Theorem xrge0gsumle 21929

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 11742	. . . . . . 7 |- ( 0 [,] +oo ) C_ RR*
2		xrge0gsumle.g	. . . . . . . . . 10 |- G = ( RR*s ↾s ( 0 [,] +oo ) )
3		xrsbas 19061	. . . . . . . . . 10 |- RR* = ( Base ` RR*s )
4	2, 3	ressbas2 15258	. . . . . . . . 9 |- ( ( 0 [,] +oo ) C_ RR* -> ( 0 [,] +oo ) = ( Base ` G ) )
5	1, 4	ax-mp 5	. . . . . . . 8 |- ( 0 [,] +oo ) = ( Base ` G )
6		eqid 2471	. . . . . . . . . 10 |- ( RR*s ↾s ( RR* \ { -oo } ) ) = ( RR*s ↾s ( RR* \ { -oo } ) )
7	6	xrge0subm 19086	. . . . . . . . 9 |- ( 0 [,] +oo ) e. (SubMnd ` ( RR*s ↾s ( RR* \ { -oo } ) ) )
8		xrex 11322	. . . . . . . . . . . . 13 |- RR* e. _V
9		difexg 4545	. . . . . . . . . . . . 13 |- ( RR* e. _V -> ( RR* \ { -oo } ) e. _V )
10	8, 9	ax-mp 5	. . . . . . . . . . . 12 |- ( RR* \ { -oo } ) e. _V
11		simpl 464	. . . . . . . . . . . . . . 15 |- ( ( x e. RR* /\ 0 <_ x ) -> x e. RR* )
12		ge0nemnf 11491	. . . . . . . . . . . . . . 15 |- ( ( x e. RR* /\ 0 <_ x ) -> x =/= -oo )
13	11, 12	jca 541	. . . . . . . . . . . . . 14 |- ( ( x e. RR* /\ 0 <_ x ) -> ( x e. RR* /\ x =/= -oo ) )
14		elxrge0 11767	. . . . . . . . . . . . . 14 |- ( x e. ( 0 [,] +oo ) <-> ( x e. RR* /\ 0 <_ x ) )
15		eldifsn 4088	. . . . . . . . . . . . . 14 |- ( x e. ( RR* \ { -oo } ) <-> ( x e. RR* /\ x =/= -oo ) )
16	13, 14, 15	3imtr4i 274	. . . . . . . . . . . . 13 |- ( x e. ( 0 [,] +oo ) -> x e. ( RR* \ { -oo } ) )
17	16	ssriv 3422	. . . . . . . . . . . 12 |- ( 0 [,] +oo ) C_ ( RR* \ { -oo } )
18		ressabs 15266	. . . . . . . . . . . 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 686	. . . . . . . . . . 11 |- ( ( RR*s ↾s ( RR* \ { -oo } ) ) ↾s ( 0 [,] +oo ) ) = ( RR*s ↾s ( 0 [,] +oo ) )
20	2, 19	eqtr4i 2496	. . . . . . . . . 10 |- G = ( ( RR*s ↾s ( RR* \ { -oo } ) ) ↾s ( 0 [,] +oo ) )
21	6	xrs10 19084	. . . . . . . . . 10 |- 0 = ( 0g ` ( RR*s ↾s ( RR* \ { -oo } ) ) )
22	20, 21	subm0 16681	. . . . . . . . 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 19087	. . . . . . . . . 10 |- ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd
25	2, 24	eqeltri 2545	. . . . . . . . 9 |- G e. CMnd
26	25	a1i 11	. . . . . . . 8 |- ( ( ph /\ s e. ( ~P A i^i Fin ) ) -> G e. CMnd )
27		elfpw 7894	. . . . . . . . . 10 |- ( s e. ( ~P A i^i Fin ) <-> ( s C_ A /\ s e. Fin ) )
28	27	simprbi 471	. . . . . . . . 9 |- ( s e. ( ~P A i^i Fin ) -> s e. Fin )
29	28	adantl 473	. . . . . . . 8 |- ( ( ph /\ s e. ( ~P A i^i Fin ) ) -> s e. Fin )
30		xrge0gsumle.f	. . . . . . . . 9 |- ( ph -> F : A --> ( 0 [,] +oo ) )
31	27	simplbi 467	. . . . . . . . 9 |- ( s e. ( ~P A i^i Fin ) -> s C_ A )
32		fssres 5761	. . . . . . . . 9 |- ( ( F : A --> ( 0 [,] +oo ) /\ s C_ A ) -> ( F |` s ) : s --> ( 0 [,] +oo ) )
33	30, 31, 32	syl2an 485	. . . . . . . 8 |- ( ( ph /\ s e. ( ~P A i^i Fin ) ) -> ( F |` s ) : s --> ( 0 [,] +oo ) )
34		c0ex 9655	. . . . . . . . . 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 7911	. . . . . . . 8 |- ( ( ph /\ s e. ( ~P A i^i Fin ) ) -> ( F |` s ) finSupp 0 )
37	5, 23, 26, 29, 33, 36	gsumcl 17627	. . . . . . 7 |- ( ( ph /\ s e. ( ~P A i^i Fin ) ) -> ( G gsumg ( F |` s ) ) e. ( 0 [,] +oo ) )
38	1, 37	sseldi 3416	. . . . . 6 |- ( ( ph /\ s e. ( ~P A i^i Fin ) ) -> ( G gsumg ( F |` s ) ) e. RR* )
39		eqid 2471	. . . . . 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 6061	. . . . 5 |- ( ph -> ( s e. ( ~P A i^i Fin ) |-> ( G gsumg ( F |` s ) ) ) : ( ~P A i^i Fin ) --> RR* )
41		frn 5747	. . . . 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 3766	. . . . . . 7 |- (/) C_ A
44		0fin 7817	. . . . . . 7 |- (/) e. Fin
45		elfpw 7894	. . . . . . 7 |- ( (/) e. ( ~P A i^i Fin ) <-> ( (/) C_ A /\ (/) e. Fin ) )
46	43, 44, 45	mpbir2an 934	. . . . . 6 |- (/) e. ( ~P A i^i Fin )
47		0cn 9653	. . . . . 6 |- 0 e. CC
48		reseq2 5106	. . . . . . . . . 10 |- ( s = (/) -> ( F |` s ) = ( F |` (/) ) )
49		res0 5115	. . . . . . . . . 10 |- ( F |` (/) ) = (/)
50	48, 49	syl6eq 2521	. . . . . . . . 9 |- ( s = (/) -> ( F |` s ) = (/) )
51	50	oveq2d 6324	. . . . . . . 8 |- ( s = (/) -> ( G gsumg ( F |` s ) ) = ( G gsumg (/) ) )
52	23	gsum0 16599	. . . . . . . 8 |- ( G gsumg (/) ) = 0
53	51, 52	syl6eq 2521	. . . . . . 7 |- ( s = (/) -> ( G gsumg ( F |` s ) ) = 0 )
54	39, 53	elrnmpt1s 5088	. . . . . 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 686	. . . . 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 3419	. . 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 7894	. . . . . . . 8 |- ( B e. ( ~P A i^i Fin ) <-> ( B C_ A /\ B e. Fin ) )
61	60	simprbi 471	. . . . . . 7 |- ( B e. ( ~P A i^i Fin ) -> B e. Fin )
62	59, 61	syl 17	. . . . . 6 |- ( ph -> B e. Fin )
63		diffi 7821	. . . . . 6 |- ( B e. Fin -> ( B \ C ) e. Fin )
64	62, 63	syl 17	. . . . 5 |- ( ph -> ( B \ C ) e. Fin )
65	60	simplbi 467	. . . . . . . 8 |- ( B e. ( ~P A i^i Fin ) -> B C_ A )
66	59, 65	syl 17	. . . . . . 7 |- ( ph -> B C_ A )
67	66	ssdifssd 3560	. . . . . 6 |- ( ph -> ( B \ C ) C_ A )
68	30, 67	fssresd 5762	. . . . 5 |- ( ph -> ( F |` ( B \ C ) ) : ( B \ C ) --> ( 0 [,] +oo ) )
69	34	a1i 11	. . . . . 6 |- ( ph -> 0 e. _V )
70	68, 64, 69	fdmfifsupp 7911	. . . . 5 |- ( ph -> ( F |` ( B \ C ) ) finSupp 0 )
71	5, 23, 58, 64, 68, 70	gsumcl 17627	. . . 4 |- ( ph -> ( G gsumg ( F |` ( B \ C ) ) ) e. ( 0 [,] +oo ) )
72	1, 71	sseldi 3416	. . 3 |- ( ph -> ( G gsumg ( F |` ( B \ C ) ) ) e. RR* )
73		xrge0gsumle.c	. . . . . 6 |- ( ph -> C C_ B )
74		ssfi 7810	. . . . . 6 |- ( ( B e. Fin /\ C C_ B ) -> C e. Fin )
75	62, 73, 74	syl2anc 673	. . . . 5 |- ( ph -> C e. Fin )
76	73, 66	sstrd 3428	. . . . . 6 |- ( ph -> C C_ A )
77	30, 76	fssresd 5762	. . . . 5 |- ( ph -> ( F |` C ) : C --> ( 0 [,] +oo ) )
78	77, 75, 69	fdmfifsupp 7911	. . . . 5 |- ( ph -> ( F |` C ) finSupp 0 )
79	5, 23, 58, 75, 77, 78	gsumcl 17627	. . . 4 |- ( ph -> ( G gsumg ( F |` C ) ) e. ( 0 [,] +oo ) )
80	1, 79	sseldi 3416	. . 3 |- ( ph -> ( G gsumg ( F |` C ) ) e. RR* )
81		elxrge0 11767	. . . . 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 471	. . . 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 11565	. . 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 1295	. 2 |- ( ph -> ( ( G gsumg ( F |` C ) ) +e 0 ) <_ ( ( G gsumg ( F |` C ) ) +e ( G gsumg ( F |` ( B \ C ) ) ) ) )
86		xaddid1 11556	. . 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 6336	. . . . 5 |- ( 0 [,] +oo ) e. _V
89		xrsadd 19062	. . . . . 6 |- +e = ( +g ` RR*s )
90	2, 89	ressplusg 15317	. . . . 5 |- ( ( 0 [,] +oo ) e. _V -> +e = ( +g ` G ) )
91	88, 90	ax-mp 5	. . . 4 |- +e = ( +g ` G )
92	30, 66	fssresd 5762	. . . 4 |- ( ph -> ( F |` B ) : B --> ( 0 [,] +oo ) )
93	92, 62, 69	fdmfifsupp 7911	. . . 4 |- ( ph -> ( F |` B ) finSupp 0 )
94		disjdif 3830	. . . . 5 |- ( C i^i ( B \ C ) ) = (/)
95	94	a1i 11	. . . 4 |- ( ph -> ( C i^i ( B \ C ) ) = (/) )
96		undif2 3834	. . . . 5 |- ( C u. ( B \ C ) ) = ( C u. B )
97		ssequn1 3595	. . . . . 6 |- ( C C_ B <-> ( C u. B ) = B )
98	73, 97	sylib 201	. . . . 5 |- ( ph -> ( C u. B ) = B )
99	96, 98	syl5req 2518	. . . 4 |- ( ph -> B = ( C u. ( B \ C ) ) )
100	5, 23, 91, 58, 59, 92, 93, 95, 99	gsumsplit 17639	. . 3 |- ( ph -> ( G gsumg ( F |` B ) ) = ( ( G gsumg ( ( F |` B ) |` C ) ) +e ( G gsumg ( ( F |` B ) |` ( B \ C ) ) ) ) )
101	73	resabs1d 5140	. . . . 5 |- ( ph -> ( ( F |` B ) |` C ) = ( F |` C ) )
102	101	oveq2d 6324	. . . 4 |- ( ph -> ( G gsumg ( ( F |` B ) |` C ) ) = ( G gsumg ( F |` C ) ) )
103		difss 3549	. . . . . 6 |- ( B \ C ) C_ B
104		resabs1 5139	. . . . . 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 6324	. . . 4 |- ( ph -> ( G gsumg ( ( F |` B ) |` ( B \ C ) ) ) = ( G gsumg ( F |` ( B \ C ) ) ) )
107	102, 106	oveq12d 6326	. . 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 2506	. 2 |- ( ph -> ( ( G gsumg ( F |` C ) ) +e ( G gsumg ( F |` ( B \ C ) ) ) ) = ( G gsumg ( F |` B ) ) )
109	85, 87, 108	3brtr3d 4425	1 |- ( ph -> ( G gsumg ( F |` C ) ) <_ ( G gsumg ( F |` B ) ) )

Syntax hints:   -> wi 4   /\ wa 376   = wceq 1452   e. wcel 1904   =/= wne 2641   _Vcvv 3031   \ cdif 3387   u. cun 3388   i^i cin 3389   C_ wss 3390   (/)c0 3722   ~Pcpw 3942   {csn 3959   class class class wbr 4395   |-> cmpt 4454   ran crn 4840   |` cres 4841   -->wf 5585   ` cfv 5589  (class class class)co 6308   Fincfn 7587   CCcc 9555   0cc0 9557   +oocpnf 9690   -oocmnf 9691   RR*cxr 9692   <_ cle 9694   +ecxad 11430   [,]cicc 11663   Basecbs 15199   ↾_s cress 15200   +g cplusg 15268   0gc0g 15416   gsum_g cgsu 15417   RR*scxrs 15476  SubMndcsubmnd 16659  CMndccmn 17508

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-xadd 11433  df-icc 11667  df-fz 11811  df-fzo 11943  df-seq 12252  df-hash 12554  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-tset 15287  df-ple 15288  df-ds 15290  df-0g 15418  df-gsum 15419  df-xrs 15478  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-cntz 17049  df-cmn 17510

This theorem is referenced by:  xrge0tsms  21930  xrge0tsmsd  28622