Theorem xrge0gsumle 22636

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 12256	. . . . . . 7 |- ( 0 [,] +oo ) C_ RR*
2		xrge0gsumle.g	. . . . . . . . . 10 |- G = ( RR*s ↾s ( 0 [,] +oo ) )
3		xrsbas 19762	. . . . . . . . . 10 |- RR* = ( Base ` RR*s )
4	2, 3	ressbas2 15931	. . . . . . . . 9 |- ( ( 0 [,] +oo ) C_ RR* -> ( 0 [,] +oo ) = ( Base ` G ) )
5	1, 4	ax-mp 5	. . . . . . . 8 |- ( 0 [,] +oo ) = ( Base ` G )
6		eqid 2622	. . . . . . . . . 10 |- ( RR*s ↾s ( RR* \ { -oo } ) ) = ( RR*s ↾s ( RR* \ { -oo } ) )
7	6	xrge0subm 19787	. . . . . . . . 9 |- ( 0 [,] +oo ) e. (SubMnd ` ( RR*s ↾s ( RR* \ { -oo } ) ) )
8		xrex 11829	. . . . . . . . . . . . 13 |- RR* e. _V
9		difexg 4808	. . . . . . . . . . . . 13 |- ( RR* e. _V -> ( RR* \ { -oo } ) e. _V )
10	8, 9	ax-mp 5	. . . . . . . . . . . 12 |- ( RR* \ { -oo } ) e. _V
11		simpl 473	. . . . . . . . . . . . . . 15 |- ( ( x e. RR* /\ 0 <_ x ) -> x e. RR* )
12		ge0nemnf 12004	. . . . . . . . . . . . . . 15 |- ( ( x e. RR* /\ 0 <_ x ) -> x =/= -oo )
13	11, 12	jca 554	. . . . . . . . . . . . . 14 |- ( ( x e. RR* /\ 0 <_ x ) -> ( x e. RR* /\ x =/= -oo ) )
14		elxrge0 12281	. . . . . . . . . . . . . 14 |- ( x e. ( 0 [,] +oo ) <-> ( x e. RR* /\ 0 <_ x ) )
15		eldifsn 4317	. . . . . . . . . . . . . 14 |- ( x e. ( RR* \ { -oo } ) <-> ( x e. RR* /\ x =/= -oo ) )
16	13, 14, 15	3imtr4i 281	. . . . . . . . . . . . 13 |- ( x e. ( 0 [,] +oo ) -> x e. ( RR* \ { -oo } ) )
17	16	ssriv 3607	. . . . . . . . . . . 12 |- ( 0 [,] +oo ) C_ ( RR* \ { -oo } )
18		ressabs 15939	. . . . . . . . . . . 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 708	. . . . . . . . . . 11 |- ( ( RR*s ↾s ( RR* \ { -oo } ) ) ↾s ( 0 [,] +oo ) ) = ( RR*s ↾s ( 0 [,] +oo ) )
20	2, 19	eqtr4i 2647	. . . . . . . . . 10 |- G = ( ( RR*s ↾s ( RR* \ { -oo } ) ) ↾s ( 0 [,] +oo ) )
21	6	xrs10 19785	. . . . . . . . . 10 |- 0 = ( 0g ` ( RR*s ↾s ( RR* \ { -oo } ) ) )
22	20, 21	subm0 17356	. . . . . . . . 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 19788	. . . . . . . . . 10 |- ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd
25	2, 24	eqeltri 2697	. . . . . . . . 9 |- G e. CMnd
26	25	a1i 11	. . . . . . . 8 |- ( ( ph /\ s e. ( ~P A i^i Fin ) ) -> G e. CMnd )
27		elfpw 8268	. . . . . . . . . 10 |- ( s e. ( ~P A i^i Fin ) <-> ( s C_ A /\ s e. Fin ) )
28	27	simprbi 480	. . . . . . . . 9 |- ( s e. ( ~P A i^i Fin ) -> s e. Fin )
29	28	adantl 482	. . . . . . . 8 |- ( ( ph /\ s e. ( ~P A i^i Fin ) ) -> s e. Fin )
30		xrge0gsumle.f	. . . . . . . . 9 |- ( ph -> F : A --> ( 0 [,] +oo ) )
31	27	simplbi 476	. . . . . . . . 9 |- ( s e. ( ~P A i^i Fin ) -> s C_ A )
32		fssres 6070	. . . . . . . . 9 |- ( ( F : A --> ( 0 [,] +oo ) /\ s C_ A ) -> ( F |` s ) : s --> ( 0 [,] +oo ) )
33	30, 31, 32	syl2an 494	. . . . . . . 8 |- ( ( ph /\ s e. ( ~P A i^i Fin ) ) -> ( F |` s ) : s --> ( 0 [,] +oo ) )
34		c0ex 10034	. . . . . . . . . 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 8285	. . . . . . . 8 |- ( ( ph /\ s e. ( ~P A i^i Fin ) ) -> ( F |` s ) finSupp 0 )
37	5, 23, 26, 29, 33, 36	gsumcl 18316	. . . . . . 7 |- ( ( ph /\ s e. ( ~P A i^i Fin ) ) -> ( G gsumg ( F |` s ) ) e. ( 0 [,] +oo ) )
38	1, 37	sseldi 3601	. . . . . 6 |- ( ( ph /\ s e. ( ~P A i^i Fin ) ) -> ( G gsumg ( F |` s ) ) e. RR* )
39		eqid 2622	. . . . . 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 6385	. . . . 5 |- ( ph -> ( s e. ( ~P A i^i Fin ) |-> ( G gsumg ( F |` s ) ) ) : ( ~P A i^i Fin ) --> RR* )
41		frn 6053	. . . . 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 17	. . . 4 |- ( ph -> ran ( s e. ( ~P A i^i Fin ) |-> ( G gsumg ( F |` s ) ) ) C_ RR* )
43		0ss 3972	. . . . . . 7 |- (/) C_ A
44		0fin 8188	. . . . . . 7 |- (/) e. Fin
45		elfpw 8268	. . . . . . 7 |- ( (/) e. ( ~P A i^i Fin ) <-> ( (/) C_ A /\ (/) e. Fin ) )
46	43, 44, 45	mpbir2an 955	. . . . . 6 |- (/) e. ( ~P A i^i Fin )
47		0cn 10032	. . . . . 6 |- 0 e. CC
48		reseq2 5391	. . . . . . . . . 10 |- ( s = (/) -> ( F |` s ) = ( F |` (/) ) )
49		res0 5400	. . . . . . . . . 10 |- ( F |` (/) ) = (/)
50	48, 49	syl6eq 2672	. . . . . . . . 9 |- ( s = (/) -> ( F |` s ) = (/) )
51	50	oveq2d 6666	. . . . . . . 8 |- ( s = (/) -> ( G gsumg ( F |` s ) ) = ( G gsumg (/) ) )
52	23	gsum0 17278	. . . . . . . 8 |- ( G gsumg (/) ) = 0
53	51, 52	syl6eq 2672	. . . . . . 7 |- ( s = (/) -> ( G gsumg ( F |` s ) ) = 0 )
54	39, 53	elrnmpt1s 5373	. . . . . 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 708	. . . . 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 3604	. . 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 8268	. . . . . . . 8 |- ( B e. ( ~P A i^i Fin ) <-> ( B C_ A /\ B e. Fin ) )
61	60	simprbi 480	. . . . . . 7 |- ( B e. ( ~P A i^i Fin ) -> B e. Fin )
62	59, 61	syl 17	. . . . . 6 |- ( ph -> B e. Fin )
63		diffi 8192	. . . . . 6 |- ( B e. Fin -> ( B \ C ) e. Fin )
64	62, 63	syl 17	. . . . 5 |- ( ph -> ( B \ C ) e. Fin )
65	60	simplbi 476	. . . . . . . 8 |- ( B e. ( ~P A i^i Fin ) -> B C_ A )
66	59, 65	syl 17	. . . . . . 7 |- ( ph -> B C_ A )
67	66	ssdifssd 3748	. . . . . 6 |- ( ph -> ( B \ C ) C_ A )
68	30, 67	fssresd 6071	. . . . 5 |- ( ph -> ( F |` ( B \ C ) ) : ( B \ C ) --> ( 0 [,] +oo ) )
69	34	a1i 11	. . . . . 6 |- ( ph -> 0 e. _V )
70	68, 64, 69	fdmfifsupp 8285	. . . . 5 |- ( ph -> ( F |` ( B \ C ) ) finSupp 0 )
71	5, 23, 58, 64, 68, 70	gsumcl 18316	. . . 4 |- ( ph -> ( G gsumg ( F |` ( B \ C ) ) ) e. ( 0 [,] +oo ) )
72	1, 71	sseldi 3601	. . 3 |- ( ph -> ( G gsumg ( F |` ( B \ C ) ) ) e. RR* )
73		xrge0gsumle.c	. . . . . 6 |- ( ph -> C C_ B )
74		ssfi 8180	. . . . . 6 |- ( ( B e. Fin /\ C C_ B ) -> C e. Fin )
75	62, 73, 74	syl2anc 693	. . . . 5 |- ( ph -> C e. Fin )
76	73, 66	sstrd 3613	. . . . . 6 |- ( ph -> C C_ A )
77	30, 76	fssresd 6071	. . . . 5 |- ( ph -> ( F |` C ) : C --> ( 0 [,] +oo ) )
78	77, 75, 69	fdmfifsupp 8285	. . . . 5 |- ( ph -> ( F |` C ) finSupp 0 )
79	5, 23, 58, 75, 77, 78	gsumcl 18316	. . . 4 |- ( ph -> ( G gsumg ( F |` C ) ) e. ( 0 [,] +oo ) )
80	1, 79	sseldi 3601	. . 3 |- ( ph -> ( G gsumg ( F |` C ) ) e. RR* )
81		elxrge0 12281	. . . . 5 |- ( ( G gsumg ( F |` ( B \ C ) ) ) e. ( 0 [,] +oo ) <-> ( ( G gsumg ( F |` ( B \ C ) ) ) e. RR* /\ 0 <_ ( G gsumg ( F |` ( B \ C ) ) ) ) )
82	81	simprbi 480	. . . 4 |- ( ( G gsumg ( F |` ( B \ C ) ) ) e. ( 0 [,] +oo ) -> 0 <_ ( G gsumg ( F |` ( B \ C ) ) ) )
83	71, 82	syl 17	. . 3 |- ( ph -> 0 <_ ( G gsumg ( F |` ( B \ C ) ) ) )
84		xleadd2a 12084	. . 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 ) ) ) ) )
85	57, 72, 80, 83, 84	syl31anc 1329	. 2 |- ( ph -> ( ( G gsumg ( F |` C ) ) +e 0 ) <_ ( ( G gsumg ( F |` C ) ) +e ( G gsumg ( F |` ( B \ C ) ) ) ) )
86		xaddid1 12072	. . 3 |- ( ( G gsumg ( F |` C ) ) e. RR* -> ( ( G gsumg ( F |` C ) ) +e 0 ) = ( G gsumg ( F |` C ) ) )
87	80, 86	syl 17	. 2 |- ( ph -> ( ( G gsumg ( F |` C ) ) +e 0 ) = ( G gsumg ( F |` C ) ) )
88		ovex 6678	. . . . 5 |- ( 0 [,] +oo ) e. _V
89		xrsadd 19763	. . . . . 6 |- +e = ( +g ` RR*s )
90	2, 89	ressplusg 15993	. . . . 5 |- ( ( 0 [,] +oo ) e. _V -> +e = ( +g ` G ) )
91	88, 90	ax-mp 5	. . . 4 |- +e = ( +g ` G )
92	30, 66	fssresd 6071	. . . 4 |- ( ph -> ( F |` B ) : B --> ( 0 [,] +oo ) )
93	92, 62, 69	fdmfifsupp 8285	. . . 4 |- ( ph -> ( F |` B ) finSupp 0 )
94		disjdif 4040	. . . . 5 |- ( C i^i ( B \ C ) ) = (/)
95	94	a1i 11	. . . 4 |- ( ph -> ( C i^i ( B \ C ) ) = (/) )
96		undif2 4044	. . . . 5 |- ( C u. ( B \ C ) ) = ( C u. B )
97		ssequn1 3783	. . . . . 6 |- ( C C_ B <-> ( C u. B ) = B )
98	73, 97	sylib 208	. . . . 5 |- ( ph -> ( C u. B ) = B )
99	96, 98	syl5req 2669	. . . 4 |- ( ph -> B = ( C u. ( B \ C ) ) )
100	5, 23, 91, 58, 59, 92, 93, 95, 99	gsumsplit 18328	. . 3 |- ( ph -> ( G gsumg ( F |` B ) ) = ( ( G gsumg ( ( F |` B ) |` C ) ) +e ( G gsumg ( ( F |` B ) |` ( B \ C ) ) ) ) )
101	73	resabs1d 5428	. . . . 5 |- ( ph -> ( ( F |` B ) |` C ) = ( F |` C ) )
102	101	oveq2d 6666	. . . 4 |- ( ph -> ( G gsumg ( ( F |` B ) |` C ) ) = ( G gsumg ( F |` C ) ) )
103		difss 3737	. . . . . 6 |- ( B \ C ) C_ B
104		resabs1 5427	. . . . . 6 |- ( ( B \ C ) C_ B -> ( ( F |` B ) |` ( B \ C ) ) = ( F |` ( B \ C ) ) )
105	103, 104	mp1i 13	. . . . 5 |- ( ph -> ( ( F |` B ) |` ( B \ C ) ) = ( F |` ( B \ C ) ) )
106	105	oveq2d 6666	. . . 4 |- ( ph -> ( G gsumg ( ( F |` B ) |` ( B \ C ) ) ) = ( G gsumg ( F |` ( B \ C ) ) ) )
107	102, 106	oveq12d 6668	. . 3 |- ( ph -> ( ( G gsumg ( ( F |` B ) |` C ) ) +e ( G gsumg ( ( F |` B ) |` ( B \ C ) ) ) ) = ( ( G gsumg ( F |` C ) ) +e ( G gsumg ( F |` ( B \ C ) ) ) ) )
108	100, 107	eqtr2d 2657	. 2 |- ( ph -> ( ( G gsumg ( F |` C ) ) +e ( G gsumg ( F |` ( B \ C ) ) ) ) = ( G gsumg ( F |` B ) ) )
109	85, 87, 108	3brtr3d 4684	1 |- ( ph -> ( G gsumg ( F |` C ) ) <_ ( G gsumg ( F |` B ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   0cc0 9936   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   <_ cle 10075   +ecxad 11944   [,]cicc 12178   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941   0gc0g 16100   gsum_g cgsu 16101   RR*scxrs 16160  SubMndcsubmnd 17334  CMndccmn 18193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-xadd 11947  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-tset 15960  df-ple 15961  df-ds 15964  df-0g 16102  df-gsum 16103  df-xrs 16162  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-cntz 17750  df-cmn 18195

This theorem is referenced by:  xrge0tsms  22637  xrge0tsmsd  29785