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Theorem xrge00 28497
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 2462 . . 3  |-  ( RR*ss  ( RR*  \  { -oo } ) )  =  (
RR*ss  ( RR*  \  { -oo } ) )
21xrs1mnd 19055 . 2  |-  ( RR*ss  ( RR*  \  { -oo } ) )  e.  Mnd
3 xrge0cmn 19059 . . 3  |-  ( RR*ss  ( 0 [,] +oo ) )  e. CMnd
4 cmnmnd 17494 . . 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 11456 . . . . . . 7  |- -oo  <  0
7 mnfxr 11443 . . . . . . . 8  |- -oo  e.  RR*
8 0xr 9713 . . . . . . . 8  |-  0  e.  RR*
9 xrltnle 9727 . . . . . . . 8  |-  ( ( -oo  e.  RR*  /\  0  e.  RR* )  ->  ( -oo  <  0  <->  -.  0  <_ -oo ) )
107, 8, 9mp2an 683 . . . . . . 7  |-  ( -oo  <  0  <->  -.  0  <_ -oo )
116, 10mpbi 213 . . . . . 6  |-  -.  0  <_ -oo
1211intnan 930 . . . . 5  |-  -.  ( -oo  e.  RR*  /\  0  <_ -oo )
13 elxrge0 11770 . . . . 5  |-  ( -oo  e.  ( 0 [,] +oo ) 
<->  ( -oo  e.  RR*  /\  0  <_ -oo )
)
1412, 13mtbir 305 . . . 4  |-  -. -oo  e.  ( 0 [,] +oo )
15 difsn 4119 . . . 4  |-  ( -. -oo  e.  ( 0 [,] +oo )  ->  ( ( 0 [,] +oo )  \  { -oo } )  =  ( 0 [,] +oo ) )
1614, 15ax-mp 5 . . 3  |-  ( ( 0 [,] +oo )  \  { -oo } )  =  ( 0 [,] +oo )
17 iccssxr 11746 . . . 4  |-  ( 0 [,] +oo )  C_  RR*
18 ssdif 3580 . . . 4  |-  ( ( 0 [,] +oo )  C_ 
RR*  ->  ( ( 0 [,] +oo )  \  { -oo } )  C_  ( RR*  \  { -oo } ) )
1917, 18ax-mp 5 . . 3  |-  ( ( 0 [,] +oo )  \  { -oo } ) 
C_  ( RR*  \  { -oo } )
2016, 19eqsstr3i 3475 . 2  |-  ( 0 [,] +oo )  C_  ( RR*  \  { -oo } )
21 0e0iccpnf 11772 . 2  |-  0  e.  ( 0 [,] +oo )
22 difss 3572 . . . . 5  |-  ( RR*  \  { -oo } ) 
C_  RR*
23 df-ss 3430 . . . . 5  |-  ( (
RR*  \  { -oo }
)  C_  RR*  <->  ( ( RR*  \  { -oo }
)  i^i  RR* )  =  ( RR*  \  { -oo } ) )
2422, 23mpbi 213 . . . 4  |-  ( (
RR*  \  { -oo }
)  i^i  RR* )  =  ( RR*  \  { -oo } )
25 xrex 11328 . . . . . 6  |-  RR*  e.  _V
26 difexg 4565 . . . . . 6  |-  ( RR*  e.  _V  ->  ( RR*  \  { -oo } )  e.  _V )
2725, 26ax-mp 5 . . . . 5  |-  ( RR*  \  { -oo } )  e.  _V
28 xrsbas 19033 . . . . . 6  |-  RR*  =  ( Base `  RR*s )
291, 28ressbas 15228 . . . . 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 2486 . . 3  |-  ( RR*  \  { -oo } )  =  ( Base `  ( RR*ss  ( RR*  \  { -oo } ) ) )
321xrs10 19056 . . 3  |-  0  =  ( 0g `  ( RR*ss  ( RR*  \  { -oo } ) ) )
33 ovex 6343 . . . . 5  |-  ( 0 [,] +oo )  e. 
_V
34 ressress 15236 . . . . 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 683 . . . 4  |-  ( (
RR*ss  ( RR*  \  { -oo } ) )s  ( 0 [,] +oo ) )  =  ( RR*ss  (
( RR*  \  { -oo } )  i^i  ( 0 [,] +oo ) ) )
36 dfss 3431 . . . . . . 7  |-  ( ( 0 [,] +oo )  C_  ( RR*  \  { -oo } )  <->  ( 0 [,] +oo )  =  (
( 0 [,] +oo )  i^i  ( RR*  \  { -oo } ) ) )
3720, 36mpbi 213 . . . . . 6  |-  ( 0 [,] +oo )  =  ( ( 0 [,] +oo )  i^i  ( RR*  \  { -oo }
) )
38 incom 3637 . . . . . 6  |-  ( ( 0 [,] +oo )  i^i  ( RR*  \  { -oo } ) )  =  ( ( RR*  \  { -oo } )  i^i  ( 0 [,] +oo ) )
3937, 38eqtr2i 2485 . . . . 5  |-  ( (
RR*  \  { -oo }
)  i^i  ( 0 [,] +oo ) )  =  ( 0 [,] +oo )
4039oveq2i 6326 . . . 4  |-  ( RR*ss  ( ( RR*  \  { -oo } )  i^i  (
0 [,] +oo )
) )  =  (
RR*ss  ( 0 [,] +oo ) )
4135, 40eqtr2i 2485 . . 3  |-  ( RR*ss  ( 0 [,] +oo ) )  =  ( ( RR*ss  ( RR*  \  { -oo } ) )s  ( 0 [,] +oo ) )
4231, 32, 41submnd0 16615 . 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 684 1  |-  0  =  ( 0g `  ( RR*ss  ( 0 [,] +oo ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 189    /\ wa 375    = wceq 1455    e. wcel 1898   _Vcvv 3057    \ cdif 3413    i^i cin 3415    C_ wss 3416   {csn 3980   class class class wbr 4416   ` cfv 5601  (class class class)co 6315   0cc0 9565   +oocpnf 9698   -oocmnf 9699   RR*cxr 9700    < clt 9701    <_ cle 9702   [,]cicc 11667   Basecbs 15170   ↾s cress 15171   0gc0g 15387   RR*scxrs 15447   Mndcmnd 16584  CMndccmn 17479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-om 6720  df-1st 6820  df-2nd 6821  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-1o 7208  df-oadd 7212  df-er 7389  df-en 7596  df-dom 7597  df-sdom 7598  df-fin 7599  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-nn 10638  df-2 10696  df-3 10697  df-4 10698  df-5 10699  df-6 10700  df-7 10701  df-8 10702  df-9 10703  df-10 10704  df-n0 10899  df-z 10967  df-dec 11081  df-uz 11189  df-xadd 11439  df-icc 11671  df-fz 11814  df-struct 15172  df-ndx 15173  df-slot 15174  df-base 15175  df-sets 15176  df-ress 15177  df-plusg 15252  df-mulr 15253  df-tset 15258  df-ple 15259  df-ds 15261  df-0g 15389  df-xrs 15449  df-mgm 16537  df-sgrp 16576  df-mnd 16586  df-submnd 16632  df-cmn 17481
This theorem is referenced by:  xrge0mulgnn0  28500  xrge0slmod  28656  xrge0iifmhm  28794  esumgsum  28915  esumnul  28918  esum0  28919  gsumesum  28929  esumsnf  28934  esumss  28942  esumpfinval  28945  esumpfinvalf  28946  esumcocn  28950  sitmcl  29233
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