Theorem xrge00 27544

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 2441	. . 3 |- ( RR*s ↾s ( RR* \ { -oo } ) ) = ( RR*s ↾s ( RR* \ { -oo } ) )
2	1	xrs1mnd 18327	. 2 |- ( RR*s ↾s ( RR* \ { -oo } ) ) e. Mnd
3		xrge0cmn 18331	. . 3 |- ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd
4		cmnmnd 16684	. . 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 11340	. . . . . . 7 |- -oo < 0
7		mnfxr 11329	. . . . . . . 8 |- -oo e. RR*
8		0xr 9640	. . . . . . . 8 |- 0 e. RR*
9		xrltnle 9653	. . . . . . . 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 11635	. . . . 5 |- ( -oo e. ( 0 [,] +oo ) <-> ( -oo e. RR* /\ 0 <_ -oo ) )
14	12, 13	mtbir 299	. . . 4 |- -. -oo e. ( 0 [,] +oo )
15		difsn 4146	. . . 4 |- ( -. -oo e. ( 0 [,] +oo ) -> ( ( 0 [,] +oo ) \ { -oo } ) = ( 0 [,] +oo ) )
16	14, 15	ax-mp 5	. . 3 |- ( ( 0 [,] +oo ) \ { -oo } ) = ( 0 [,] +oo )
17		iccssxr 11613	. . . 4 |- ( 0 [,] +oo ) C_ RR*
18		ssdif 3622	. . . 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 3518	. 2 |- ( 0 [,] +oo ) C_ ( RR* \ { -oo } )
21		0e0iccpnf 11637	. 2 |- 0 e. ( 0 [,] +oo )
22		difss 3614	. . . . 5 |- ( RR* \ { -oo } ) C_ RR*
23		df-ss 3473	. . . . 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 11223	. . . . . 6 |- RR* e. _V
26		difexg 4582	. . . . . 6 |- ( RR* e. _V -> ( RR* \ { -oo } ) e. _V )
27	25, 26	ax-mp 5	. . . . 5 |- ( RR* \ { -oo } ) e. _V
28		xrsbas 18305	. . . . . 6 |- RR* = ( Base ` RR*s )
29	1, 28	ressbas 14561	. . . . 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 2472	. . 3 |- ( RR* \ { -oo } ) = ( Base ` ( RR*s ↾s ( RR* \ { -oo } ) ) )
32	1	xrs10 18328	. . 3 |- 0 = ( 0g ` ( RR*s ↾s ( RR* \ { -oo } ) ) )
33		ovex 6306	. . . . 5 |- ( 0 [,] +oo ) e. _V
34		ressress 14568	. . . . 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 3474	. . . . . . 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 3674	. . . . . 6 |- ( ( 0 [,] +oo ) i^i ( RR* \ { -oo } ) ) = ( ( RR* \ { -oo } ) i^i ( 0 [,] +oo ) )
39	37, 38	eqtr2i 2471	. . . . 5 |- ( ( RR* \ { -oo } ) i^i ( 0 [,] +oo ) ) = ( 0 [,] +oo )
40	39	oveq2i 6289	. . . 4 |- ( RR*s ↾s ( ( RR* \ { -oo } ) i^i ( 0 [,] +oo ) ) ) = ( RR*s ↾s ( 0 [,] +oo ) )
41	35, 40	eqtr2i 2471	. . 3 |- ( RR*s ↾s ( 0 [,] +oo ) ) = ( ( RR*s ↾s ( RR* \ { -oo } ) ) ↾s ( 0 [,] +oo ) )
42	31, 32, 41	submnd0 15821	. 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 1381   e. wcel 1802   _Vcvv 3093   \ cdif 3456   i^i cin 3458   C_ wss 3459   {csn 4011   class class class wbr 4434   ` cfv 5575  (class class class)co 6278   0cc0 9492   +oocpnf 9625   -oocmnf 9626   RR*cxr 9627   < clt 9628   <_ cle 9629   [,]cicc 11538   Basecbs 14506   ↾_s cress 14507   0gc0g 14711   RR*scxrs 14771   Mndcmnd 15790  CMndccmn 16669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-pss 3475  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-tp 4016  df-op 4018  df-uni 4232  df-int 4269  df-iun 4314  df-br 4435  df-opab 4493  df-mpt 4494  df-tr 4528  df-eprel 4778  df-id 4782  df-po 4787  df-so 4788  df-fr 4825  df-we 4827  df-ord 4868  df-on 4869  df-lim 4870  df-suc 4871  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-riota 6239  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6683  df-1st 6782  df-2nd 6783  df-recs 7041  df-rdg 7075  df-1o 7129  df-oadd 7133  df-er 7310  df-en 7516  df-dom 7517  df-sdom 7518  df-fin 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9809  df-neg 9810  df-nn 10540  df-2 10597  df-3 10598  df-4 10599  df-5 10600  df-6 10601  df-7 10602  df-8 10603  df-9 10604  df-10 10605  df-n0 10799  df-z 10868  df-dec 10982  df-uz 11088  df-xadd 11325  df-icc 11542  df-fz 11679  df-struct 14508  df-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-ress 14513  df-plusg 14584  df-mulr 14585  df-tset 14590  df-ple 14591  df-ds 14593  df-0g 14713  df-xrs 14773  df-mgm 15743  df-sgrp 15782  df-mnd 15792  df-submnd 15838  df-cmn 16671

This theorem is referenced by:  xrge0mulgnn0  27547  xrge0slmod  27704  xrge0iifmhm  27791  esumnul  27929  esum0  27930  gsumesum  27937  esumsn  27942  esumss  27948  esumpfinval  27951  esumpfinvalf  27952  esumcocn  27956