 . . . 4  |-  ( ( 0 [,] +oo )  C_ 
RR*  ->  ( ( 0 [,] +oo )  \  { -oo } )  C_  ( RR*  \  { -oo } ) )
1917, 18ax-mp 5 . . 3  |-  ( ( 0 [,] +oo )  \  { -oo } ) 
C_  ( RR*  \  { -oo } )
2016, 19eqsstr3i 3386 . 2  |-  ( 0 [,] +oo )  C_  ( RR*  \  { -oo } )
21 0e0iccpnf 11395 . 2  |-  0  e.  ( 0 [,] +oo )
22 difss 3482 . . . . 5  |-  ( RR*  \  { -oo } ) 
C_  RR*
23 df-ss 3341 . . . . 5  |-  ( (
RR*  \  { -oo }
)  C_  RR*  <->  ( ( RR*  \  { -oo }
)  i^i  RR* )  =  ( RR*  \  { -oo } ) )
2422, 23mpbi 208 . . . 4  |-  ( (
RR*  \  { -oo }
)  i^i  RR* )  =  ( RR*  \  { -oo } )
25 xrex 10987 . . . . . 6  |-  RR*  e.  _V
26 difexg 4439 . . . . . 6  |-  ( RR*  e.  _V  ->  ( RR*  \  { -oo } )  e.  _V )
2725, 26ax-mp 5 . . . . 5  |-  ( RR*  \  { -oo } )  e.  _V
28 xrsbas 17831 . . . . . 6  |-  RR*  =  ( Base `  RR*s )
291, 28ressbas 14227 . . . . 5  |-  ( (
RR*  \  { -oo }
)  e.  _V  ->  ( ( RR*  \  { -oo } )  i^i  RR* )  =  ( Base `  ( RR*ss  ( RR*  \  { -oo } ) ) ) )
3027, 29ax-mp 5 . . . 4  |-  ( (
RR*  \  { -oo }
)  i^i  RR* )  =  ( Base `  ( RR*ss  ( RR*  \  { -oo } ) ) )
3124, 30eqtr3i 2464 . . 3  |-  ( RR*  \  { -oo } )  =  ( Base `  ( RR*ss  ( RR*  \  { -oo } ) ) )
321xrs10 17851 . . 3  |-  0  =  ( 0g `  ( RR*ss  ( RR*  \  { -oo } ) ) )
33 ovex 6115 . . . . 5  |-  ( 0 [,] +oo )  e. 
_V
34 ressress 14234 . . . . 5  |-  ( ( ( RR*  \  { -oo } )  e.  _V  /\  ( 0 [,] +oo )  e.  _V )  ->  ( ( RR*ss  ( RR*  \  { -oo }
) )s  ( 0 [,] +oo ) )  =  (
RR*ss  ( ( RR*  \  { -oo } )  i^i  ( 0 [,] +oo ) ) ) )
3527, 33, 34mp2an 672 . . . 4  |-  ( (
RR*ss  ( RR*  \  { -oo } ) )s  ( 0 [,] +oo ) )  =  ( RR*ss  (
( RR*  \  { -oo } )  i^i  ( 0 [,] +oo ) ) )
36 dfss 3342 . . . . . . 7  |-  ( ( 0 [,] +oo )  C_  ( RR*  \  { -oo } )  <->  ( 0 [,] +oo )  =  (
( 0 [,] +oo )  i^i  ( RR*  \  { -oo } ) ) )
3720, 36mpbi 208 . . . . . 6  |-  ( 0 [,] +oo )  =  ( ( 0 [,] +oo )  i^i  ( RR*  \  { -oo }
) )
38 incom 3542 . . . . . 6  |-  ( ( 0 [,] +oo )  i^i  ( RR*  \  { -oo } ) )  =  ( ( RR*  \  { -oo } )  i^i  ( 0 [,] +oo ) )
3937, 38eqtr2i 2463 . . . . 5  |-  ( (
RR*  \  { -oo }
)  i^i  ( 0 [,] +oo ) )  =  ( 0 [,] +oo )
4039oveq2i 6101 . . . 4  |-  ( RR*ss  ( ( RR*  \  { -oo } )  i^i  (
0 [,] +oo )
) )  =  (
RR*ss  ( 0 [,] +oo ) )
4135, 40eqtr2i 2463 . . 3  |-  ( RR*ss  ( 0 [,] +oo ) )  =  ( ( RR*ss  ( RR*  \  { -oo } ) )s  ( 0 [,] +oo ) )
4231, 32, 41submnd0 15450 . 2  |-  ( ( ( ( RR*ss  ( RR*  \  { -oo }
) )  e.  Mnd  /\  ( RR*ss  ( 0 [,] +oo ) )  e.  Mnd )  /\  ( ( 0 [,] +oo )  C_  ( RR*  \  { -oo } )  /\  0  e.  ( 0 [,] +oo )
) )  ->  0  =  ( 0g `  ( RR*ss  ( 0 [,] +oo ) ) ) )
432, 5, 20, 21, 42mp4an 673 1  |-  0  =  ( 0g `  ( RR*ss  ( 0 [,] +oo ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2971    \ cdif 3324    i^i cin 3326    C_ wss 3327   {csn 3876   class class class wbr 4291   ` cfv 5417  (class class class)co 6090   0cc0 9281   +oocpnf 9414   -oocmnf 9415   RR*cxr 9416    < clt 9417    <_ cle 9418   [,]cicc 11302   Basecbs 14173   ↾s cress 14174   0gc0g 14377   RR*scxrs 14437   Mndcmnd 15408  CMndccmn 16276
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-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358
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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-nn 10322  df-2 10379  df-3 10380  df-4 10381  df-5 10382  df-6 10383  df-7 10384  df-8 10385  df-9 10386  df-10 10387  df-n0 10579  df-z 10646  df-dec 10755  df-uz 10861  df-xadd 11089  df-icc 11306  df-fz 11437  df-struct 14175  df-ndx 14176  df-slot 14177  df-base 14178  df-sets 14179  df-ress 14180  df-plusg 14250  df-mulr 14251  df-tset 14256  df-ple 14257  df-ds 14259  df-0g 14379  df-xrs 14439  df-mnd 15414  df-submnd 15464  df-cmn 16278
This theorem is referenced by:  xrge0mulgnn0  26149  xrge0slmod  26311  xrge0iifmhm  26368  esumnul  26501  esum0  26502  gsumesum  26509  esumlub  26510  esumsn  26514  esumss  26520  esumpfinval  26523  esumpfinvalf  26524  esumcocn  26528
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