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Theorem xrge00 27544
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 2441 . . 3  |-  ( RR*ss  ( RR*  \  { -oo } ) )  =  (
RR*ss  ( RR*  \  { -oo } ) )
21xrs1mnd 18327 . 2  |-  ( RR*ss  ( RR*  \  { -oo } ) )  e.  Mnd
3 xrge0cmn 18331 . . 3  |-  ( RR*ss  ( 0 [,] +oo ) )  e. CMnd
4 cmnmnd 16684 . . 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 11340 . . . . . . 7  |- -oo  <  0
7 mnfxr 11329 . . . . . . . 8  |- -oo  e.  RR*
8 0xr 9640 . . . . . . . 8  |-  0  e.  RR*
9 xrltnle 9653 . . . . . . . 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 912 . . . . 5  |-  -.  ( -oo  e.  RR*  /\  0  <_ -oo )
13 elxrge0 11635 . . . . 5  |-  ( -oo  e.  ( 0 [,] +oo ) 
<->  ( -oo  e.  RR*  /\  0  <_ -oo )
)
1412, 13mtbir 299 . . . 4  |-  -. -oo  e.  ( 0 [,] +oo )
15 difsn 4146 . . . 4  |-  ( -. -oo  e.  ( 0 [,] +oo )  ->  ( ( 0 [,] +oo )  \  { -oo } )  =  ( 0 [,] +oo ) )
1614, 15ax-mp 5 . . 3  |-  ( ( 0 [,] +oo )  \  { -oo } )  =  ( 0 [,] +oo )
17 iccssxr 11613 . . . 4  |-  ( 0 [,] +oo )  C_  RR*
18 ssdif 3622 . . . 4  |-  ( ( 0 [,] +oo )  C_ 
RR*  ->  ( ( 0 [,] +oo )  \  { -oo } )  C_  ( RR*  \  { -oo } ) )
1917, 18ax-mp 5 . . 3  |-  ( ( 0 [,] +oo )  \  { -oo } ) 
C_  ( RR*  \  { -oo } )
2016, 19eqsstr3i 3518 . 2  |-  ( 0 [,] +oo )  C_  ( RR*  \  { -oo } )
21 0e0iccpnf 11637 . 2  |-  0  e.  ( 0 [,] +oo )
22 difss 3614 . . . . 5  |-  ( RR*  \  { -oo } ) 
C_  RR*
23 df-ss 3473 . . . . 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 11223 . . . . . 6  |-  RR*  e.  _V
26 difexg 4582 . . . . . 6  |-  ( RR*  e.  _V  ->  ( RR*  \  { -oo } )  e.  _V )
2725, 26ax-mp 5 . . . . 5  |-  ( RR*  \  { -oo } )  e.  _V
28 xrsbas 18305 . . . . . 6  |-  RR*  =  ( Base `  RR*s )
291, 28ressbas 14561 . . . . 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 2472 . . 3  |-  ( RR*  \  { -oo } )  =  ( Base `  ( RR*ss  ( RR*  \  { -oo } ) ) )
321xrs10 18328 . . 3  |-  0  =  ( 0g `  ( RR*ss  ( RR*  \  { -oo } ) ) )
33 ovex 6306 . . . . 5  |-  ( 0 [,] +oo )  e. 
_V
34 ressress 14568 . . . . 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 3474 . . . . . . 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 3674 . . . . . 6  |-  ( ( 0 [,] +oo )  i^i  ( RR*  \  { -oo } ) )  =  ( ( RR*  \  { -oo } )  i^i  ( 0 [,] +oo ) )
3937, 38eqtr2i 2471 . . . . 5  |-  ( (
RR*  \  { -oo }
)  i^i  ( 0 [,] +oo ) )  =  ( 0 [,] +oo )
4039oveq2i 6289 . . . 4  |-  ( RR*ss  ( ( RR*  \  { -oo } )  i^i  (
0 [,] +oo )
) )  =  (
RR*ss  ( 0 [,] +oo ) )
4135, 40eqtr2i 2471 . . 3  |-  ( RR*ss  ( 0 [,] +oo ) )  =  ( ( RR*ss  ( RR*  \  { -oo } ) )s  ( 0 [,] +oo ) )
4231, 32, 41submnd0 15821 . 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 1381    e. wcel 1802   _Vcvv 3093    \ cdif 3456    i^i cin 3458    C_ wss 3459   {csn 4011   class class class wbr 4434   ` cfv 5575  (class class class)co 6278   0cc0 9492   +oocpnf 9625   -oocmnf 9626   RR*cxr 9627    < clt 9628    <_ cle 9629   [,]cicc 11538   Basecbs 14506   ↾s cress 14507   0gc0g 14711   RR*scxrs 14771   Mndcmnd 15790  CMndccmn 16669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-pss 3475  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-tp 4016  df-op 4018  df-uni 4232  df-int 4269  df-iun 4314  df-br 4435  df-opab 4493  df-mpt 4494  df-tr 4528  df-eprel 4778  df-id 4782  df-po 4787  df-so 4788  df-fr 4825  df-we 4827  df-ord 4868  df-on 4869  df-lim 4870  df-suc 4871  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-riota 6239  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6683  df-1st 6782  df-2nd 6783  df-recs 7041  df-rdg 7075  df-1o 7129  df-oadd 7133  df-er 7310  df-en 7516  df-dom 7517  df-sdom 7518  df-fin 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9809  df-neg 9810  df-nn 10540  df-2 10597  df-3 10598  df-4 10599  df-5 10600  df-6 10601  df-7 10602  df-8 10603  df-9 10604  df-10 10605  df-n0 10799  df-z 10868  df-dec 10982  df-uz 11088  df-xadd 11325  df-icc 11542  df-fz 11679  df-struct 14508  df-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-ress 14513  df-plusg 14584  df-mulr 14585  df-tset 14590  df-ple 14591  df-ds 14593  df-0g 14713  df-xrs 14773  df-mgm 15743  df-sgrp 15782  df-mnd 15792  df-submnd 15838  df-cmn 16671
This theorem is referenced by:  xrge0mulgnn0  27547  xrge0slmod  27704  xrge0iifmhm  27791  esumnul  27929  esum0  27930  gsumesum  27937  esumsn  27942  esumss  27948  esumpfinval  27951  esumpfinvalf  27952  esumcocn  27956
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