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Theorem xrge00 29686
Description: The zero of the extended nonnegative real numbers monoid. (Contributed by Thierry Arnoux, 30-Jan-2017.)
Assertion
Ref Expression
xrge00 |- 0 = ( 0g ` ( RR*s ↾s ( 0 [,] +oo ) ) )
Proof of Theorem xrge00
StepHypRef Expression
1 eqid 2622 . . 3 |- ( RR*s ↾s ( RR* \ { -oo } ) ) = ( RR*s ↾s ( RR* \ { -oo } ) )
21xrs1mnd 19784 . 2 |- ( RR*s ↾s ( RR* \ { -oo } ) ) e. Mnd
3 xrge0cmn 19788 . . 3 |- ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd
4 cmnmnd 18208 . . 3 |- ( ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd -> ( RR*s ↾s ( 0 [,] +oo ) ) e. Mnd )
53, 4ax-mp 5 . 2 |- ( RR*s ↾s ( 0 [,] +oo ) ) e. Mnd
6 mnflt0 11959 . . . . . . 7 |- -oo < 0
7 mnfxr 10096 . . . . . . . 8 |- -oo e. RR*
8 0xr 10086 . . . . . . . 8 |- 0 e. RR*
9 xrltnle 10105 . . . . . . . 8 |- ( ( -oo e. RR* /\ 0 e. RR* ) -> ( -oo < 0 <-> -. 0 <_ -oo ) )
107, 8, 9mp2an 708 . . . . . . 7 |- ( -oo < 0 <-> -. 0 <_ -oo )
116, 10mpbi 220 . . . . . 6 |- -. 0 <_ -oo
1211intnan 960 . . . . 5 |- -. ( -oo e. RR* /\ 0 <_ -oo )
13 elxrge0 12281 . . . . 5 |- ( -oo e. ( 0 [,] +oo ) <-> ( -oo e. RR* /\ 0 <_ -oo ) )
1412, 13mtbir 313 . . . 4 |- -. -oo e. ( 0 [,] +oo )
15 difsn 4328 . . . 4 |- ( -. -oo e. ( 0 [,] +oo ) -> ( ( 0 [,] +oo ) \ { -oo } ) = ( 0 [,] +oo ) )
1614, 15ax-mp 5 . . 3 |- ( ( 0 [,] +oo ) \ { -oo } ) = ( 0 [,] +oo )
17 iccssxr 12256 . . . 4 |- ( 0 [,] +oo ) C_ RR*
18 ssdif 3745 . . . 4 |- ( ( 0 [,] +oo ) C_ RR* -> ( ( 0 [,] +oo ) \ { -oo } ) C_ ( RR* \ { -oo } ) )
1917, 18ax-mp 5 . . 3 |- ( ( 0 [,] +oo ) \ { -oo } ) C_ ( RR* \ { -oo } )
2016, 19eqsstr3i 3636 . 2 |- ( 0 [,] +oo ) C_ ( RR* \ { -oo } )
21 0e0iccpnf 12283 . 2 |- 0 e. ( 0 [,] +oo )
22 difss 3737 . . . . 5 |- ( RR* \ { -oo } ) C_ RR*
23 df-ss 3588 . . . . 5 |- ( ( RR* \ { -oo } ) C_ RR* <-> ( ( RR* \ { -oo } ) i^i RR* ) = ( RR* \ { -oo } ) )
2422, 23mpbi 220 . . . 4 |- ( ( RR* \ { -oo } ) i^i RR* ) = ( RR* \ { -oo } )
25 xrex 11829 . . . . . 6 |- RR* e. _V
26 difexg 4808 . . . . . 6 |- ( RR* e. _V -> ( RR* \ { -oo } ) e. _V )
2725, 26ax-mp 5 . . . . 5 |- ( RR* \ { -oo } ) e. _V
28 xrsbas 19762 . . . . . 6 |- RR* = ( Base ` RR*s )
291, 28ressbas 15930 . . . . 5 |- ( ( RR* \ { -oo } ) e. _V -> ( ( RR* \ { -oo } ) i^i RR* ) = ( Base ` ( RR*s ↾s ( RR* \ { -oo } ) ) ) )
3027, 29ax-mp 5 . . . 4 |- ( ( RR* \ { -oo } ) i^i RR* ) = ( Base ` ( RR*s ↾s ( RR* \ { -oo } ) ) )
3124, 30eqtr3i 2646 . . 3 |- ( RR* \ { -oo } ) = ( Base ` ( RR*s ↾s ( RR* \ { -oo } ) ) )
321xrs10 19785 . . 3 |- 0 = ( 0g ` ( RR*s ↾s ( RR* \ { -oo } ) ) )
33 ovex 6678 . . . . 5 |- ( 0 [,] +oo ) e. _V
34 ressress 15938 . . . . 5 |- ( ( ( RR* \ { -oo } ) e. _V /\ ( 0 [,] +oo ) e. _V ) -> ( ( RR*s ↾s ( RR* \ { -oo } ) ) ↾s ( 0 [,] +oo ) ) = ( RR*s ↾s ( ( RR* \ { -oo } ) i^i ( 0 [,] +oo ) ) ) )
3527, 33, 34mp2an 708 . . . 4 |- ( ( RR*s ↾s ( RR* \ { -oo } ) ) ↾s ( 0 [,] +oo ) ) = ( RR*s ↾s ( ( RR* \ { -oo } ) i^i ( 0 [,] +oo ) ) )
36 dfss 3589 . . . . . . 7 |- ( ( 0 [,] +oo ) C_ ( RR* \ { -oo } ) <-> ( 0 [,] +oo ) = ( ( 0 [,] +oo ) i^i ( RR* \ { -oo } ) ) )
3720, 36mpbi 220 . . . . . 6 |- ( 0 [,] +oo ) = ( ( 0 [,] +oo ) i^i ( RR* \ { -oo } ) )
38 incom 3805 . . . . . 6 |- ( ( 0 [,] +oo ) i^i ( RR* \ { -oo } ) ) = ( ( RR* \ { -oo } ) i^i ( 0 [,] +oo ) )
3937, 38eqtr2i 2645 . . . . 5 |- ( ( RR* \ { -oo } ) i^i ( 0 [,] +oo ) ) = ( 0 [,] +oo )
4039oveq2i 6661 . . . 4 |- ( RR*s ↾s ( ( RR* \ { -oo } ) i^i ( 0 [,] +oo ) ) ) = ( RR*s ↾s ( 0 [,] +oo ) )
4135, 40eqtr2i 2645 . . 3 |- ( RR*s ↾s ( 0 [,] +oo ) ) = ( ( RR*s ↾s ( RR* \ { -oo } ) ) ↾s ( 0 [,] +oo ) )
4231, 32, 41submnd0 17320 . 2 |- ( ( ( ( RR*s ↾s ( RR* \ { -oo } ) ) e. Mnd /\ ( RR*s ↾s ( 0 [,] +oo ) ) e. Mnd ) /\ ( ( 0 [,] +oo ) C_ ( RR* \ { -oo } ) /\ 0 e. ( 0 [,] +oo ) ) ) -> 0 = ( 0g ` ( RR*s ↾s ( 0 [,] +oo ) ) ) )
432, 5, 20, 21, 42mp4an 709 1 |- 0 = ( 0g ` ( RR*s ↾s ( 0 [,] +oo ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   {csn 4177   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   0cc0 9936   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   < clt 10074   <_ cle 10075   [,]cicc 12178   Basecbs 15857   ↾s cress 15858   0gc0g 16100   RR*scxrs 16160   Mndcmnd 17294  CMndccmn 18193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-xadd 11947  df-icc 12182  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-tset 15960  df-ple 15961  df-ds 15964  df-0g 16102  df-xrs 16162  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-cmn 18195
This theorem is referenced by:  xrge0mulgnn0  29689  xrge0slmod  29844  xrge0iifmhm  29985  esumgsum  30107  esumnul  30110  esum0  30111  gsumesum  30121  esumsnf  30126  esumss  30134  esumpfinval  30137  esumpfinvalf  30138  esumcocn  30142  sitmcl  30413
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