Theorem esumpfinvallem 30136

Description: Lemma for esumpfinval 30137. (Contributed by Thierry Arnoux, 28-Jun-2017.)

Assertion

Ref	Expression
esumpfinvallem	|- ( ( A e. V /\ F : A --> ( 0 [,) +oo ) ) -> (ℂfld gsumg F ) = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg F ) )

Proof of Theorem esumpfinvallem

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

Step	Hyp	Ref	Expression
1		fex 6490	. . . 4 |- ( ( F : A --> ( 0 [,) +oo ) /\ A e. V ) -> F e. _V )
2	1	ancoms 469	. . 3 |- ( ( A e. V /\ F : A --> ( 0 [,) +oo ) ) -> F e. _V )
3		ovexd 6680	. . 3 |- ( ( A e. V /\ F : A --> ( 0 [,) +oo ) ) -> (ℂfld ↾s ( 0 [,) +oo ) ) e. _V )
4		ovexd 6680	. . 3 |- ( ( A e. V /\ F : A --> ( 0 [,) +oo ) ) -> ( RR*s ↾s ( 0 [,) +oo ) ) e. _V )
5		rge0ssre 12280	. . . . . . 7 |- ( 0 [,) +oo ) C_ RR
6		ax-resscn 9993	. . . . . . 7 |- RR C_ CC
7	5, 6	sstri 3612	. . . . . 6 |- ( 0 [,) +oo ) C_ CC
8		eqid 2622	. . . . . . 7 |- (ℂfld ↾s ( 0 [,) +oo ) ) = (ℂfld ↾s ( 0 [,) +oo ) )
9		cnfldbas 19750	. . . . . . 7 |- CC = ( Base ` ℂfld )
10	8, 9	ressbas2 15931	. . . . . 6 |- ( ( 0 [,) +oo ) C_ CC -> ( 0 [,) +oo ) = ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
11	7, 10	ax-mp 5	. . . . 5 |- ( 0 [,) +oo ) = ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) )
12		icossxr 12258	. . . . . 6 |- ( 0 [,) +oo ) C_ RR*
13		eqid 2622	. . . . . . 7 |- ( RR*s ↾s ( 0 [,) +oo ) ) = ( RR*s ↾s ( 0 [,) +oo ) )
14		xrsbas 19762	. . . . . . 7 |- RR* = ( Base ` RR*s )
15	13, 14	ressbas2 15931	. . . . . 6 |- ( ( 0 [,) +oo ) C_ RR* -> ( 0 [,) +oo ) = ( Base ` ( RR*s ↾s ( 0 [,) +oo ) ) ) )
16	12, 15	ax-mp 5	. . . . 5 |- ( 0 [,) +oo ) = ( Base ` ( RR*s ↾s ( 0 [,) +oo ) ) )
17	11, 16	eqtr3i 2646	. . . 4 |- ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) = ( Base ` ( RR*s ↾s ( 0 [,) +oo ) ) )
18	17	a1i 11	. . 3 |- ( ( A e. V /\ F : A --> ( 0 [,) +oo ) ) -> ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) = ( Base ` ( RR*s ↾s ( 0 [,) +oo ) ) ) )
19		simprl 794	. . . . 5 |- ( ( ( A e. V /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) /\ y e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) ) ) -> x e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
20	19, 11	syl6eleqr 2712	. . . 4 |- ( ( ( A e. V /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) /\ y e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) ) ) -> x e. ( 0 [,) +oo ) )
21		simprr 796	. . . . 5 |- ( ( ( A e. V /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) /\ y e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) ) ) -> y e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
22	21, 11	syl6eleqr 2712	. . . 4 |- ( ( ( A e. V /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) /\ y e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) ) ) -> y e. ( 0 [,) +oo ) )
23		ge0addcl 12284	. . . . 5 |- ( ( x e. ( 0 [,) +oo ) /\ y e. ( 0 [,) +oo ) ) -> ( x + y ) e. ( 0 [,) +oo ) )
24		ovex 6678	. . . . . . 7 |- ( 0 [,) +oo ) e. _V
25		cnfldadd 19751	. . . . . . . 8 |- + = ( +g ` ℂfld )
26	8, 25	ressplusg 15993	. . . . . . 7 |- ( ( 0 [,) +oo ) e. _V -> + = ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
27	24, 26	ax-mp 5	. . . . . 6 |- + = ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) )
28	27	oveqi 6663	. . . . 5 |- ( x + y ) = ( x ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) ) y )
29	23, 28, 11	3eltr3g 2717	. . . 4 |- ( ( x e. ( 0 [,) +oo ) /\ y e. ( 0 [,) +oo ) ) -> ( x ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) ) y ) e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
30	20, 22, 29	syl2anc 693	. . 3 |- ( ( ( A e. V /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) /\ y e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) ) ) -> ( x ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) ) y ) e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
31		simpl 473	. . . . . . 7 |- ( ( x e. ( 0 [,) +oo ) /\ y e. ( 0 [,) +oo ) ) -> x e. ( 0 [,) +oo ) )
32	5, 31	sseldi 3601	. . . . . 6 |- ( ( x e. ( 0 [,) +oo ) /\ y e. ( 0 [,) +oo ) ) -> x e. RR )
33		simpr 477	. . . . . . 7 |- ( ( x e. ( 0 [,) +oo ) /\ y e. ( 0 [,) +oo ) ) -> y e. ( 0 [,) +oo ) )
34	5, 33	sseldi 3601	. . . . . 6 |- ( ( x e. ( 0 [,) +oo ) /\ y e. ( 0 [,) +oo ) ) -> y e. RR )
35		rexadd 12063	. . . . . . 7 |- ( ( x e. RR /\ y e. RR ) -> ( x +e y ) = ( x + y ) )
36	35	eqcomd 2628	. . . . . 6 |- ( ( x e. RR /\ y e. RR ) -> ( x + y ) = ( x +e y ) )
37	32, 34, 36	syl2anc 693	. . . . 5 |- ( ( x e. ( 0 [,) +oo ) /\ y e. ( 0 [,) +oo ) ) -> ( x + y ) = ( x +e y ) )
38		xrsadd 19763	. . . . . . . 8 |- +e = ( +g ` RR*s )
39	13, 38	ressplusg 15993	. . . . . . 7 |- ( ( 0 [,) +oo ) e. _V -> +e = ( +g ` ( RR*s ↾s ( 0 [,) +oo ) ) ) )
40	24, 39	ax-mp 5	. . . . . 6 |- +e = ( +g ` ( RR*s ↾s ( 0 [,) +oo ) ) )
41	40	oveqi 6663	. . . . 5 |- ( x +e y ) = ( x ( +g ` ( RR*s ↾s ( 0 [,) +oo ) ) ) y )
42	37, 28, 41	3eqtr3g 2679	. . . 4 |- ( ( x e. ( 0 [,) +oo ) /\ y e. ( 0 [,) +oo ) ) -> ( x ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) ) y ) = ( x ( +g ` ( RR*s ↾s ( 0 [,) +oo ) ) ) y ) )
43	20, 22, 42	syl2anc 693	. . 3 |- ( ( ( A e. V /\ F : A --> ( 0 [,) +oo ) ) /\ ( x e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) /\ y e. ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) ) ) -> ( x ( +g ` (ℂfld ↾s ( 0 [,) +oo ) ) ) y ) = ( x ( +g ` ( RR*s ↾s ( 0 [,) +oo ) ) ) y ) )
44		simpr 477	. . . 4 |- ( ( A e. V /\ F : A --> ( 0 [,) +oo ) ) -> F : A --> ( 0 [,) +oo ) )
45		ffun 6048	. . . 4 |- ( F : A --> ( 0 [,) +oo ) -> Fun F )
46	44, 45	syl 17	. . 3 |- ( ( A e. V /\ F : A --> ( 0 [,) +oo ) ) -> Fun F )
47		frn 6053	. . . . 5 |- ( F : A --> ( 0 [,) +oo ) -> ran F C_ ( 0 [,) +oo ) )
48	44, 47	syl 17	. . . 4 |- ( ( A e. V /\ F : A --> ( 0 [,) +oo ) ) -> ran F C_ ( 0 [,) +oo ) )
49	48, 11	syl6sseq 3651	. . 3 |- ( ( A e. V /\ F : A --> ( 0 [,) +oo ) ) -> ran F C_ ( Base ` (ℂfld ↾s ( 0 [,) +oo ) ) ) )
50	2, 3, 4, 18, 30, 43, 46, 49	gsumpropd2 17274	. 2 |- ( ( A e. V /\ F : A --> ( 0 [,) +oo ) ) -> ( (ℂfld ↾s ( 0 [,) +oo ) ) gsumg F ) = ( ( RR*s ↾s ( 0 [,) +oo ) ) gsumg F ) )
51		cnfldex 19749	. . . 4 |- ℂfld e. _V
52	51	a1i 11	. . 3 |- ( ( A e. V /\ F : A --> ( 0 [,) +oo ) ) -> ℂfld e. _V )
53		simpl 473	. . 3 |- ( ( A e. V /\ F : A --> ( 0 [,) +oo ) ) -> A e. V )
54	7	a1i 11	. . 3 |- ( ( A e. V /\ F : A --> ( 0 [,) +oo ) ) -> ( 0 [,) +oo ) C_ CC )
55		0e0icopnf 12282	. . . 4 |- 0 e. ( 0 [,) +oo )
56	55	a1i 11	. . 3 |- ( ( A e. V /\ F : A --> ( 0 [,) +oo ) ) -> 0 e. ( 0 [,) +oo ) )
57		simpr 477	. . . . 5 |- ( ( ( A e. V /\ F : A --> ( 0 [,) +oo ) ) /\ x e. CC ) -> x e. CC )
58	57	addid2d 10237	. . . 4 |- ( ( ( A e. V /\ F : A --> ( 0 [,) +oo ) ) /\ x e. CC ) -> ( 0 + x ) = x )
59	57	addid1d 10236	. . . 4 |- ( ( ( A e. V /\ F : A --> ( 0 [,) +oo ) ) /\ x e. CC ) -> ( x + 0 ) = x )
60	58, 59	jca 554	. . 3 |- ( ( ( A e. V /\ F : A --> ( 0 [,) +oo ) ) /\ x e. CC ) -> ( ( 0 + x ) = x /\ ( x + 0 ) = x ) )
61	9, 25, 8, 52, 53, 54, 44, 56, 60	gsumress 17276	. 2 |- ( ( A e. V /\ F : A --> ( 0 [,) +oo ) ) -> (ℂfld gsumg F ) = ( (ℂfld ↾s ( 0 [,) +oo ) ) gsumg F ) )
62		xrge0base 29685	. . 3 |- ( 0 [,] +oo ) = ( Base ` ( RR*s ↾s ( 0 [,] +oo ) ) )
63		xrge0plusg 29687	. . 3 |- +e = ( +g ` ( RR*s ↾s ( 0 [,] +oo ) ) )
64		ovex 6678	. . . . 5 |- ( 0 [,] +oo ) e. _V
65		ressress 15938	. . . . 5 |- ( ( ( 0 [,] +oo ) e. _V /\ ( 0 [,) +oo ) e. _V ) -> ( ( RR*s ↾s ( 0 [,] +oo ) ) ↾s ( 0 [,) +oo ) ) = ( RR*s ↾s ( ( 0 [,] +oo ) i^i ( 0 [,) +oo ) ) ) )
66	64, 24, 65	mp2an 708	. . . 4 |- ( ( RR*s ↾s ( 0 [,] +oo ) ) ↾s ( 0 [,) +oo ) ) = ( RR*s ↾s ( ( 0 [,] +oo ) i^i ( 0 [,) +oo ) ) )
67		incom 3805	. . . . . 6 |- ( ( 0 [,] +oo ) i^i ( 0 [,) +oo ) ) = ( ( 0 [,) +oo ) i^i ( 0 [,] +oo ) )
68		icossicc 12260	. . . . . . 7 |- ( 0 [,) +oo ) C_ ( 0 [,] +oo )
69		dfss 3589	. . . . . . 7 |- ( ( 0 [,) +oo ) C_ ( 0 [,] +oo ) <-> ( 0 [,) +oo ) = ( ( 0 [,) +oo ) i^i ( 0 [,] +oo ) ) )
70	68, 69	mpbi 220	. . . . . 6 |- ( 0 [,) +oo ) = ( ( 0 [,) +oo ) i^i ( 0 [,] +oo ) )
71	67, 70	eqtr4i 2647	. . . . 5 |- ( ( 0 [,] +oo ) i^i ( 0 [,) +oo ) ) = ( 0 [,) +oo )
72	71	oveq2i 6661	. . . 4 |- ( RR*s ↾s ( ( 0 [,] +oo ) i^i ( 0 [,) +oo ) ) ) = ( RR*s ↾s ( 0 [,) +oo ) )
73	66, 72	eqtr2i 2645	. . 3 |- ( RR*s ↾s ( 0 [,) +oo ) ) = ( ( RR*s ↾s ( 0 [,] +oo ) ) ↾s ( 0 [,) +oo ) )
74		ovexd 6680	. . 3 |- ( ( A e. V /\ F : A --> ( 0 [,) +oo ) ) -> ( RR*s ↾s ( 0 [,] +oo ) ) e. _V )
75	68	a1i 11	. . 3 |- ( ( A e. V /\ F : A --> ( 0 [,) +oo ) ) -> ( 0 [,) +oo ) C_ ( 0 [,] +oo ) )
76		iccssxr 12256	. . . . . 6 |- ( 0 [,] +oo ) C_ RR*
77		simpr 477	. . . . . 6 |- ( ( ( A e. V /\ F : A --> ( 0 [,) +oo ) ) /\ x e. ( 0 [,] +oo ) ) -> x e. ( 0 [,] +oo ) )
78	76, 77	sseldi 3601	. . . . 5 |- ( ( ( A e. V /\ F : A --> ( 0 [,) +oo ) ) /\ x e. ( 0 [,] +oo ) ) -> x e. RR* )
79		xaddid2 12073	. . . . 5 |- ( x e. RR* -> ( 0 +e x ) = x )
80	78, 79	syl 17	. . . 4 |- ( ( ( A e. V /\ F : A --> ( 0 [,) +oo ) ) /\ x e. ( 0 [,] +oo ) ) -> ( 0 +e x ) = x )
81		xaddid1 12072	. . . . 5 |- ( x e. RR* -> ( x +e 0 ) = x )
82	78, 81	syl 17	. . . 4 |- ( ( ( A e. V /\ F : A --> ( 0 [,) +oo ) ) /\ x e. ( 0 [,] +oo ) ) -> ( x +e 0 ) = x )
83	80, 82	jca 554	. . 3 |- ( ( ( A e. V /\ F : A --> ( 0 [,) +oo ) ) /\ x e. ( 0 [,] +oo ) ) -> ( ( 0 +e x ) = x /\ ( x +e 0 ) = x ) )
84	62, 63, 73, 74, 53, 75, 44, 56, 83	gsumress 17276	. 2 |- ( ( A e. V /\ F : A --> ( 0 [,) +oo ) ) -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg F ) = ( ( RR*s ↾s ( 0 [,) +oo ) ) gsumg F ) )
85	50, 61, 84	3eqtr4d 2666	1 |- ( ( A e. V /\ F : A --> ( 0 [,) +oo ) ) -> (ℂfld gsumg F ) = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg F ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ran crn 5115   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   +oocpnf 10071   RR*cxr 10073   +ecxad 11944   [,)cico 12177   [,]cicc 12178   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941   gsum_g cgsu 16101   RR*scxrs 16160  ℂ_fldccnfld 19746

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-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-addf 10015

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-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-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-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-xadd 11947  df-ico 12181  df-icc 12182  df-fz 12327  df-seq 12802  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-0g 16102  df-gsum 16103  df-xrs 16162  df-cnfld 19747

This theorem is referenced by:  esumpfinval  30137  esumpfinvalf  30138  esumpcvgval  30140  esumcvg  30148