Theorem sge0resrnlem 40620

Description: The sum of nonnegative extended reals restricted to the range of a function is less or equal to the sum of the composition of the two functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
sge0resrnlem.a	|- ( ph -> A e. V )
sge0resrnlem.f	|- ( ph -> F : B --> ( 0 [,] +oo ) )
sge0resrnlem.g	|- ( ph -> G : A --> B )
sge0resrnlem.x	|- ( ph -> X e. ~P A )
sge0resrnlem.f1o	|- ( ph -> ( G |` X ) : X -1-1-onto-> ran G )

Assertion

Ref	Expression
sge0resrnlem	|- ( ph -> (Σ^ ` ( F |` ran G ) ) <_ (Σ^ ` ( F o. G ) ) )

Proof of Theorem sge0resrnlem

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1843	. . . 4 |- F/ y ph
2		nfv 1843	. . . 4 |- F/ x ph
3		fveq2 6191	. . . 4 |- ( y = ( G ` x ) -> ( F ` y ) = ( F ` ( G ` x ) ) )
4		sge0resrnlem.x	. . . 4 |- ( ph -> X e. ~P A )
5		sge0resrnlem.f1o	. . . 4 |- ( ph -> ( G |` X ) : X -1-1-onto-> ran G )
6		fvres 6207	. . . . 5 |- ( x e. X -> ( ( G |` X ) ` x ) = ( G ` x ) )
7	6	adantl 482	. . . 4 |- ( ( ph /\ x e. X ) -> ( ( G |` X ) ` x ) = ( G ` x ) )
8		sge0resrnlem.f	. . . . . 6 |- ( ph -> F : B --> ( 0 [,] +oo ) )
9	8	adantr 481	. . . . 5 |- ( ( ph /\ y e. ran G ) -> F : B --> ( 0 [,] +oo ) )
10		sge0resrnlem.g	. . . . . . . 8 |- ( ph -> G : A --> B )
11		frn 6053	. . . . . . . 8 |- ( G : A --> B -> ran G C_ B )
12	10, 11	syl 17	. . . . . . 7 |- ( ph -> ran G C_ B )
13	12	adantr 481	. . . . . 6 |- ( ( ph /\ y e. ran G ) -> ran G C_ B )
14		simpr 477	. . . . . 6 |- ( ( ph /\ y e. ran G ) -> y e. ran G )
15	13, 14	sseldd 3604	. . . . 5 |- ( ( ph /\ y e. ran G ) -> y e. B )
16	9, 15	ffvelrnd 6360	. . . 4 |- ( ( ph /\ y e. ran G ) -> ( F ` y ) e. ( 0 [,] +oo ) )
17	1, 2, 3, 4, 5, 7, 16	sge0f1o 40599	. . 3 |- ( ph -> (Σ^ ` ( y e. ran G |-> ( F ` y ) ) ) = (Σ^ ` ( x e. X |-> ( F ` ( G ` x ) ) ) ) )
18	8, 12	feqresmpt 6250	. . . 4 |- ( ph -> ( F |` ran G ) = ( y e. ran G |-> ( F ` y ) ) )
19	18	fveq2d 6195	. . 3 |- ( ph -> (Σ^ ` ( F |` ran G ) ) = (Σ^ ` ( y e. ran G |-> ( F ` y ) ) ) )
20		fcompt 6400	. . . . . . 7 |- ( ( F : B --> ( 0 [,] +oo ) /\ G : A --> B ) -> ( F o. G ) = ( x e. A |-> ( F ` ( G ` x ) ) ) )
21	8, 10, 20	syl2anc 693	. . . . . 6 |- ( ph -> ( F o. G ) = ( x e. A |-> ( F ` ( G ` x ) ) ) )
22	21	reseq1d 5395	. . . . 5 |- ( ph -> ( ( F o. G ) |` X ) = ( ( x e. A |-> ( F ` ( G ` x ) ) ) |` X ) )
23	4	elpwid 4170	. . . . . 6 |- ( ph -> X C_ A )
24	23	resmptd 5452	. . . . 5 |- ( ph -> ( ( x e. A |-> ( F ` ( G ` x ) ) ) |` X ) = ( x e. X |-> ( F ` ( G ` x ) ) ) )
25	22, 24	eqtrd 2656	. . . 4 |- ( ph -> ( ( F o. G ) |` X ) = ( x e. X |-> ( F ` ( G ` x ) ) ) )
26	25	fveq2d 6195	. . 3 |- ( ph -> (Σ^ ` ( ( F o. G ) |` X ) ) = (Σ^ ` ( x e. X |-> ( F ` ( G ` x ) ) ) ) )
27	17, 19, 26	3eqtr4d 2666	. 2 |- ( ph -> (Σ^ ` ( F |` ran G ) ) = (Σ^ ` ( ( F o. G ) |` X ) ) )
28		sge0resrnlem.a	. . 3 |- ( ph -> A e. V )
29		fco 6058	. . . 4 |- ( ( F : B --> ( 0 [,] +oo ) /\ G : A --> B ) -> ( F o. G ) : A --> ( 0 [,] +oo ) )
30	8, 10, 29	syl2anc 693	. . 3 |- ( ph -> ( F o. G ) : A --> ( 0 [,] +oo ) )
31	28, 30	sge0less 40609	. 2 |- ( ph -> (Σ^ ` ( ( F o. G ) |` X ) ) <_ (Σ^ ` ( F o. G ) ) )
32	27, 31	eqbrtrd 4675	1 |- ( ph -> (Σ^ ` ( F |` ran G ) ) <_ (Σ^ ` ( F o. G ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   |` cres 5116   o. ccom 5118   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   0cc0 9936   +oocpnf 10071   <_ cle 10075   [,]cicc 12178  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:  sge0resrn  40621