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*s ↾s ( 0 [,] +oo ) ) )

Proof of Theorem xrge00

Step	Hyp	Ref	Expression
1		eqid 2467	. . 3 |- ( RR*s ↾s ( RR* \ { -oo } ) ) = ( RR*s ↾s ( RR* \ { -oo } ) )
2	1	xrs1mnd 18326	. 2 |- ( RR*s ↾s ( RR* \ { -oo } ) ) e. Mnd
3		xrge0cmn 18330	. . 3 |- ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd
4		cmnmnd 16686	. . 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 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 ) )
10	7, 8, 9	mp2an 672	. . . . . . 7 |- ( -oo < 0 <-> -. 0 <_ -oo )
11	6, 10	mpbi 208	. . . . . 6 |- -. 0 <_ -oo
12	11	intnan 912	. . . . 5 |- -. ( -oo e. RR* /\ 0 <_ -oo )
13		elxrge0 11641	. . . . 5 |- ( -oo e. ( 0 [,] +oo ) <-> ( -oo e. RR* /\ 0 <_ -oo ) )
14	12, 13	mtbir 299	. . . 4 |- -. -oo e. ( 0 [,] +oo )
15		difsn 4167	. . . 4 |- ( -. -oo e. ( 0 [,] +oo ) -> ( ( 0 [,] +oo ) \ { -oo } ) = ( 0 [,] +oo ) )
16	14, 15	ax-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 } ) )
19	17, 18	ax-mp 5	. . 3 |- ( ( 0 [,] +oo ) \ { -oo } ) C_ ( RR* \ { -oo } )
20	16, 19	eqsstr3i 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 } ) )
24	22, 23	mpbi 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 )
27	25, 26	ax-mp 5	. . . . 5 |- ( RR* \ { -oo } ) e. _V
28		xrsbas 18304	. . . . . 6 |- RR* = ( Base ` RR*s )
29	1, 28	ressbas 14562	. . . . 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 2498	. . 3 |- ( RR* \ { -oo } ) = ( Base ` ( RR*s ↾s ( RR* \ { -oo } ) ) )
32	1	xrs10 18327	. . 3 |- 0 = ( 0g ` ( RR*s ↾s ( RR* \ { -oo } ) ) )
33		ovex 6320	. . . . 5 |- ( 0 [,] +oo ) e. _V
34		ressress 14569	. . . . 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 3496	. . . . . . 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 3696	. . . . . 6 |- ( ( 0 [,] +oo ) i^i ( RR* \ { -oo } ) ) = ( ( RR* \ { -oo } ) i^i ( 0 [,] +oo ) )
39	37, 38	eqtr2i 2497	. . . . 5 |- ( ( RR* \ { -oo } ) i^i ( 0 [,] +oo ) ) = ( 0 [,] +oo )
40	39	oveq2i 6306	. . . 4 |- ( RR*s ↾s ( ( RR* \ { -oo } ) i^i ( 0 [,] +oo ) ) ) = ( RR*s ↾s ( 0 [,] +oo ) )
41	35, 40	eqtr2i 2497	. . 3 |- ( RR*s ↾s ( 0 [,] +oo ) ) = ( ( RR*s ↾s ( RR* \ { -oo } ) ) ↾s ( 0 [,] +oo ) )
42	31, 32, 41	submnd0 15823	. 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 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