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Theorem xrge00 27827
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 2382 . . 3  |-  ( RR*ss  ( RR*  \  { -oo } ) )  =  (
RR*ss  ( RR*  \  { -oo } ) )
21xrs1mnd 18569 . 2  |-  ( RR*ss  ( RR*  \  { -oo } ) )  e.  Mnd
3 xrge0cmn 18573 . . 3  |-  ( RR*ss  ( 0 [,] +oo ) )  e. CMnd
4 cmnmnd 16930 . . 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 11255 . . . . . . 7  |- -oo  <  0
7 mnfxr 11244 . . . . . . . 8  |- -oo  e.  RR*
8 0xr 9551 . . . . . . . 8  |-  0  e.  RR*
9 xrltnle 9564 . . . . . . . 8  |-  ( ( -oo  e.  RR*  /\  0  e.  RR* )  ->  ( -oo  <  0  <->  -.  0  <_ -oo ) )
107, 8, 9mp2an 670 . . . . . . 7  |-  ( -oo  <  0  <->  -.  0  <_ -oo )
116, 10mpbi 208 . . . . . 6  |-  -.  0  <_ -oo
1211intnan 912 . . . . 5  |-  -.  ( -oo  e.  RR*  /\  0  <_ -oo )
13 elxrge0 11550 . . . . 5  |-  ( -oo  e.  ( 0 [,] +oo ) 
<->  ( -oo  e.  RR*  /\  0  <_ -oo )
)
1412, 13mtbir 297 . . . 4  |-  -. -oo  e.  ( 0 [,] +oo )
15 difsn 4078 . . . 4  |-  ( -. -oo  e.  ( 0 [,] +oo )  ->  ( ( 0 [,] +oo )  \  { -oo } )  =  ( 0 [,] +oo ) )
1614, 15ax-mp 5 . . 3  |-  ( ( 0 [,] +oo )  \  { -oo } )  =  ( 0 [,] +oo )
17 iccssxr 11528 . . . 4  |-  ( 0 [,] +oo )  C_  RR*
18 ssdif 3553 . . . 4  |-  ( ( 0 [,] +oo )  C_ 
RR*  ->  ( ( 0 [,] +oo )  \  { -oo } )  C_  ( RR*  \  { -oo } ) )
1917, 18ax-mp 5 . . 3  |-  ( ( 0 [,] +oo )  \  { -oo } ) 
C_  ( RR*  \  { -oo } )
2016, 19eqsstr3i 3448 . 2  |-  ( 0 [,] +oo )  C_  ( RR*  \  { -oo } )
21 0e0iccpnf 11552 . 2  |-  0  e.  ( 0 [,] +oo )
22 difss 3545 . . . . 5  |-  ( RR*  \  { -oo } ) 
C_  RR*
23 df-ss 3403 . . . . 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 11136 . . . . . 6  |-  RR*  e.  _V
26 difexg 4513 . . . . . 6  |-  ( RR*  e.  _V  ->  ( RR*  \  { -oo } )  e.  _V )
2725, 26ax-mp 5 . . . . 5  |-  ( RR*  \  { -oo } )  e.  _V
28 xrsbas 18547 . . . . . 6  |-  RR*  =  ( Base `  RR*s )
291, 28ressbas 14691 . . . . 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 2413 . . 3  |-  ( RR*  \  { -oo } )  =  ( Base `  ( RR*ss  ( RR*  \  { -oo } ) ) )
321xrs10 18570 . . 3  |-  0  =  ( 0g `  ( RR*ss  ( RR*  \  { -oo } ) ) )
33 ovex 6224 . . . . 5  |-  ( 0 [,] +oo )  e. 
_V
34 ressress 14699 . . . . 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 670 . . . 4  |-  ( (
RR*ss  ( RR*  \  { -oo } ) )s  ( 0 [,] +oo ) )  =  ( RR*ss  (
( RR*  \  { -oo } )  i^i  ( 0 [,] +oo ) ) )
36 dfss 3404 . . . . . . 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 3605 . . . . . 6  |-  ( ( 0 [,] +oo )  i^i  ( RR*  \  { -oo } ) )  =  ( ( RR*  \  { -oo } )  i^i  ( 0 [,] +oo ) )
3937, 38eqtr2i 2412 . . . . 5  |-  ( (
RR*  \  { -oo }
)  i^i  ( 0 [,] +oo ) )  =  ( 0 [,] +oo )
4039oveq2i 6207 . . . 4  |-  ( RR*ss  ( ( RR*  \  { -oo } )  i^i  (
0 [,] +oo )
) )  =  (
RR*ss  ( 0 [,] +oo ) )
4135, 40eqtr2i 2412 . . 3  |-  ( RR*ss  ( 0 [,] +oo ) )  =  ( ( RR*ss  ( RR*  \  { -oo } ) )s  ( 0 [,] +oo ) )
4231, 32, 41submnd0 16067 . 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 671 1  |-  0  =  ( 0g `  ( RR*ss  ( 0 [,] +oo ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826   _Vcvv 3034    \ cdif 3386    i^i cin 3388    C_ wss 3389   {csn 3944   class class class wbr 4367   ` cfv 5496  (class class class)co 6196   0cc0 9403   +oocpnf 9536   -oocmnf 9537   RR*cxr 9538    < clt 9539    <_ cle 9540   [,]cicc 11453   Basecbs 14634   ↾s cress 14635   0gc0g 14847   RR*scxrs 14907   Mndcmnd 16036  CMndccmn 16915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-7 10516  df-8 10517  df-9 10518  df-10 10519  df-n0 10713  df-z 10782  df-dec 10896  df-uz 11002  df-xadd 11240  df-icc 11457  df-fz 11594  df-struct 14636  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-mulr 14716  df-tset 14721  df-ple 14722  df-ds 14724  df-0g 14849  df-xrs 14909  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-submnd 16084  df-cmn 16917
This theorem is referenced by:  xrge0mulgnn0  27830  xrge0slmod  27988  xrge0iifmhm  28075  esumgsum  28193  esumnul  28196  esum0  28197  gsumesum  28207  esumsnf  28212  esumss  28220  esumpfinval  28223  esumpfinvalf  28224  esumcocn  28228
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