Theorem xrge0gsumle 21632

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 11663	. . . . . . 7 |- ( 0 [,] +oo ) C_ RR*
2		xrge0gsumle.g	. . . . . . . . . 10 |- G = ( RR*s ↾s ( 0 [,] +oo ) )
3		xrsbas 18756	. . . . . . . . . 10 |- RR* = ( Base ` RR*s )
4	2, 3	ressbas2 14901	. . . . . . . . 9 |- ( ( 0 [,] +oo ) C_ RR* -> ( 0 [,] +oo ) = ( Base ` G ) )
5	1, 4	ax-mp 5	. . . . . . . 8 |- ( 0 [,] +oo ) = ( Base ` G )
6		eqid 2404	. . . . . . . . . 10 |- ( RR*s ↾s ( RR* \ { -oo } ) ) = ( RR*s ↾s ( RR* \ { -oo } ) )
7	6	xrge0subm 18781	. . . . . . . . 9 |- ( 0 [,] +oo ) e. (SubMnd ` ( RR*s ↾s ( RR* \ { -oo } ) ) )
8		xrex 11264	. . . . . . . . . . . . 13 |- RR* e. _V
9		difexg 4544	. . . . . . . . . . . . 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 11429	. . . . . . . . . . . . . . 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 11685	. . . . . . . . . . . . . 14 |- ( x e. ( 0 [,] +oo ) <-> ( x e. RR* /\ 0 <_ x ) )
15		eldifsn 4099	. . . . . . . . . . . . . 14 |- ( x e. ( RR* \ { -oo } ) <-> ( x e. RR* /\ x =/= -oo ) )
16	13, 14, 15	3imtr4i 268	. . . . . . . . . . . . 13 |- ( x e. ( 0 [,] +oo ) -> x e. ( RR* \ { -oo } ) )
17	16	ssriv 3448	. . . . . . . . . . . 12 |- ( 0 [,] +oo ) C_ ( RR* \ { -oo } )
18		ressabs 14909	. . . . . . . . . . . 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 2436	. . . . . . . . . 10 |- G = ( ( RR*s ↾s ( RR* \ { -oo } ) ) ↾s ( 0 [,] +oo ) )
21	6	xrs10 18779	. . . . . . . . . 10 |- 0 = ( 0g ` ( RR*s ↾s ( RR* \ { -oo } ) ) )
22	20, 21	subm0 16313	. . . . . . . . 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 18782	. . . . . . . . . 10 |- ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd
25	2, 24	eqeltri 2488	. . . . . . . . 9 |- G e. CMnd
26	25	a1i 11	. . . . . . . 8 |- ( ( ph /\ s e. ( ~P A i^i Fin ) ) -> G e. CMnd )
27		elfpw 7858	. . . . . . . . . 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 5736	. . . . . . . . 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 9622	. . . . . . . . . 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 7875	. . . . . . . 8 |- ( ( ph /\ s e. ( ~P A i^i Fin ) ) -> ( F |` s ) finSupp 0 )
37	5, 23, 26, 29, 33, 36	gsumcl 17249	. . . . . . 7 |- ( ( ph /\ s e. ( ~P A i^i Fin ) ) -> ( G gsumg ( F |` s ) ) e. ( 0 [,] +oo ) )
38	1, 37	sseldi 3442	. . . . . 6 |- ( ( ph /\ s e. ( ~P A i^i Fin ) ) -> ( G gsumg ( F |` s ) ) e. RR* )
39		eqid 2404	. . . . . 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 6035	. . . . 5 |- ( ph -> ( s e. ( ~P A i^i Fin ) |-> ( G gsumg ( F |` s ) ) ) : ( ~P A i^i Fin ) --> RR* )
41		frn 5722	. . . . 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 3770	. . . . . . 7 |- (/) C_ A
44		0fin 7784	. . . . . . 7 |- (/) e. Fin
45		elfpw 7858	. . . . . . 7 |- ( (/) e. ( ~P A i^i Fin ) <-> ( (/) C_ A /\ (/) e. Fin ) )
46	43, 44, 45	mpbir2an 923	. . . . . 6 |- (/) e. ( ~P A i^i Fin )
47		0cn 9620	. . . . . 6 |- 0 e. CC
48		reseq2 5091	. . . . . . . . . 10 |- ( s = (/) -> ( F |` s ) = ( F |` (/) ) )
49		res0 5100	. . . . . . . . . 10 |- ( F |` (/) ) = (/)
50	48, 49	syl6eq 2461	. . . . . . . . 9 |- ( s = (/) -> ( F |` s ) = (/) )
51	50	oveq2d 6296	. . . . . . . 8 |- ( s = (/) -> ( G gsumg ( F |` s ) ) = ( G gsumg (/) ) )
52	23	gsum0 16231	. . . . . . . 8 |- ( G gsumg (/) ) = 0
53	51, 52	syl6eq 2461	. . . . . . 7 |- ( s = (/) -> ( G gsumg ( F |` s ) ) = 0 )
54	39, 53	elrnmpt1s 5073	. . . . . 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 3445	. . 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 7858	. . . . . . . 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 17	. . . . . 6 |- ( ph -> B e. Fin )
63		diffi 7788	. . . . . 6 |- ( B e. Fin -> ( B \ C ) e. Fin )
64	62, 63	syl 17	. . . . 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 17	. . . . . . 7 |- ( ph -> B C_ A )
67	66	ssdifssd 3583	. . . . . 6 |- ( ph -> ( B \ C ) C_ A )
68	30, 67	fssresd 5737	. . . . 5 |- ( ph -> ( F |` ( B \ C ) ) : ( B \ C ) --> ( 0 [,] +oo ) )
69	34	a1i 11	. . . . . 6 |- ( ph -> 0 e. _V )
70	68, 64, 69	fdmfifsupp 7875	. . . . 5 |- ( ph -> ( F |` ( B \ C ) ) finSupp 0 )
71	5, 23, 58, 64, 68, 70	gsumcl 17249	. . . 4 |- ( ph -> ( G gsumg ( F |` ( B \ C ) ) ) e. ( 0 [,] +oo ) )
72	1, 71	sseldi 3442	. . 3 |- ( ph -> ( G gsumg ( F |` ( B \ C ) ) ) e. RR* )
73		xrge0gsumle.c	. . . . . 6 |- ( ph -> C C_ B )
74		ssfi 7777	. . . . . 6 |- ( ( B e. Fin /\ C C_ B ) -> C e. Fin )
75	62, 73, 74	syl2anc 661	. . . . 5 |- ( ph -> C e. Fin )
76	73, 66	sstrd 3454	. . . . . 6 |- ( ph -> C C_ A )
77	30, 76	fssresd 5737	. . . . 5 |- ( ph -> ( F |` C ) : C --> ( 0 [,] +oo ) )
78	77, 75, 69	fdmfifsupp 7875	. . . . 5 |- ( ph -> ( F |` C ) finSupp 0 )
79	5, 23, 58, 75, 77, 78	gsumcl 17249	. . . 4 |- ( ph -> ( G gsumg ( F |` C ) ) e. ( 0 [,] +oo ) )
80	1, 79	sseldi 3442	. . 3 |- ( ph -> ( G gsumg ( F |` C ) ) e. RR* )
81		elxrge0 11685	. . . . 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 464	. . . 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 11501	. . 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 1235	. 2 |- ( ph -> ( ( G gsumg ( F |` C ) ) +e 0 ) <_ ( ( G gsumg ( F |` C ) ) +e ( G gsumg ( F |` ( B \ C ) ) ) ) )
86		xaddid1 11493	. . 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 6308	. . . . 5 |- ( 0 [,] +oo ) e. _V
89		xrsadd 18757	. . . . . 6 |- +e = ( +g ` RR*s )
90	2, 89	ressplusg 14957	. . . . 5 |- ( ( 0 [,] +oo ) e. _V -> +e = ( +g ` G ) )
91	88, 90	ax-mp 5	. . . 4 |- +e = ( +g ` G )
92	30, 66	fssresd 5737	. . . 4 |- ( ph -> ( F |` B ) : B --> ( 0 [,] +oo ) )
93	92, 62, 69	fdmfifsupp 7875	. . . 4 |- ( ph -> ( F |` B ) finSupp 0 )
94		disjdif 3846	. . . . 5 |- ( C i^i ( B \ C ) ) = (/)
95	94	a1i 11	. . . 4 |- ( ph -> ( C i^i ( B \ C ) ) = (/) )
96		undif2 3850	. . . . 5 |- ( C u. ( B \ C ) ) = ( C u. B )
97		ssequn1 3615	. . . . . 6 |- ( C C_ B <-> ( C u. B ) = B )
98	73, 97	sylib 198	. . . . 5 |- ( ph -> ( C u. B ) = B )
99	96, 98	syl5req 2458	. . . 4 |- ( ph -> B = ( C u. ( B \ C ) ) )
100	5, 23, 91, 58, 59, 92, 93, 95, 99	gsumsplit 17272	. . 3 |- ( ph -> ( G gsumg ( F |` B ) ) = ( ( G gsumg ( ( F |` B ) |` C ) ) +e ( G gsumg ( ( F |` B ) |` ( B \ C ) ) ) ) )
101	73	resabs1d 5125	. . . . 5 |- ( ph -> ( ( F |` B ) |` C ) = ( F |` C ) )
102	101	oveq2d 6296	. . . 4 |- ( ph -> ( G gsumg ( ( F |` B ) |` C ) ) = ( G gsumg ( F |` C ) ) )
103		difss 3572	. . . . . 6 |- ( B \ C ) C_ B
104		resabs1 5124	. . . . . 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 6296	. . . 4 |- ( ph -> ( G gsumg ( ( F |` B ) |` ( B \ C ) ) ) = ( G gsumg ( F |` ( B \ C ) ) ) )
107	102, 106	oveq12d 6298	. . 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 2446	. 2 |- ( ph -> ( ( G gsumg ( F |` C ) ) +e ( G gsumg ( F |` ( B \ C ) ) ) ) = ( G gsumg ( F |` B ) ) )
109	85, 87, 108	3brtr3d 4426	1 |- ( ph -> ( G gsumg ( F |` C ) ) <_ ( G gsumg ( F |` B ) ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1407   e. wcel 1844   =/= wne 2600   _Vcvv 3061   \ cdif 3413   u. cun 3414   i^i cin 3415   C_ wss 3416   (/)c0 3740   ~Pcpw 3957   {csn 3974   class class class wbr 4397   |-> cmpt 4455   ran crn 4826   |` cres 4827   -->wf 5567   ` cfv 5571  (class class class)co 6280   Fincfn 7556   CCcc 9522   0cc0 9524   +oocpnf 9657   -oocmnf 9658   RR*cxr 9659   <_ cle 9661   +ecxad 11371   [,]cicc 11587   Basecbs 14843   ↾_s cress 14844   +g cplusg 14911   0gc0g 15056   gsum_g cgsu 15057   RR*scxrs 15116  SubMndcsubmnd 16291  CMndccmn 17124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601

This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-iin 4276  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-se 4785  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-isom 5580  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-of 6523  df-om 6686  df-1st 6786  df-2nd 6787  df-supp 6905  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-fsupp 7866  df-oi 7971  df-card 8354  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-nn 10579  df-2 10637  df-3 10638  df-4 10639  df-5 10640  df-6 10641  df-7 10642  df-8 10643  df-9 10644  df-10 10645  df-n0 10839  df-z 10908  df-dec 11022  df-uz 11130  df-xadd 11374  df-icc 11591  df-fz 11729  df-fzo 11857  df-seq 12154  df-hash 12455  df-struct 14845  df-ndx 14846  df-slot 14847  df-base 14848  df-sets 14849  df-ress 14850  df-plusg 14924  df-mulr 14925  df-tset 14930  df-ple 14931  df-ds 14933  df-0g 15058  df-gsum 15059  df-xrs 15118  df-mre 15202  df-mrc 15203  df-acs 15205  df-mgm 16198  df-sgrp 16237  df-mnd 16247  df-submnd 16293  df-cntz 16681  df-cmn 17126

This theorem is referenced by:  xrge0tsms  21633  xrge0tsmsd  28241