Theorem xrge00 26146

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

Step	Hyp	Ref	Expression
1		eqid 2442	. . 3 |- ( RR*s ↾s ( RR* \ { -oo } ) ) = ( RR*s ↾s ( RR* \ { -oo } ) )
2	1	xrs1mnd 17850	. 2 |- ( RR*s ↾s ( RR* \ { -oo } ) ) e. Mnd
3		xrge0cmn 17854	. . 3 |- ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd
4		cmnmnd 16291	. . 3 |- ( ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd -> ( RR*s ↾s ( 0 [,] +oo ) ) e. Mnd )
5	3, 4	ax-mp 5	. 2 |- ( RR*s ↾s ( 0 [,] +oo ) ) e. Mnd
6		mnflt0 11104	. . . . . . 7 |- -oo < 0
7		mnfxr 11093	. . . . . . . 8 |- -oo e. RR*
8		0xr 9429	. . . . . . . 8 |- 0 e. RR*
9		xrltnle 9442	. . . . . . . 8 |- ( ( -oo e. RR* /\ 0 e. RR* ) -> ( -oo < 0 <-> -. 0 <_ -oo ) )
10	7, 8, 9	mp2an 672	. . . . . . 7 |- ( -oo < 0 <-> -. 0 <_ -oo )
11	6, 10	mpbi 208	. . . . . 6 |- -. 0 <_ -oo
12	11	intnan 905	. . . . 5 |- -. ( -oo e. RR* /\ 0 <_ -oo )
13		elxrge0 11393	. . . . 5 |- ( -oo e. ( 0 [,] +oo ) <-> ( -oo e. RR* /\ 0 <_ -oo ) )
14	12, 13	mtbir 299	. . . 4 |- -. -oo e. ( 0 [,] +oo )
15		difsn 4007	. . . 4 |- ( -. -oo e. ( 0 [,] +oo ) -> ( ( 0 [,] +oo ) \ { -oo } ) = ( 0 [,] +oo ) )
16	14, 15	ax-mp 5	. . 3 |- ( ( 0 [,] +oo ) \ { -oo } ) = ( 0 [,] +oo )
17		iccssxr 11377	. . . 4 |- ( 0 [,] +oo ) C_ RR*
18		ssdif 3490	. . . 4 |- ( ( 0 [,] +oo ) C_ RR* -> ( ( 0 [,] +oo ) \ { -oo } ) C_ ( RR* \ { -oo } ) )
19	17, 18	ax-mp 5	. . 3 |- ( ( 0 [,] +oo ) \ { -oo } ) C_ ( RR* \ { -oo } )
20	16, 19	eqsstr3i 3386	. 2 |- ( 0 [,] +oo ) C_ ( RR* \ { -oo } )
21		0e0iccpnf 11395	. 2 |- 0 e. ( 0 [,] +oo )
22		difss 3482	. . . . 5 |- ( RR* \ { -oo } ) C_ RR*
23		df-ss 3341	. . . . 5 |- ( ( RR* \ { -oo } ) C_ RR* <-> ( ( RR* \ { -oo } ) i^i RR* ) = ( RR* \ { -oo } ) )
24	22, 23	mpbi 208	. . . 4 |- ( ( RR* \ { -oo } ) i^i RR* ) = ( RR* \ { -oo } )
25		xrex 10987	. . . . . 6 |- RR* e. _V
26		difexg 4439	. . . . . 6 |- ( RR* e. _V -> ( RR* \ { -oo } ) e. _V )
27	25, 26	ax-mp 5	. . . . 5 |- ( RR* \ { -oo } ) e. _V
28		xrsbas 17831	. . . . . 6 |- RR* = ( Base ` RR*s )
29	1, 28	ressbas 14227	. . . . 5 |- ( ( RR* \ { -oo } ) e. _V -> ( ( RR* \ { -oo } ) i^i RR* ) = ( Base ` ( RR*s ↾s ( RR* \ { -oo } ) ) ) )
30	27, 29	ax-mp 5	. . . 4 |- ( ( RR* \ { -oo } ) i^i RR* ) = ( Base ` ( RR*s ↾s ( RR* \ { -oo } ) ) )
31	24, 30	eqtr3i 2464	. . 3 |- ( RR* \ { -oo } ) = ( Base ` ( RR*s ↾s ( RR* \ { -oo } ) ) )
32	1	xrs10 17851	. . 3 |- 0 = ( 0g ` ( RR*s ↾s ( RR* \ { -oo } ) ) )
33		ovex 6115	. . . . 5 |- ( 0 [,] +oo ) e. _V
34		ressress 14234	. . . . 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 ) ) ) )
35	27, 33, 34	mp2an 672	. . . 4 |- ( ( RR*s ↾s ( RR* \ { -oo } ) ) ↾s ( 0 [,] +oo ) ) = ( RR*s ↾s ( ( RR* \ { -oo } ) i^i ( 0 [,] +oo ) ) )
36		dfss 3342	. . . . . . 7 |- ( ( 0 [,] +oo ) C_ ( RR* \ { -oo } ) <-> ( 0 [,] +oo ) = ( ( 0 [,] +oo ) i^i ( RR* \ { -oo } ) ) )
37	20, 36	mpbi 208	. . . . . 6 |- ( 0 [,] +oo ) = ( ( 0 [,] +oo ) i^i ( RR* \ { -oo } ) )
38		incom 3542	. . . . . 6 |- ( ( 0 [,] +oo ) i^i ( RR* \ { -oo } ) ) = ( ( RR* \ { -oo } ) i^i ( 0 [,] +oo ) )
39	37, 38	eqtr2i 2463	. . . . 5 |- ( ( RR* \ { -oo } ) i^i ( 0 [,] +oo ) ) = ( 0 [,] +oo )
40	39	oveq2i 6101	. . . 4 |- ( RR*s ↾s ( ( RR* \ { -oo } ) i^i ( 0 [,] +oo ) ) ) = ( RR*s ↾s ( 0 [,] +oo ) )
41	35, 40	eqtr2i 2463	. . 3 |- ( RR*s ↾s ( 0 [,] +oo ) ) = ( ( RR*s ↾s ( RR* \ { -oo } ) ) ↾s ( 0 [,] +oo ) )
42	31, 32, 41	submnd0 15450	. 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 ) ) ) )
43	2, 5, 20, 21, 42	mp4an 673	1 |- 0 = ( 0g ` ( RR*s ↾s ( 0 [,] +oo ) ) )

Syntax hints:   -. wn 3   <-> wb 184   /\ wa 369   = wceq 1369   e. wcel 1756   _Vcvv 2971   \ cdif 3324   i^i cin 3326   C_ wss 3327   {csn 3876   class class class wbr 4291   ` cfv 5417  (class class class)co 6090   0cc0 9281   +oocpnf 9414   -oocmnf 9415   RR*cxr 9416   < clt 9417   <_ cle 9418   [,]cicc 11302   Basecbs 14173   ↾_s cress 14174   0gc0g 14377   RR*scxrs 14437   Mndcmnd 15408  CMndccmn 16276

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-nn 10322  df-2 10379  df-3 10380  df-4 10381  df-5 10382  df-6 10383  df-7 10384  df-8 10385  df-9 10386  df-10 10387  df-n0 10579  df-z 10646  df-dec 10755  df-uz 10861  df-xadd 11089  df-icc 11306  df-fz 11437  df-struct 14175  df-ndx 14176  df-slot 14177  df-base 14178  df-sets 14179  df-ress 14180  df-plusg 14250  df-mulr 14251  df-tset 14256  df-ple 14257  df-ds 14259  df-0g 14379  df-xrs 14439  df-mnd 15414  df-submnd 15464  df-cmn 16278

This theorem is referenced by:  xrge0mulgnn0  26149  xrge0slmod  26311  xrge0iifmhm  26368  esumnul  26501  esum0  26502  gsumesum  26509  esumlub  26510  esumsn  26514  esumss  26520  esumpfinval  26523  esumpfinvalf  26524  esumcocn  26528