Theorem sge0le 40624

Description: If all of the terms of sums compare, so do the sums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
sge0le.x	|- ( ph -> X e. V )
sge0le.F	|- ( ph -> F : X --> ( 0 [,] +oo ) )
sge0le.g	|- ( ph -> G : X --> ( 0 [,] +oo ) )
sge0le.le	|- ( ( ph /\ x e. X ) -> ( F ` x ) <_ ( G ` x ) )

Assertion

Ref	Expression
sge0le	|- ( ph -> (Σ^ ` F ) <_ (Σ^ ` G ) )

Distinct variable groups:   x, F   x, G   x, X   ph, x

Allowed substitution hint:   V( x)

Proof of Theorem sge0le

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sge0le.x	. . . . . 6 |- ( ph -> X e. V )
2		sge0le.F	. . . . . 6 |- ( ph -> F : X --> ( 0 [,] +oo ) )
3	1, 2	sge0xrcl 40602	. . . . 5 |- ( ph -> (Σ^ ` F ) e. RR* )
4		pnfge 11964	. . . . 5 |- ( (Σ^ ` F ) e. RR* -> (Σ^ ` F ) <_ +oo )
5	3, 4	syl 17	. . . 4 |- ( ph -> (Σ^ ` F ) <_ +oo )
6	5	adantr 481	. . 3 |- ( ( ph /\ (Σ^ ` G ) = +oo ) -> (Σ^ ` F ) <_ +oo )
7		id 22	. . . . 5 |- ( (Σ^ ` G ) = +oo -> (Σ^ ` G ) = +oo )
8	7	eqcomd 2628	. . . 4 |- ( (Σ^ ` G ) = +oo -> +oo = (Σ^ ` G ) )
9	8	adantl 482	. . 3 |- ( ( ph /\ (Σ^ ` G ) = +oo ) -> +oo = (Σ^ ` G ) )
10	6, 9	breqtrd 4679	. 2 |- ( ( ph /\ (Σ^ ` G ) = +oo ) -> (Σ^ ` F ) <_ (Σ^ ` G ) )
11		elinel2 3800	. . . . . . . 8 |- ( y e. ( ~P X i^i Fin ) -> y e. Fin )
12	11	adantl 482	. . . . . . 7 |- ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) -> y e. Fin )
13	2	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ -. (Σ^ ` G ) = +oo ) -> F : X --> ( 0 [,] +oo ) )
14	1	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ +oo e. ran F ) -> X e. V )
15		sge0le.g	. . . . . . . . . . . . . 14 |- ( ph -> G : X --> ( 0 [,] +oo ) )
16	15	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ +oo e. ran F ) -> G : X --> ( 0 [,] +oo ) )
17		simpr 477	. . . . . . . . . . . . . . 15 |- ( ( ph /\ +oo e. ran F ) -> +oo e. ran F )
18	2	ffnd 6046	. . . . . . . . . . . . . . . . 17 |- ( ph -> F Fn X )
19		fvelrnb 6243	. . . . . . . . . . . . . . . . 17 |- ( F Fn X -> ( +oo e. ran F <-> E. x e. X ( F ` x ) = +oo ) )
20	18, 19	syl 17	. . . . . . . . . . . . . . . 16 |- ( ph -> ( +oo e. ran F <-> E. x e. X ( F ` x ) = +oo ) )
21	20	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ph /\ +oo e. ran F ) -> ( +oo e. ran F <-> E. x e. X ( F ` x ) = +oo ) )
22	17, 21	mpbid 222	. . . . . . . . . . . . . 14 |- ( ( ph /\ +oo e. ran F ) -> E. x e. X ( F ` x ) = +oo )
23		iccssxr 12256	. . . . . . . . . . . . . . . . . . . . . 22 |- ( 0 [,] +oo ) C_ RR*
24	15	ffvelrnda 6359	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ph /\ x e. X ) -> ( G ` x ) e. ( 0 [,] +oo ) )
25	23, 24	sseldi 3601	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ x e. X ) -> ( G ` x ) e. RR* )
26	25	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ x e. X ) /\ ( F ` x ) = +oo ) -> ( G ` x ) e. RR* )
27		id 22	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( F ` x ) = +oo -> ( F ` x ) = +oo )
28	27	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( F ` x ) = +oo -> +oo = ( F ` x ) )
29	28	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ph /\ x e. X ) /\ ( F ` x ) = +oo ) -> +oo = ( F ` x ) )
30		sge0le.le	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ph /\ x e. X ) -> ( F ` x ) <_ ( G ` x ) )
31	30	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ph /\ x e. X ) /\ ( F ` x ) = +oo ) -> ( F ` x ) <_ ( G ` x ) )
32	29, 31	eqbrtrd 4675	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ x e. X ) /\ ( F ` x ) = +oo ) -> +oo <_ ( G ` x ) )
33	26, 32	xrgepnfd 39547	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ x e. X ) /\ ( F ` x ) = +oo ) -> ( G ` x ) = +oo )
34	33	eqcomd 2628	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ x e. X ) /\ ( F ` x ) = +oo ) -> +oo = ( G ` x ) )
35	15	ffnd 6046	. . . . . . . . . . . . . . . . . . . . 21 |- ( ph -> G Fn X )
36	35	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ x e. X ) -> G Fn X )
37		simpr 477	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ x e. X ) -> x e. X )
38		fnfvelrn 6356	. . . . . . . . . . . . . . . . . . . 20 |- ( ( G Fn X /\ x e. X ) -> ( G ` x ) e. ran G )
39	36, 37, 38	syl2anc 693	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ x e. X ) -> ( G ` x ) e. ran G )
40	39	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ x e. X ) /\ ( F ` x ) = +oo ) -> ( G ` x ) e. ran G )
41	34, 40	eqeltrd 2701	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ x e. X ) /\ ( F ` x ) = +oo ) -> +oo e. ran G )
42	41	ex 450	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. X ) -> ( ( F ` x ) = +oo -> +oo e. ran G ) )
43	42	adantlr 751	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ +oo e. ran F ) /\ x e. X ) -> ( ( F ` x ) = +oo -> +oo e. ran G ) )
44	43	rexlimdva 3031	. . . . . . . . . . . . . 14 |- ( ( ph /\ +oo e. ran F ) -> ( E. x e. X ( F ` x ) = +oo -> +oo e. ran G ) )
45	22, 44	mpd 15	. . . . . . . . . . . . 13 |- ( ( ph /\ +oo e. ran F ) -> +oo e. ran G )
46	14, 16, 45	sge0pnfval 40590	. . . . . . . . . . . 12 |- ( ( ph /\ +oo e. ran F ) -> (Σ^ ` G ) = +oo )
47	46	adantlr 751	. . . . . . . . . . 11 |- ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ +oo e. ran F ) -> (Σ^ ` G ) = +oo )
48		simplr 792	. . . . . . . . . . 11 |- ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ +oo e. ran F ) -> -. (Σ^ ` G ) = +oo )
49	47, 48	pm2.65da 600	. . . . . . . . . 10 |- ( ( ph /\ -. (Σ^ ` G ) = +oo ) -> -. +oo e. ran F )
50	13, 49	fge0iccico 40587	. . . . . . . . 9 |- ( ( ph /\ -. (Σ^ ` G ) = +oo ) -> F : X --> ( 0 [,) +oo ) )
51	50	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) -> F : X --> ( 0 [,) +oo ) )
52		elpwinss 39216	. . . . . . . . 9 |- ( y e. ( ~P X i^i Fin ) -> y C_ X )
53	52	adantl 482	. . . . . . . 8 |- ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) -> y C_ X )
54	51, 53	fssresd 6071	. . . . . . 7 |- ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) -> ( F |` y ) : y --> ( 0 [,) +oo ) )
55	12, 54	sge0fsum 40604	. . . . . 6 |- ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) -> (Σ^ ` ( F |` y ) ) = sum_ x e. y ( ( F |` y ) ` x ) )
56		rge0ssre 12280	. . . . . . . 8 |- ( 0 [,) +oo ) C_ RR
57	54	ffvelrnda 6359	. . . . . . . 8 |- ( ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) /\ x e. y ) -> ( ( F |` y ) ` x ) e. ( 0 [,) +oo ) )
58	56, 57	sseldi 3601	. . . . . . 7 |- ( ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) /\ x e. y ) -> ( ( F |` y ) ` x ) e. RR )
59	12, 58	fsumrecl 14465	. . . . . 6 |- ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) -> sum_ x e. y ( ( F |` y ) ` x ) e. RR )
60	55, 59	eqeltrd 2701	. . . . 5 |- ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) -> (Σ^ ` ( F |` y ) ) e. RR )
61	15	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ -. (Σ^ ` G ) = +oo ) -> G : X --> ( 0 [,] +oo ) )
62	1	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ -. (Σ^ ` G ) = +oo ) -> X e. V )
63		simpr 477	. . . . . . . . . . . 12 |- ( ( ph /\ -. (Σ^ ` G ) = +oo ) -> -. (Σ^ ` G ) = +oo )
64	62, 61	sge0repnf 40603	. . . . . . . . . . . 12 |- ( ( ph /\ -. (Σ^ ` G ) = +oo ) -> ( (Σ^ ` G ) e. RR <-> -. (Σ^ ` G ) = +oo ) )
65	63, 64	mpbird 247	. . . . . . . . . . 11 |- ( ( ph /\ -. (Σ^ ` G ) = +oo ) -> (Σ^ ` G ) e. RR )
66	62, 61, 65	sge0rern 40605	. . . . . . . . . 10 |- ( ( ph /\ -. (Σ^ ` G ) = +oo ) -> -. +oo e. ran G )
67	61, 66	fge0iccico 40587	. . . . . . . . 9 |- ( ( ph /\ -. (Σ^ ` G ) = +oo ) -> G : X --> ( 0 [,) +oo ) )
68	67	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) -> G : X --> ( 0 [,) +oo ) )
69	68, 53	fssresd 6071	. . . . . . 7 |- ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) -> ( G |` y ) : y --> ( 0 [,) +oo ) )
70	12, 69	sge0fsum 40604	. . . . . 6 |- ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) -> (Σ^ ` ( G |` y ) ) = sum_ x e. y ( ( G |` y ) ` x ) )
71	69	ffvelrnda 6359	. . . . . . . 8 |- ( ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) /\ x e. y ) -> ( ( G |` y ) ` x ) e. ( 0 [,) +oo ) )
72	56, 71	sseldi 3601	. . . . . . 7 |- ( ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) /\ x e. y ) -> ( ( G |` y ) ` x ) e. RR )
73	12, 72	fsumrecl 14465	. . . . . 6 |- ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) -> sum_ x e. y ( ( G |` y ) ` x ) e. RR )
74	70, 73	eqeltrd 2701	. . . . 5 |- ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) -> (Σ^ ` ( G |` y ) ) e. RR )
75	65	adantr 481	. . . . 5 |- ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) -> (Σ^ ` G ) e. RR )
76		simplll 798	. . . . . . . . 9 |- ( ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) /\ x e. y ) -> ph )
77	53	sselda 3603	. . . . . . . . 9 |- ( ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) /\ x e. y ) -> x e. X )
78	76, 77, 30	syl2anc 693	. . . . . . . 8 |- ( ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) /\ x e. y ) -> ( F ` x ) <_ ( G ` x ) )
79		fvres 6207	. . . . . . . . . 10 |- ( x e. y -> ( ( F |` y ) ` x ) = ( F ` x ) )
80	79	adantl 482	. . . . . . . . 9 |- ( ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) /\ x e. y ) -> ( ( F |` y ) ` x ) = ( F ` x ) )
81		fvres 6207	. . . . . . . . . 10 |- ( x e. y -> ( ( G |` y ) ` x ) = ( G ` x ) )
82	81	adantl 482	. . . . . . . . 9 |- ( ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) /\ x e. y ) -> ( ( G |` y ) ` x ) = ( G ` x ) )
83	80, 82	breq12d 4666	. . . . . . . 8 |- ( ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) /\ x e. y ) -> ( ( ( F |` y ) ` x ) <_ ( ( G |` y ) ` x ) <-> ( F ` x ) <_ ( G ` x ) ) )
84	78, 83	mpbird 247	. . . . . . 7 |- ( ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) /\ x e. y ) -> ( ( F |` y ) ` x ) <_ ( ( G |` y ) ` x ) )
85	12, 58, 72, 84	fsumle 14531	. . . . . 6 |- ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) -> sum_ x e. y ( ( F |` y ) ` x ) <_ sum_ x e. y ( ( G |` y ) ` x ) )
86	55, 70	breq12d 4666	. . . . . 6 |- ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) -> ( (Σ^ ` ( F |` y ) ) <_ (Σ^ ` ( G |` y ) ) <-> sum_ x e. y ( ( F |` y ) ` x ) <_ sum_ x e. y ( ( G |` y ) ` x ) ) )
87	85, 86	mpbird 247	. . . . 5 |- ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) -> (Σ^ ` ( F |` y ) ) <_ (Σ^ ` ( G |` y ) ) )
88	1	adantr 481	. . . . . . 7 |- ( ( ph /\ y e. ( ~P X i^i Fin ) ) -> X e. V )
89	15	adantr 481	. . . . . . 7 |- ( ( ph /\ y e. ( ~P X i^i Fin ) ) -> G : X --> ( 0 [,] +oo ) )
90	88, 89	sge0less 40609	. . . . . 6 |- ( ( ph /\ y e. ( ~P X i^i Fin ) ) -> (Σ^ ` ( G |` y ) ) <_ (Σ^ ` G ) )
91	90	adantlr 751	. . . . 5 |- ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) -> (Σ^ ` ( G |` y ) ) <_ (Σ^ ` G ) )
92	60, 74, 75, 87, 91	letrd 10194	. . . 4 |- ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) -> (Σ^ ` ( F |` y ) ) <_ (Σ^ ` G ) )
93	92	ralrimiva 2966	. . 3 |- ( ( ph /\ -. (Σ^ ` G ) = +oo ) -> A. y e. ( ~P X i^i Fin ) (Σ^ ` ( F |` y ) ) <_ (Σ^ ` G ) )
94	62, 61	sge0xrcl 40602	. . . 4 |- ( ( ph /\ -. (Σ^ ` G ) = +oo ) -> (Σ^ ` G ) e. RR* )
95	62, 13, 94	sge0lefi 40615	. . 3 |- ( ( ph /\ -. (Σ^ ` G ) = +oo ) -> ( (Σ^ ` F ) <_ (Σ^ ` G ) <-> A. y e. ( ~P X i^i Fin ) (Σ^ ` ( F |` y ) ) <_ (Σ^ ` G ) ) )
96	93, 95	mpbird 247	. 2 |- ( ( ph /\ -. (Σ^ ` G ) = +oo ) -> (Σ^ ` F ) <_ (Σ^ ` G ) )
97	10, 96	pm2.61dan 832	1 |- ( ph -> (Σ^ ` F ) <_ (Σ^ ` G ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   ran crn 5115   |` cres 5116   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955   RRcr 9935   0cc0 9936   +oocpnf 10071   RR*cxr 10073   <_ cle 10075   [,)cico 12177   [,]cicc 12178   sum_csu 14416  Σ^{^}csumge0 40579

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-inf2 8538  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  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-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-om 7066  df-1st 7168  df-2nd 7169  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-sup 8348  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-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-sumge0 40580

This theorem is referenced by:  sge0lempt  40627