Theorem sge0le 38363

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 38341	. . . . 5 |- ( ph -> (Σ^ ` F ) e. RR* )
4		pnfge 11455	. . . . 5 |- ( (Σ^ ` F ) e. RR* -> (Σ^ ` F ) <_ +oo )
5	3, 4	syl 17	. . . 4 |- ( ph -> (Σ^ ` F ) <_ +oo )
6	5	adantr 472	. . 3 |- ( ( ph /\ (Σ^ ` G ) = +oo ) -> (Σ^ ` F ) <_ +oo )
7		id 22	. . . . 5 |- ( (Σ^ ` G ) = +oo -> (Σ^ ` G ) = +oo )
8	7	eqcomd 2477	. . . 4 |- ( (Σ^ ` G ) = +oo -> +oo = (Σ^ ` G ) )
9	8	adantl 473	. . 3 |- ( ( ph /\ (Σ^ ` G ) = +oo ) -> +oo = (Σ^ ` G ) )
10	6, 9	breqtrd 4420	. 2 |- ( ( ph /\ (Σ^ ` G ) = +oo ) -> (Σ^ ` F ) <_ (Σ^ ` G ) )
11		elinel2 3611	. . . . . . . 8 |- ( y e. ( ~P X i^i Fin ) -> y e. Fin )
12	11	adantl 473	. . . . . . 7 |- ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) -> y e. Fin )
13	2	adantr 472	. . . . . . . . . 10 |- ( ( ph /\ -. (Σ^ ` G ) = +oo ) -> F : X --> ( 0 [,] +oo ) )
14	1	adantr 472	. . . . . . . . . . . . 13 |- ( ( ph /\ +oo e. ran F ) -> X e. V )
15		sge0le.g	. . . . . . . . . . . . . 14 |- ( ph -> G : X --> ( 0 [,] +oo ) )
16	15	adantr 472	. . . . . . . . . . . . 13 |- ( ( ph /\ +oo e. ran F ) -> G : X --> ( 0 [,] +oo ) )
17		simpr 468	. . . . . . . . . . . . . . 15 |- ( ( ph /\ +oo e. ran F ) -> +oo e. ran F )
18	2	ffnd 5740	. . . . . . . . . . . . . . . . 17 |- ( ph -> F Fn X )
19		fvelrnb 5926	. . . . . . . . . . . . . . . . 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 472	. . . . . . . . . . . . . . 15 |- ( ( ph /\ +oo e. ran F ) -> ( +oo e. ran F <-> E. x e. X ( F ` x ) = +oo ) )
22	17, 21	mpbid 215	. . . . . . . . . . . . . 14 |- ( ( ph /\ +oo e. ran F ) -> E. x e. X ( F ` x ) = +oo )
23		iccssxr 11742	. . . . . . . . . . . . . . . . . . . . . 22 |- ( 0 [,] +oo ) C_ RR*
24	15	ffvelrnda 6037	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ph /\ x e. X ) -> ( G ` x ) e. ( 0 [,] +oo ) )
25	23, 24	sseldi 3416	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ x e. X ) -> ( G ` x ) e. RR* )
26	25	adantr 472	. . . . . . . . . . . . . . . . . . . 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 2477	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( F ` x ) = +oo -> +oo = ( F ` x ) )
29	28	adantl 473	. . . . . . . . . . . . . . . . . . . . 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 472	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ph /\ x e. X ) /\ ( F ` x ) = +oo ) -> ( F ` x ) <_ ( G ` x ) )
32	29, 31	eqbrtrd 4416	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ph /\ x e. X ) /\ ( F ` x ) = +oo ) -> +oo <_ ( G ` x ) )
33	26, 32	xrgepnfd 37641	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ x e. X ) /\ ( F ` x ) = +oo ) -> ( G ` x ) = +oo )
34	33	eqcomd 2477	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ x e. X ) /\ ( F ` x ) = +oo ) -> +oo = ( G ` x ) )
35	15	ffnd 5740	. . . . . . . . . . . . . . . . . . . . 21 |- ( ph -> G Fn X )
36	35	adantr 472	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ x e. X ) -> G Fn X )
37		simpr 468	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ x e. X ) -> x e. X )
38		fnfvelrn 6034	. . . . . . . . . . . . . . . . . . . 20 |- ( ( G Fn X /\ x e. X ) -> ( G ` x ) e. ran G )
39	36, 37, 38	syl2anc 673	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ x e. X ) -> ( G ` x ) e. ran G )
40	39	adantr 472	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ x e. X ) /\ ( F ` x ) = +oo ) -> ( G ` x ) e. ran G )
41	34, 40	eqeltrd 2549	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ x e. X ) /\ ( F ` x ) = +oo ) -> +oo e. ran G )
42	41	ex 441	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. X ) -> ( ( F ` x ) = +oo -> +oo e. ran G ) )
43	42	adantlr 729	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ +oo e. ran F ) /\ x e. X ) -> ( ( F ` x ) = +oo -> +oo e. ran G ) )
44	43	rexlimdva 2871	. . . . . . . . . . . . . 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 38329	. . . . . . . . . . . 12 |- ( ( ph /\ +oo e. ran F ) -> (Σ^ ` G ) = +oo )
47	46	adantlr 729	. . . . . . . . . . 11 |- ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ +oo e. ran F ) -> (Σ^ ` G ) = +oo )
48		simplr 770	. . . . . . . . . . 11 |- ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ +oo e. ran F ) -> -. (Σ^ ` G ) = +oo )
49	47, 48	pm2.65da 586	. . . . . . . . . 10 |- ( ( ph /\ -. (Σ^ ` G ) = +oo ) -> -. +oo e. ran F )
50	13, 49	fge0iccico 38326	. . . . . . . . 9 |- ( ( ph /\ -. (Σ^ ` G ) = +oo ) -> F : X --> ( 0 [,) +oo ) )
51	50	adantr 472	. . . . . . . 8 |- ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) -> F : X --> ( 0 [,) +oo ) )
52		elpwinss 37446	. . . . . . . . 9 |- ( y e. ( ~P X i^i Fin ) -> y C_ X )
53	52	adantl 473	. . . . . . . 8 |- ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) -> y C_ X )
54	51, 53	fssresd 5762	. . . . . . 7 |- ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) -> ( F |` y ) : y --> ( 0 [,) +oo ) )
55	12, 54	sge0fsum 38343	. . . . . 6 |- ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) -> (Σ^ ` ( F |` y ) ) = sum_ x e. y ( ( F |` y ) ` x ) )
56		rge0ssre 11766	. . . . . . . 8 |- ( 0 [,) +oo ) C_ RR
57	54	ffvelrnda 6037	. . . . . . . 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 3416	. . . . . . 7 |- ( ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) /\ x e. y ) -> ( ( F |` y ) ` x ) e. RR )
59	12, 58	fsumrecl 13877	. . . . . 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 2549	. . . . 5 |- ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) -> (Σ^ ` ( F |` y ) ) e. RR )
61	15	adantr 472	. . . . . . . . . 10 |- ( ( ph /\ -. (Σ^ ` G ) = +oo ) -> G : X --> ( 0 [,] +oo ) )
62	1	adantr 472	. . . . . . . . . . 11 |- ( ( ph /\ -. (Σ^ ` G ) = +oo ) -> X e. V )
63		simpr 468	. . . . . . . . . . . 12 |- ( ( ph /\ -. (Σ^ ` G ) = +oo ) -> -. (Σ^ ` G ) = +oo )
64	62, 61	sge0repnf 38342	. . . . . . . . . . . 12 |- ( ( ph /\ -. (Σ^ ` G ) = +oo ) -> ( (Σ^ ` G ) e. RR <-> -. (Σ^ ` G ) = +oo ) )
65	63, 64	mpbird 240	. . . . . . . . . . 11 |- ( ( ph /\ -. (Σ^ ` G ) = +oo ) -> (Σ^ ` G ) e. RR )
66	62, 61, 65	sge0rern 38344	. . . . . . . . . 10 |- ( ( ph /\ -. (Σ^ ` G ) = +oo ) -> -. +oo e. ran G )
67	61, 66	fge0iccico 38326	. . . . . . . . 9 |- ( ( ph /\ -. (Σ^ ` G ) = +oo ) -> G : X --> ( 0 [,) +oo ) )
68	67	adantr 472	. . . . . . . 8 |- ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) -> G : X --> ( 0 [,) +oo ) )
69	68, 53	fssresd 5762	. . . . . . 7 |- ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) -> ( G |` y ) : y --> ( 0 [,) +oo ) )
70	12, 69	sge0fsum 38343	. . . . . 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 6037	. . . . . . . 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 3416	. . . . . . 7 |- ( ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) /\ x e. y ) -> ( ( G |` y ) ` x ) e. RR )
73	12, 72	fsumrecl 13877	. . . . . 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 2549	. . . . 5 |- ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) -> (Σ^ ` ( G |` y ) ) e. RR )
75	65	adantr 472	. . . . 5 |- ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) -> (Σ^ ` G ) e. RR )
76		simplll 776	. . . . . . . . 9 |- ( ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) /\ x e. y ) -> ph )
77	53	sselda 3418	. . . . . . . . 9 |- ( ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) /\ x e. y ) -> x e. X )
78	76, 77, 30	syl2anc 673	. . . . . . . 8 |- ( ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) /\ x e. y ) -> ( F ` x ) <_ ( G ` x ) )
79		fvres 5893	. . . . . . . . . 10 |- ( x e. y -> ( ( F |` y ) ` x ) = ( F ` x ) )
80	79	adantl 473	. . . . . . . . 9 |- ( ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) /\ x e. y ) -> ( ( F |` y ) ` x ) = ( F ` x ) )
81		fvres 5893	. . . . . . . . . 10 |- ( x e. y -> ( ( G |` y ) ` x ) = ( G ` x ) )
82	81	adantl 473	. . . . . . . . 9 |- ( ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) /\ x e. y ) -> ( ( G |` y ) ` x ) = ( G ` x ) )
83	80, 82	breq12d 4408	. . . . . . . 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 240	. . . . . . 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 13936	. . . . . 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 4408	. . . . . 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 240	. . . . 5 |- ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) -> (Σ^ ` ( F |` y ) ) <_ (Σ^ ` ( G |` y ) ) )
88	1	adantr 472	. . . . . . 7 |- ( ( ph /\ y e. ( ~P X i^i Fin ) ) -> X e. V )
89	15	adantr 472	. . . . . . 7 |- ( ( ph /\ y e. ( ~P X i^i Fin ) ) -> G : X --> ( 0 [,] +oo ) )
90	88, 89	sge0less 38348	. . . . . 6 |- ( ( ph /\ y e. ( ~P X i^i Fin ) ) -> (Σ^ ` ( G |` y ) ) <_ (Σ^ ` G ) )
91	90	adantlr 729	. . . . 5 |- ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) -> (Σ^ ` ( G |` y ) ) <_ (Σ^ ` G ) )
92	60, 74, 75, 87, 91	letrd 9809	. . . 4 |- ( ( ( ph /\ -. (Σ^ ` G ) = +oo ) /\ y e. ( ~P X i^i Fin ) ) -> (Σ^ ` ( F |` y ) ) <_ (Σ^ ` G ) )
93	92	ralrimiva 2809	. . 3 |- ( ( ph /\ -. (Σ^ ` G ) = +oo ) -> A. y e. ( ~P X i^i Fin ) (Σ^ ` ( F |` y ) ) <_ (Σ^ ` G ) )
94	62, 61	sge0xrcl 38341	. . . 4 |- ( ( ph /\ -. (Σ^ ` G ) = +oo ) -> (Σ^ ` G ) e. RR* )
95	62, 13, 94	sge0lefi 38354	. . 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 240	. 2 |- ( ( ph /\ -. (Σ^ ` G ) = +oo ) -> (Σ^ ` F ) <_ (Σ^ ` G ) )
97	10, 96	pm2.61dan 808	1 |- ( ph -> (Σ^ ` F ) <_ (Σ^ ` G ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 376   = wceq 1452   e. wcel 1904   A.wral 2756   E.wrex 2757   i^i cin 3389   C_ wss 3390   ~Pcpw 3942   class class class wbr 4395   ran crn 4840   |` cres 4841   Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   Fincfn 7587   RRcr 9556   0cc0 9557   +oocpnf 9690   RR*cxr 9692   <_ cle 9694   [,)cico 11662   [,]cicc 11663   sum_csu 13829  Σ^{^}csumge0 38318

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-inf2 8164  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  ax-pre-sup 9635

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  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-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-om 6712  df-1st 6812  df-2nd 6813  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-sup 7974  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-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-sum 13830  df-sumge0 38319

This theorem is referenced by:  sge0lempt  38366