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Theorem xrge00 27498
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 2467 . . 3  |-  ( RR*ss  ( RR*  \  { -oo } ) )  =  (
RR*ss  ( RR*  \  { -oo } ) )
21xrs1mnd 18326 . 2  |-  ( RR*ss  ( RR*  \  { -oo } ) )  e.  Mnd
3 xrge0cmn 18330 . . 3  |-  ( RR*ss  ( 0 [,] +oo ) )  e. CMnd
4 cmnmnd 16686 . . 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 11346 . . . . . . 7  |- -oo  <  0
7 mnfxr 11335 . . . . . . . 8  |- -oo  e.  RR*
8 0xr 9652 . . . . . . . 8  |-  0  e.  RR*
9 xrltnle 9665 . . . . . . . 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 11641 . . . . 5  |-  ( -oo  e.  ( 0 [,] +oo ) 
<->  ( -oo  e.  RR*  /\  0  <_ -oo )
)
1412, 13mtbir 299 . . . 4  |-  -. -oo  e.  ( 0 [,] +oo )
15 difsn 4167 . . . 4  |-  ( -. -oo  e.  ( 0 [,] +oo )  ->  ( ( 0 [,] +oo )  \  { -oo } )  =  ( 0 [,] +oo ) )
1614, 15ax-mp 5 . . 3  |-  ( ( 0 [,] +oo )  \  { -oo } )  =  ( 0 [,] +oo )
17 iccssxr 11619 . . . 4  |-  ( 0 [,] +oo )  C_  RR*
18 ssdif 3644 . . . 4  |-  ( ( 0 [,] +oo )  C_ 
RR*  ->  ( ( 0 [,] +oo )  \  { -oo } )  C_  ( RR*  \  { -oo } ) )
1917, 18ax-mp 5 . . 3  |-  ( ( 0 [,] +oo )  \  { -oo } ) 
C_  ( RR*  \  { -oo } )
2016, 19eqsstr3i 3540 . 2  |-  ( 0 [,] +oo )  C_  ( RR*  \  { -oo } )
21 0e0iccpnf 11643 . 2  |-  0  e.  ( 0 [,] +oo )
22 difss 3636 . . . . 5  |-  ( RR*  \  { -oo } ) 
C_  RR*
23 df-ss 3495 . . . . 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 11229 . . . . . 6  |-  RR*  e.  _V
26 difexg 4601 . . . . . 6  |-  ( RR*  e.  _V  ->  ( RR*  \  { -oo } )  e.  _V )
2725, 26ax-mp 5 . . . . 5  |-  ( RR*  \  { -oo } )  e.  _V
28 xrsbas 18304 . . . . . 6  |-  RR*  =  ( Base `  RR*s )
291, 28ressbas 14562 . . . . 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 2498 . . 3  |-  ( RR*  \  { -oo } )  =  ( Base `  ( RR*ss  ( RR*  \  { -oo } ) ) )
321xrs10 18327 . . 3  |-  0  =  ( 0g `  ( RR*ss  ( RR*  \  { -oo } ) ) )
33 ovex 6320 . . . . 5  |-  ( 0 [,] +oo )  e. 
_V
34 ressress 14569 . . . . 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 3496 . . . . . . 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 3696 . . . . . 6  |-  ( ( 0 [,] +oo )  i^i  ( RR*  \  { -oo } ) )  =  ( ( RR*  \  { -oo } )  i^i  ( 0 [,] +oo ) )
3937, 38eqtr2i 2497 . . . . 5  |-  ( (
RR*  \  { -oo }
)  i^i  ( 0 [,] +oo ) )  =  ( 0 [,] +oo )
4039oveq2i 6306 . . . 4  |-  ( RR*ss  ( ( RR*  \  { -oo } )  i^i  (
0 [,] +oo )
) )  =  (
RR*ss  ( 0 [,] +oo ) )
4135, 40eqtr2i 2497 . . 3  |-  ( RR*ss  ( 0 [,] +oo ) )  =  ( ( RR*ss  ( RR*  \  { -oo } ) )s  ( 0 [,] +oo ) )
4231, 32, 41submnd0 15823 . 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 1379    e. wcel 1767   _Vcvv 3118    \ cdif 3478    i^i cin 3480    C_ wss 3481   {csn 4033   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   0cc0 9504   +oocpnf 9637   -oocmnf 9638   RR*cxr 9639    < clt 9640    <_ cle 9641   [,]cicc 11544   Basecbs 14507   ↾s cress 14508   0gc0g 14712   RR*scxrs 14772   Mndcmnd 15793  CMndccmn 16671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-xadd 11331  df-icc 11548  df-fz 11685  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-tset 14591  df-ple 14592  df-ds 14594  df-0g 14714  df-xrs 14774  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-cmn 16673
This theorem is referenced by:  xrge0mulgnn0  27501  xrge0slmod  27659  xrge0iifmhm  27746  esumnul  27884  esum0  27885  gsumesum  27892  esumsn  27897  esumss  27903  esumpfinval  27906  esumpfinvalf  27907  esumcocn  27911
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