Theorem xrge0gsumle 20429

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 11397	. . . . . . 7 |- ( 0 [,] +oo ) C_ RR*
2		xrge0gsumle.g	. . . . . . . . . 10 |- G = ( RR*s ↾s ( 0 [,] +oo ) )
3		xrsbas 17851	. . . . . . . . . 10 |- RR* = ( Base ` RR*s )
4	2, 3	ressbas2 14248	. . . . . . . . 9 |- ( ( 0 [,] +oo ) C_ RR* -> ( 0 [,] +oo ) = ( Base ` G ) )
5	1, 4	ax-mp 5	. . . . . . . 8 |- ( 0 [,] +oo ) = ( Base ` G )
6		eqid 2443	. . . . . . . . . 10 |- ( RR*s ↾s ( RR* \ { -oo } ) ) = ( RR*s ↾s ( RR* \ { -oo } ) )
7	6	xrge0subm 17873	. . . . . . . . 9 |- ( 0 [,] +oo ) e. (SubMnd ` ( RR*s ↾s ( RR* \ { -oo } ) ) )
8		xrex 11007	. . . . . . . . . . . . 13 |- RR* e. _V
9		difexg 4459	. . . . . . . . . . . . 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 11164	. . . . . . . . . . . . . . 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 11413	. . . . . . . . . . . . . 14 |- ( x e. ( 0 [,] +oo ) <-> ( x e. RR* /\ 0 <_ x ) )
15		eldifsn 4019	. . . . . . . . . . . . . 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 3379	. . . . . . . . . . . 12 |- ( 0 [,] +oo ) C_ ( RR* \ { -oo } )
18		ressabs 14255	. . . . . . . . . . . 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 2466	. . . . . . . . . 10 |- G = ( ( RR*s ↾s ( RR* \ { -oo } ) ) ↾s ( 0 [,] +oo ) )
21	6	xrs10 17871	. . . . . . . . . 10 |- 0 = ( 0g ` ( RR*s ↾s ( RR* \ { -oo } ) ) )
22	20, 21	subm0 15503	. . . . . . . . 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 17874	. . . . . . . . . 10 |- ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd
25	2, 24	eqeltri 2513	. . . . . . . . 9 |- G e. CMnd
26	25	a1i 11	. . . . . . . 8 |- ( ( ph /\ s e. ( ~P A i^i Fin ) ) -> G e. CMnd )
27		elfpw 7632	. . . . . . . . . 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 5597	. . . . . . . . 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 9399	. . . . . . . . . 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 7649	. . . . . . . 8 |- ( ( ph /\ s e. ( ~P A i^i Fin ) ) -> ( F |` s ) finSupp 0 )
37	5, 23, 26, 29, 33, 36	gsumcl 16416	. . . . . . 7 |- ( ( ph /\ s e. ( ~P A i^i Fin ) ) -> ( G gsumg ( F |` s ) ) e. ( 0 [,] +oo ) )
38	1, 37	sseldi 3373	. . . . . 6 |- ( ( ph /\ s e. ( ~P A i^i Fin ) ) -> ( G gsumg ( F |` s ) ) e. RR* )
39		eqid 2443	. . . . . 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 5886	. . . . 5 |- ( ph -> ( s e. ( ~P A i^i Fin ) |-> ( G gsumg ( F |` s ) ) ) : ( ~P A i^i Fin ) --> RR* )
41		frn 5584	. . . . 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 3685	. . . . . . 7 |- (/) C_ A
44		0fin 7559	. . . . . . 7 |- (/) e. Fin
45		elfpw 7632	. . . . . . 7 |- ( (/) e. ( ~P A i^i Fin ) <-> ( (/) C_ A /\ (/) e. Fin ) )
46	43, 44, 45	mpbir2an 911	. . . . . 6 |- (/) e. ( ~P A i^i Fin )
47		0cn 9397	. . . . . 6 |- 0 e. CC
48		reseq2 5124	. . . . . . . . . 10 |- ( s = (/) -> ( F |` s ) = ( F |` (/) ) )
49		res0 5134	. . . . . . . . . 10 |- ( F |` (/) ) = (/)
50	48, 49	syl6eq 2491	. . . . . . . . 9 |- ( s = (/) -> ( F |` s ) = (/) )
51	50	oveq2d 6126	. . . . . . . 8 |- ( s = (/) -> ( G gsumg ( F |` s ) ) = ( G gsumg (/) ) )
52	23	gsum0 15529	. . . . . . . 8 |- ( G gsumg (/) ) = 0
53	51, 52	syl6eq 2491	. . . . . . 7 |- ( s = (/) -> ( G gsumg ( F |` s ) ) = 0 )
54	39, 53	elrnmpt1s 5106	. . . . . 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 3376	. . 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 7632	. . . . . . . 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 7562	. . . . . 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 3513	. . . . . 6 |- ( ph -> ( B \ C ) C_ A )
68		fssres 5597	. . . . . 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 7649	. . . . 5 |- ( ph -> ( F |` ( B \ C ) ) finSupp 0 )
72	5, 23, 58, 64, 69, 71	gsumcl 16416	. . . 4 |- ( ph -> ( G gsumg ( F |` ( B \ C ) ) ) e. ( 0 [,] +oo ) )
73	1, 72	sseldi 3373	. . 3 |- ( ph -> ( G gsumg ( F |` ( B \ C ) ) ) e. RR* )
74		xrge0gsumle.c	. . . . . 6 |- ( ph -> C C_ B )
75		ssfi 7552	. . . . . 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 3385	. . . . . 6 |- ( ph -> C C_ A )
78		fssres 5597	. . . . . 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 7649	. . . . 5 |- ( ph -> ( F |` C ) finSupp 0 )
81	5, 23, 58, 76, 79, 80	gsumcl 16416	. . . 4 |- ( ph -> ( G gsumg ( F |` C ) ) e. ( 0 [,] +oo ) )
82	1, 81	sseldi 3373	. . 3 |- ( ph -> ( G gsumg ( F |` C ) ) e. RR* )
83		elxrge0 11413	. . . . 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 11236	. . 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 1221	. 2 |- ( ph -> ( ( G gsumg ( F |` C ) ) +e 0 ) <_ ( ( G gsumg ( F |` C ) ) +e ( G gsumg ( F |` ( B \ C ) ) ) ) )
88		xaddid1 11228	. . 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 6135	. . . . 5 |- ( 0 [,] +oo ) e. _V
91		xrsadd 17852	. . . . . 6 |- +e = ( +g ` RR*s )
92	2, 91	ressplusg 14299	. . . . 5 |- ( ( 0 [,] +oo ) e. _V -> +e = ( +g ` G ) )
93	90, 92	ax-mp 5	. . . 4 |- +e = ( +g ` G )
94		fssres 5597	. . . . 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 7649	. . . 4 |- ( ph -> ( F |` B ) finSupp 0 )
97		disjdif 3770	. . . . 5 |- ( C i^i ( B \ C ) ) = (/)
98	97	a1i 11	. . . 4 |- ( ph -> ( C i^i ( B \ C ) ) = (/) )
99		undif2 3774	. . . . 5 |- ( C u. ( B \ C ) ) = ( C u. B )
100		ssequn1 3545	. . . . . 6 |- ( C C_ B <-> ( C u. B ) = B )
101	74, 100	sylib 196	. . . . 5 |- ( ph -> ( C u. B ) = B )
102	99, 101	syl5req 2488	. . . 4 |- ( ph -> B = ( C u. ( B \ C ) ) )
103	5, 23, 93, 58, 59, 95, 96, 98, 102	gsumsplit 16439	. . 3 |- ( ph -> ( G gsumg ( F |` B ) ) = ( ( G gsumg ( ( F |` B ) |` C ) ) +e ( G gsumg ( ( F |` B ) |` ( B \ C ) ) ) ) )
104		resabs1 5158	. . . . . 6 |- ( C C_ B -> ( ( F |` B ) |` C ) = ( F |` C ) )
105	74, 104	syl 16	. . . . 5 |- ( ph -> ( ( F |` B ) |` C ) = ( F |` C ) )
106	105	oveq2d 6126	. . . 4 |- ( ph -> ( G gsumg ( ( F |` B ) |` C ) ) = ( G gsumg ( F |` C ) ) )
107		difss 3502	. . . . . 6 |- ( B \ C ) C_ B
108		resabs1 5158	. . . . . 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 6126	. . . 4 |- ( ph -> ( G gsumg ( ( F |` B ) |` ( B \ C ) ) ) = ( G gsumg ( F |` ( B \ C ) ) ) )
111	106, 110	oveq12d 6128	. . 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 2476	. 2 |- ( ph -> ( ( G gsumg ( F |` C ) ) +e ( G gsumg ( F |` ( B \ C ) ) ) ) = ( G gsumg ( F |` B ) ) )
113	87, 89, 112	3brtr3d 4340	1 |- ( ph -> ( G gsumg ( F |` C ) ) <_ ( G gsumg ( F |` B ) ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1369   e. wcel 1756   =/= wne 2620   _Vcvv 2991   \ cdif 3344   u. cun 3345   i^i cin 3346   C_ wss 3347   (/)c0 3656   ~Pcpw 3879   {csn 3896   class class class wbr 4311   e. cmpt 4369   ran crn 4860   |` cres 4861   -->wf 5433   ` cfv 5437  (class class class)co 6110   Fincfn 7329   CCcc 9299   0cc0 9301   +oocpnf 9434   -oocmnf 9435   RR*cxr 9436   <_ cle 9438   +ecxad 11106   [,]cicc 11322   Basecbs 14193   ↾_s cress 14194   +g cplusg 14257   0gc0g 14397   gsum_g cgsu 14398   RR*scxrs 14457  SubMndcsubmnd 15482  CMndccmn 16296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-cnex 9357  ax-resscn 9358  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-addrcl 9362  ax-mulcl 9363  ax-mulrcl 9364  ax-mulcom 9365  ax-addass 9366  ax-mulass 9367  ax-distr 9368  ax-i2m1 9369  ax-1ne0 9370  ax-1rid 9371  ax-rnegex 9372  ax-rrecex 9373  ax-cnre 9374  ax-pre-lttri 9375  ax-pre-lttrn 9376  ax-pre-ltadd 9377  ax-pre-mulgt0 9378

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2739  df-rex 2740  df-reu 2741  df-rmo 2742  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-int 4148  df-iun 4192  df-iin 4193  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-se 4699  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-isom 5446  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-of 6339  df-om 6496  df-1st 6596  df-2nd 6597  df-supp 6710  df-recs 6851  df-rdg 6885  df-1o 6939  df-oadd 6943  df-er 7120  df-en 7330  df-dom 7331  df-sdom 7332  df-fin 7333  df-fsupp 7640  df-oi 7743  df-card 8128  df-pnf 9439  df-mnf 9440  df-xr 9441  df-ltxr 9442  df-le 9443  df-sub 9616  df-neg 9617  df-nn 10342  df-2 10399  df-3 10400  df-4 10401  df-5 10402  df-6 10403  df-7 10404  df-8 10405  df-9 10406  df-10 10407  df-n0 10599  df-z 10666  df-dec 10775  df-uz 10881  df-xadd 11109  df-icc 11326  df-fz 11457  df-fzo 11568  df-seq 11826  df-hash 12123  df-struct 14195  df-ndx 14196  df-slot 14197  df-base 14198  df-sets 14199  df-ress 14200  df-plusg 14270  df-mulr 14271  df-tset 14276  df-ple 14277  df-ds 14279  df-0g 14399  df-gsum 14400  df-xrs 14459  df-mre 14543  df-mrc 14544  df-acs 14546  df-mnd 15434  df-submnd 15484  df-cntz 15854  df-cmn 16298

This theorem is referenced by:  xrge0tsms  20430  xrge0tsmsd  26272