Theorem xrge00 27827

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 2382	. . 3 |- ( RR*s ↾s ( RR* \ { -oo } ) ) = ( RR*s ↾s ( RR* \ { -oo } ) )
2	1	xrs1mnd 18569	. 2 |- ( RR*s ↾s ( RR* \ { -oo } ) ) e. Mnd
3		xrge0cmn 18573	. . 3 |- ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd
4		cmnmnd 16930	. . 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 11255	. . . . . . 7 |- -oo < 0
7		mnfxr 11244	. . . . . . . 8 |- -oo e. RR*
8		0xr 9551	. . . . . . . 8 |- 0 e. RR*
9		xrltnle 9564	. . . . . . . 8 |- ( ( -oo e. RR* /\ 0 e. RR* ) -> ( -oo < 0 <-> -. 0 <_ -oo ) )
10	7, 8, 9	mp2an 670	. . . . . . 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 11550	. . . . 5 |- ( -oo e. ( 0 [,] +oo ) <-> ( -oo e. RR* /\ 0 <_ -oo ) )
14	12, 13	mtbir 297	. . . 4 |- -. -oo e. ( 0 [,] +oo )
15		difsn 4078	. . . 4 |- ( -. -oo e. ( 0 [,] +oo ) -> ( ( 0 [,] +oo ) \ { -oo } ) = ( 0 [,] +oo ) )
16	14, 15	ax-mp 5	. . 3 |- ( ( 0 [,] +oo ) \ { -oo } ) = ( 0 [,] +oo )
17		iccssxr 11528	. . . 4 |- ( 0 [,] +oo ) C_ RR*
18		ssdif 3553	. . . 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 3448	. 2 |- ( 0 [,] +oo ) C_ ( RR* \ { -oo } )
21		0e0iccpnf 11552	. 2 |- 0 e. ( 0 [,] +oo )
22		difss 3545	. . . . 5 |- ( RR* \ { -oo } ) C_ RR*
23		df-ss 3403	. . . . 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 11136	. . . . . 6 |- RR* e. _V
26		difexg 4513	. . . . . 6 |- ( RR* e. _V -> ( RR* \ { -oo } ) e. _V )
27	25, 26	ax-mp 5	. . . . 5 |- ( RR* \ { -oo } ) e. _V
28		xrsbas 18547	. . . . . 6 |- RR* = ( Base ` RR*s )
29	1, 28	ressbas 14691	. . . . 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 2413	. . 3 |- ( RR* \ { -oo } ) = ( Base ` ( RR*s ↾s ( RR* \ { -oo } ) ) )
32	1	xrs10 18570	. . 3 |- 0 = ( 0g ` ( RR*s ↾s ( RR* \ { -oo } ) ) )
33		ovex 6224	. . . . 5 |- ( 0 [,] +oo ) e. _V
34		ressress 14699	. . . . 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 670	. . . 4 |- ( ( RR*s ↾s ( RR* \ { -oo } ) ) ↾s ( 0 [,] +oo ) ) = ( RR*s ↾s ( ( RR* \ { -oo } ) i^i ( 0 [,] +oo ) ) )
36		dfss 3404	. . . . . . 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 3605	. . . . . 6 |- ( ( 0 [,] +oo ) i^i ( RR* \ { -oo } ) ) = ( ( RR* \ { -oo } ) i^i ( 0 [,] +oo ) )
39	37, 38	eqtr2i 2412	. . . . 5 |- ( ( RR* \ { -oo } ) i^i ( 0 [,] +oo ) ) = ( 0 [,] +oo )
40	39	oveq2i 6207	. . . 4 |- ( RR*s ↾s ( ( RR* \ { -oo } ) i^i ( 0 [,] +oo ) ) ) = ( RR*s ↾s ( 0 [,] +oo ) )
41	35, 40	eqtr2i 2412	. . 3 |- ( RR*s ↾s ( 0 [,] +oo ) ) = ( ( RR*s ↾s ( RR* \ { -oo } ) ) ↾s ( 0 [,] +oo ) )
42	31, 32, 41	submnd0 16067	. 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 671	1 |- 0 = ( 0g ` ( RR*s ↾s ( 0 [,] +oo ) ) )

Syntax hints:   -. wn 3   <-> wb 184   /\ wa 367   = wceq 1399   e. wcel 1826   _Vcvv 3034   \ cdif 3386   i^i cin 3388   C_ wss 3389   {csn 3944   class class class wbr 4367   ` cfv 5496  (class class class)co 6196   0cc0 9403   +oocpnf 9536   -oocmnf 9537   RR*cxr 9538   < clt 9539   <_ cle 9540   [,]cicc 11453   Basecbs 14634   ↾_s cress 14635   0gc0g 14847   RR*scxrs 14907   Mndcmnd 16036  CMndccmn 16915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-7 10516  df-8 10517  df-9 10518  df-10 10519  df-n0 10713  df-z 10782  df-dec 10896  df-uz 11002  df-xadd 11240  df-icc 11457  df-fz 11594  df-struct 14636  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-mulr 14716  df-tset 14721  df-ple 14722  df-ds 14724  df-0g 14849  df-xrs 14909  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-submnd 16084  df-cmn 16917

This theorem is referenced by:  xrge0mulgnn0  27830  xrge0slmod  27988  xrge0iifmhm  28075  esumgsum  28193  esumnul  28196  esum0  28197  gsumesum  28207  esumsnf  28212  esumss  28220  esumpfinval  28223  esumpfinvalf  28224  esumcocn  28228