Theorem xrge0gsumle 21206

Description: A finite sum in the nonnegative extended reals is monotonic in the support. (Contributed by Mario Carneiro, 13-Sep-2015.)

Hypotheses

Ref	Expression
xrge0gsumle.g	|- G = ( RR*s ↾s ( 0 [,] +oo ) )
xrge0gsumle.a	|- ( ph -> A e. V )
xrge0gsumle.f	|- ( ph -> F : A --> ( 0 [,] +oo ) )
xrge0gsumle.b	|- ( ph -> B e. ( ~P A i^i Fin ) )
xrge0gsumle.c	|- ( ph -> C C_ B )

Assertion

Ref	Expression
xrge0gsumle	|- ( ph -> ( G gsumg ( F |` C ) ) <_ ( G gsumg ( F |` B ) ) )

Proof of Theorem xrge0gsumle

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iccssxr 11619	. . . . . . 7 |- ( 0 [,] +oo ) C_ RR*
2		xrge0gsumle.g	. . . . . . . . . 10 |- G = ( RR*s ↾s ( 0 [,] +oo ) )
3		xrsbas 18304	. . . . . . . . . 10 |- RR* = ( Base ` RR*s )
4	2, 3	ressbas2 14563	. . . . . . . . 9 |- ( ( 0 [,] +oo ) C_ RR* -> ( 0 [,] +oo ) = ( Base ` G ) )
5	1, 4	ax-mp 5	. . . . . . . 8 |- ( 0 [,] +oo ) = ( Base ` G )
6		eqid 2467	. . . . . . . . . 10 |- ( RR*s ↾s ( RR* \ { -oo } ) ) = ( RR*s ↾s ( RR* \ { -oo } ) )
7	6	xrge0subm 18329	. . . . . . . . 9 |- ( 0 [,] +oo ) e. (SubMnd ` ( RR*s ↾s ( RR* \ { -oo } ) ) )
8		xrex 11229	. . . . . . . . . . . . 13 |- RR* e. _V
9		difexg 4601	. . . . . . . . . . . . 13 |- ( RR* e. _V -> ( RR* \ { -oo } ) e. _V )
10	8, 9	ax-mp 5	. . . . . . . . . . . 12 |- ( RR* \ { -oo } ) e. _V
11		simpl 457	. . . . . . . . . . . . . . 15 |- ( ( x e. RR* /\ 0 <_ x ) -> x e. RR* )
12		ge0nemnf 11386	. . . . . . . . . . . . . . 15 |- ( ( x e. RR* /\ 0 <_ x ) -> x =/= -oo )
13	11, 12	jca 532	. . . . . . . . . . . . . 14 |- ( ( x e. RR* /\ 0 <_ x ) -> ( x e. RR* /\ x =/= -oo ) )
14		elxrge0 11641	. . . . . . . . . . . . . 14 |- ( x e. ( 0 [,] +oo ) <-> ( x e. RR* /\ 0 <_ x ) )
15		eldifsn 4158	. . . . . . . . . . . . . 14 |- ( x e. ( RR* \ { -oo } ) <-> ( x e. RR* /\ x =/= -oo ) )
16	13, 14, 15	3imtr4i 266	. . . . . . . . . . . . 13 |- ( x e. ( 0 [,] +oo ) -> x e. ( RR* \ { -oo } ) )
17	16	ssriv 3513	. . . . . . . . . . . 12 |- ( 0 [,] +oo ) C_ ( RR* \ { -oo } )
18		ressabs 14570	. . . . . . . . . . . 12 |- ( ( ( RR* \ { -oo } ) e. _V /\ ( 0 [,] +oo ) C_ ( RR* \ { -oo } ) ) -> ( ( RR*s ↾s ( RR* \ { -oo } ) ) ↾s ( 0 [,] +oo ) ) = ( RR*s ↾s ( 0 [,] +oo ) ) )
19	10, 17, 18	mp2an 672	. . . . . . . . . . 11 |- ( ( RR*s ↾s ( RR* \ { -oo } ) ) ↾s ( 0 [,] +oo ) ) = ( RR*s ↾s ( 0 [,] +oo ) )
20	2, 19	eqtr4i 2499	. . . . . . . . . 10 |- G = ( ( RR*s ↾s ( RR* \ { -oo } ) ) ↾s ( 0 [,] +oo ) )
21	6	xrs10 18327	. . . . . . . . . 10 |- 0 = ( 0g ` ( RR*s ↾s ( RR* \ { -oo } ) ) )
22	20, 21	subm0 15859	. . . . . . . . 9 |- ( ( 0 [,] +oo ) e. (SubMnd ` ( RR*s ↾s ( RR* \ { -oo } ) ) ) -> 0 = ( 0g ` G ) )
23	7, 22	ax-mp 5	. . . . . . . 8 |- 0 = ( 0g ` G )
24		xrge0cmn 18330	. . . . . . . . . 10 |- ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd
25	2, 24	eqeltri 2551	. . . . . . . . 9 |- G e. CMnd
26	25	a1i 11	. . . . . . . 8 |- ( ( ph /\ s e. ( ~P A i^i Fin ) ) -> G e. CMnd )
27		elfpw 7834	. . . . . . . . . 10 |- ( s e. ( ~P A i^i Fin ) <-> ( s C_ A /\ s e. Fin ) )
28	27	simprbi 464	. . . . . . . . 9 |- ( s e. ( ~P A i^i Fin ) -> s e. Fin )
29	28	adantl 466	. . . . . . . 8 |- ( ( ph /\ s e. ( ~P A i^i Fin ) ) -> s e. Fin )
30		xrge0gsumle.f	. . . . . . . . 9 |- ( ph -> F : A --> ( 0 [,] +oo ) )
31	27	simplbi 460	. . . . . . . . 9 |- ( s e. ( ~P A i^i Fin ) -> s C_ A )
32		fssres 5757	. . . . . . . . 9 |- ( ( F : A --> ( 0 [,] +oo ) /\ s C_ A ) -> ( F |` s ) : s --> ( 0 [,] +oo ) )
33	30, 31, 32	syl2an 477	. . . . . . . 8 |- ( ( ph /\ s e. ( ~P A i^i Fin ) ) -> ( F |` s ) : s --> ( 0 [,] +oo ) )
34		c0ex 9602	. . . . . . . . . 10 |- 0 e. _V
35	34	a1i 11	. . . . . . . . 9 |- ( ( ph /\ s e. ( ~P A i^i Fin ) ) -> 0 e. _V )
36	33, 29, 35	fdmfifsupp 7851	. . . . . . . 8 |- ( ( ph /\ s e. ( ~P A i^i Fin ) ) -> ( F |` s ) finSupp 0 )
37	5, 23, 26, 29, 33, 36	gsumcl 16796	. . . . . . 7 |- ( ( ph /\ s e. ( ~P A i^i Fin ) ) -> ( G gsumg ( F |` s ) ) e. ( 0 [,] +oo ) )
38	1, 37	sseldi 3507	. . . . . 6 |- ( ( ph /\ s e. ( ~P A i^i Fin ) ) -> ( G gsumg ( F |` s ) ) e. RR* )
39		eqid 2467	. . . . . 6 |- ( s e. ( ~P A i^i Fin ) |-> ( G gsumg ( F |` s ) ) ) = ( s e. ( ~P A i^i Fin ) |-> ( G gsumg ( F |` s ) ) )
40	38, 39	fmptd 6056	. . . . 5 |- ( ph -> ( s e. ( ~P A i^i Fin ) |-> ( G gsumg ( F |` s ) ) ) : ( ~P A i^i Fin ) --> RR* )
41		frn 5743	. . . . 5 |- ( ( s e. ( ~P A i^i Fin ) |-> ( G gsumg ( F |` s ) ) ) : ( ~P A i^i Fin ) --> RR* -> ran ( s e. ( ~P A i^i Fin ) |-> ( G gsumg ( F |` s ) ) ) C_ RR* )
42	40, 41	syl 16	. . . 4 |- ( ph -> ran ( s e. ( ~P A i^i Fin ) |-> ( G gsumg ( F |` s ) ) ) C_ RR* )
43		0ss 3819	. . . . . . 7 |- (/) C_ A
44		0fin 7759	. . . . . . 7 |- (/) e. Fin
45		elfpw 7834	. . . . . . 7 |- ( (/) e. ( ~P A i^i Fin ) <-> ( (/) C_ A /\ (/) e. Fin ) )
46	43, 44, 45	mpbir2an 918	. . . . . 6 |- (/) e. ( ~P A i^i Fin )
47		0cn 9600	. . . . . 6 |- 0 e. CC
48		reseq2 5274	. . . . . . . . . 10 |- ( s = (/) -> ( F |` s ) = ( F |` (/) ) )
49		res0 5284	. . . . . . . . . 10 |- ( F |` (/) ) = (/)
50	48, 49	syl6eq 2524	. . . . . . . . 9 |- ( s = (/) -> ( F |` s ) = (/) )
51	50	oveq2d 6311	. . . . . . . 8 |- ( s = (/) -> ( G gsumg ( F |` s ) ) = ( G gsumg (/) ) )
52	23	gsum0 15779	. . . . . . . 8 |- ( G gsumg (/) ) = 0
53	51, 52	syl6eq 2524	. . . . . . 7 |- ( s = (/) -> ( G gsumg ( F |` s ) ) = 0 )
54	39, 53	elrnmpt1s 5256	. . . . . 6 |- ( ( (/) e. ( ~P A i^i Fin ) /\ 0 e. CC ) -> 0 e. ran ( s e. ( ~P A i^i Fin ) |-> ( G gsumg ( F |` s ) ) ) )
55	46, 47, 54	mp2an 672	. . . . 5 |- 0 e. ran ( s e. ( ~P A i^i Fin ) |-> ( G gsumg ( F |` s ) ) )
56	55	a1i 11	. . . 4 |- ( ph -> 0 e. ran ( s e. ( ~P A i^i Fin ) |-> ( G gsumg ( F |` s ) ) ) )
57	42, 56	sseldd 3510	. . 3 |- ( ph -> 0 e. RR* )
58	25	a1i 11	. . . . 5 |- ( ph -> G e. CMnd )
59		xrge0gsumle.b	. . . . . . 7 |- ( ph -> B e. ( ~P A i^i Fin ) )
60		elfpw 7834	. . . . . . . 8 |- ( B e. ( ~P A i^i Fin ) <-> ( B C_ A /\ B e. Fin ) )
61	60	simprbi 464	. . . . . . 7 |- ( B e. ( ~P A i^i Fin ) -> B e. Fin )
62	59, 61	syl 16	. . . . . 6 |- ( ph -> B e. Fin )
63		diffi 7763	. . . . . 6 |- ( B e. Fin -> ( B \ C ) e. Fin )
64	62, 63	syl 16	. . . . 5 |- ( ph -> ( B \ C ) e. Fin )
65	60	simplbi 460	. . . . . . . 8 |- ( B e. ( ~P A i^i Fin ) -> B C_ A )
66	59, 65	syl 16	. . . . . . 7 |- ( ph -> B C_ A )
67	66	ssdifssd 3647	. . . . . 6 |- ( ph -> ( B \ C ) C_ A )
68		fssres 5757	. . . . . 6 |- ( ( F : A --> ( 0 [,] +oo ) /\ ( B \ C ) C_ A ) -> ( F |` ( B \ C ) ) : ( B \ C ) --> ( 0 [,] +oo ) )
69	30, 67, 68	syl2anc 661	. . . . 5 |- ( ph -> ( F |` ( B \ C ) ) : ( B \ C ) --> ( 0 [,] +oo ) )
70	34	a1i 11	. . . . . 6 |- ( ph -> 0 e. _V )
71	69, 64, 70	fdmfifsupp 7851	. . . . 5 |- ( ph -> ( F |` ( B \ C ) ) finSupp 0 )
72	5, 23, 58, 64, 69, 71	gsumcl 16796	. . . 4 |- ( ph -> ( G gsumg ( F |` ( B \ C ) ) ) e. ( 0 [,] +oo ) )
73	1, 72	sseldi 3507	. . 3 |- ( ph -> ( G gsumg ( F |` ( B \ C ) ) ) e. RR* )
74		xrge0gsumle.c	. . . . . 6 |- ( ph -> C C_ B )
75		ssfi 7752	. . . . . 6 |- ( ( B e. Fin /\ C C_ B ) -> C e. Fin )
76	62, 74, 75	syl2anc 661	. . . . 5 |- ( ph -> C e. Fin )
77	74, 66	sstrd 3519	. . . . . 6 |- ( ph -> C C_ A )
78		fssres 5757	. . . . . 6 |- ( ( F : A --> ( 0 [,] +oo ) /\ C C_ A ) -> ( F |` C ) : C --> ( 0 [,] +oo ) )
79	30, 77, 78	syl2anc 661	. . . . 5 |- ( ph -> ( F |` C ) : C --> ( 0 [,] +oo ) )
80	79, 76, 70	fdmfifsupp 7851	. . . . 5 |- ( ph -> ( F |` C ) finSupp 0 )
81	5, 23, 58, 76, 79, 80	gsumcl 16796	. . . 4 |- ( ph -> ( G gsumg ( F |` C ) ) e. ( 0 [,] +oo ) )
82	1, 81	sseldi 3507	. . 3 |- ( ph -> ( G gsumg ( F |` C ) ) e. RR* )
83		elxrge0 11641	. . . . 5 |- ( ( G gsumg ( F |` ( B \ C ) ) ) e. ( 0 [,] +oo ) <-> ( ( G gsumg ( F |` ( B \ C ) ) ) e. RR* /\ 0 <_ ( G gsumg ( F |` ( B \ C ) ) ) ) )
84	83	simprbi 464	. . . 4 |- ( ( G gsumg ( F |` ( B \ C ) ) ) e. ( 0 [,] +oo ) -> 0 <_ ( G gsumg ( F |` ( B \ C ) ) ) )
85	72, 84	syl 16	. . 3 |- ( ph -> 0 <_ ( G gsumg ( F |` ( B \ C ) ) ) )
86		xleadd2a 11458	. . 3 |- ( ( ( 0 e. RR* /\ ( G gsumg ( F |` ( B \ C ) ) ) e. RR* /\ ( G gsumg ( F |` C ) ) e. RR* ) /\ 0 <_ ( G gsumg ( F |` ( B \ C ) ) ) ) -> ( ( G gsumg ( F |` C ) ) +e 0 ) <_ ( ( G gsumg ( F |` C ) ) +e ( G gsumg ( F |` ( B \ C ) ) ) ) )
87	57, 73, 82, 85, 86	syl31anc 1231	. 2 |- ( ph -> ( ( G gsumg ( F |` C ) ) +e 0 ) <_ ( ( G gsumg ( F |` C ) ) +e ( G gsumg ( F |` ( B \ C ) ) ) ) )
88		xaddid1 11450	. . 3 |- ( ( G gsumg ( F |` C ) ) e. RR* -> ( ( G gsumg ( F |` C ) ) +e 0 ) = ( G gsumg ( F |` C ) ) )
89	82, 88	syl 16	. 2 |- ( ph -> ( ( G gsumg ( F |` C ) ) +e 0 ) = ( G gsumg ( F |` C ) ) )
90		ovex 6320	. . . . 5 |- ( 0 [,] +oo ) e. _V
91		xrsadd 18305	. . . . . 6 |- +e = ( +g ` RR*s )
92	2, 91	ressplusg 14614	. . . . 5 |- ( ( 0 [,] +oo ) e. _V -> +e = ( +g ` G ) )
93	90, 92	ax-mp 5	. . . 4 |- +e = ( +g ` G )
94		fssres 5757	. . . . 5 |- ( ( F : A --> ( 0 [,] +oo ) /\ B C_ A ) -> ( F |` B ) : B --> ( 0 [,] +oo ) )
95	30, 66, 94	syl2anc 661	. . . 4 |- ( ph -> ( F |` B ) : B --> ( 0 [,] +oo ) )
96	95, 62, 70	fdmfifsupp 7851	. . . 4 |- ( ph -> ( F |` B ) finSupp 0 )
97		disjdif 3905	. . . . 5 |- ( C i^i ( B \ C ) ) = (/)
98	97	a1i 11	. . . 4 |- ( ph -> ( C i^i ( B \ C ) ) = (/) )
99		undif2 3909	. . . . 5 |- ( C u. ( B \ C ) ) = ( C u. B )
100		ssequn1 3679	. . . . . 6 |- ( C C_ B <-> ( C u. B ) = B )
101	74, 100	sylib 196	. . . . 5 |- ( ph -> ( C u. B ) = B )
102	99, 101	syl5req 2521	. . . 4 |- ( ph -> B = ( C u. ( B \ C ) ) )
103	5, 23, 93, 58, 59, 95, 96, 98, 102	gsumsplit 16819	. . 3 |- ( ph -> ( G gsumg ( F |` B ) ) = ( ( G gsumg ( ( F |` B ) |` C ) ) +e ( G gsumg ( ( F |` B ) |` ( B \ C ) ) ) ) )
104		resabs1 5308	. . . . . 6 |- ( C C_ B -> ( ( F |` B ) |` C ) = ( F |` C ) )
105	74, 104	syl 16	. . . . 5 |- ( ph -> ( ( F |` B ) |` C ) = ( F |` C ) )
106	105	oveq2d 6311	. . . 4 |- ( ph -> ( G gsumg ( ( F |` B ) |` C ) ) = ( G gsumg ( F |` C ) ) )
107		difss 3636	. . . . . 6 |- ( B \ C ) C_ B
108		resabs1 5308	. . . . . 6 |- ( ( B \ C ) C_ B -> ( ( F |` B ) |` ( B \ C ) ) = ( F |` ( B \ C ) ) )
109	107, 108	mp1i 12	. . . . 5 |- ( ph -> ( ( F |` B ) |` ( B \ C ) ) = ( F |` ( B \ C ) ) )
110	109	oveq2d 6311	. . . 4 |- ( ph -> ( G gsumg ( ( F |` B ) |` ( B \ C ) ) ) = ( G gsumg ( F |` ( B \ C ) ) ) )
111	106, 110	oveq12d 6313	. . 3 |- ( ph -> ( ( G gsumg ( ( F |` B ) |` C ) ) +e ( G gsumg ( ( F |` B ) |` ( B \ C ) ) ) ) = ( ( G gsumg ( F |` C ) ) +e ( G gsumg ( F |` ( B \ C ) ) ) ) )
112	103, 111	eqtr2d 2509	. 2 |- ( ph -> ( ( G gsumg ( F |` C ) ) +e ( G gsumg ( F |` ( B \ C ) ) ) ) = ( G gsumg ( F |` B ) ) )
113	87, 89, 112	3brtr3d 4482	1 |- ( ph -> ( G gsumg ( F |` C ) ) <_ ( G gsumg ( F |` B ) ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1379   e. wcel 1767   =/= wne 2662   _Vcvv 3118   \ cdif 3478   u. cun 3479   i^i cin 3480   C_ wss 3481   (/)c0 3790   ~Pcpw 4016   {csn 4033   class class class wbr 4453   |-> cmpt 4511   ran crn 5006   |` cres 5007   -->wf 5590   ` cfv 5594  (class class class)co 6295   Fincfn 7528   CCcc 9502   0cc0 9504   +oocpnf 9637   -oocmnf 9638   RR*cxr 9639   <_ cle 9641   +ecxad 11328   [,]cicc 11544   Basecbs 14507   ↾_s cress 14508   +g cplusg 14572   0gc0g 14712   gsum_g cgsu 14713   RR*scxrs 14772  SubMndcsubmnd 15838  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-rep 4564  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-iin 4334  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-se 4845  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-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  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-fsupp 7842  df-oi 7947  df-card 8332  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-fzo 11805  df-seq 12088  df-hash 12386  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-gsum 14715  df-xrs 14774  df-mre 14858  df-mrc 14859  df-acs 14861  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-cntz 16227  df-cmn 16673

This theorem is referenced by:  xrge0tsms  21207  xrge0tsmsd  27600