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*s ↾s ( 0 [,] +oo ) ) )

Proof of Theorem xrge00

Step	Hyp	Ref	Expression
1		eqid 2462	. . 3 |- ( RR*s ↾s ( RR* \ { -oo } ) ) = ( RR*s ↾s ( RR* \ { -oo } ) )
2	1	xrs1mnd 19055	. 2 |- ( RR*s ↾s ( RR* \ { -oo } ) ) e. Mnd
3		xrge0cmn 19059	. . 3 |- ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd
4		cmnmnd 17494	. . 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 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 ) )
10	7, 8, 9	mp2an 683	. . . . . . 7 |- ( -oo < 0 <-> -. 0 <_ -oo )
11	6, 10	mpbi 213	. . . . . 6 |- -. 0 <_ -oo
12	11	intnan 930	. . . . 5 |- -. ( -oo e. RR* /\ 0 <_ -oo )
13		elxrge0 11770	. . . . 5 |- ( -oo e. ( 0 [,] +oo ) <-> ( -oo e. RR* /\ 0 <_ -oo ) )
14	12, 13	mtbir 305	. . . 4 |- -. -oo e. ( 0 [,] +oo )
15		difsn 4119	. . . 4 |- ( -. -oo e. ( 0 [,] +oo ) -> ( ( 0 [,] +oo ) \ { -oo } ) = ( 0 [,] +oo ) )
16	14, 15	ax-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 } ) )
19	17, 18	ax-mp 5	. . 3 |- ( ( 0 [,] +oo ) \ { -oo } ) C_ ( RR* \ { -oo } )
20	16, 19	eqsstr3i 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 } ) )
24	22, 23	mpbi 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 )
27	25, 26	ax-mp 5	. . . . 5 |- ( RR* \ { -oo } ) e. _V
28		xrsbas 19033	. . . . . 6 |- RR* = ( Base ` RR*s )
29	1, 28	ressbas 15228	. . . . 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 2486	. . 3 |- ( RR* \ { -oo } ) = ( Base ` ( RR*s ↾s ( RR* \ { -oo } ) ) )
32	1	xrs10 19056	. . 3 |- 0 = ( 0g ` ( RR*s ↾s ( RR* \ { -oo } ) ) )
33		ovex 6343	. . . . 5 |- ( 0 [,] +oo ) e. _V
34		ressress 15236	. . . . 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 683	. . . 4 |- ( ( RR*s ↾s ( RR* \ { -oo } ) ) ↾s ( 0 [,] +oo ) ) = ( RR*s ↾s ( ( RR* \ { -oo } ) i^i ( 0 [,] +oo ) ) )
36		dfss 3431	. . . . . . 7 |- ( ( 0 [,] +oo ) C_ ( RR* \ { -oo } ) <-> ( 0 [,] +oo ) = ( ( 0 [,] +oo ) i^i ( RR* \ { -oo } ) ) )
37	20, 36	mpbi 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 ) )
39	37, 38	eqtr2i 2485	. . . . 5 |- ( ( RR* \ { -oo } ) i^i ( 0 [,] +oo ) ) = ( 0 [,] +oo )
40	39	oveq2i 6326	. . . 4 |- ( RR*s ↾s ( ( RR* \ { -oo } ) i^i ( 0 [,] +oo ) ) ) = ( RR*s ↾s ( 0 [,] +oo ) )
41	35, 40	eqtr2i 2485	. . 3 |- ( RR*s ↾s ( 0 [,] +oo ) ) = ( ( RR*s ↾s ( RR* \ { -oo } ) ) ↾s ( 0 [,] +oo ) )
42	31, 32, 41	submnd0 16615	. 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 684	1 |- 0 = ( 0g ` ( RR*s ↾s ( 0 [,] +oo ) ) )

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