Theorem gsumesum 30121

Description: Relate a group sum on ( RR*s ↾_s ( 0 [,] +oo ) ) to a finite extended sum. (Contributed by Thierry Arnoux, 19-Oct-2017.) (Proof shortened by AV, 12-Dec-2019.)

Hypotheses

Ref	Expression
gsumesum.0	|- F/ k ph
gsumesum.1	|- ( ph -> A e. Fin )
gsumesum.2	|- ( ( ph /\ k e. A ) -> B e. ( 0 [,] +oo ) )

Assertion

Ref	Expression
gsumesum	|- ( ph -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) = Σ* k e. A B )

Distinct variable group:   A, k

Allowed substitution hints:   ph( k)   B( k)

Proof of Theorem gsumesum

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

Step	Hyp	Ref	Expression
1		gsumesum.0	. . 3 |- F/ k ph
2		nfcv 2764	. . 3 |- F/_ k A
3		gsumesum.1	. . 3 |- ( ph -> A e. Fin )
4		gsumesum.2	. . 3 |- ( ( ph /\ k e. A ) -> B e. ( 0 [,] +oo ) )
5		eqidd 2623	. . 3 |- ( ( ph /\ x e. ( ~P A i^i Fin ) ) -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) )
6	1, 2, 3, 4, 5	esumval 30108	. 2 |- ( ph -> Σ* k e. A B = sup ( ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) , RR* , < ) )
7		xrltso 11974	. . . 4 |- < Or RR*
8	7	a1i 11	. . 3 |- ( ph -> < Or RR* )
9		iccssxr 12256	. . . 4 |- ( 0 [,] +oo ) C_ RR*
10		xrge0base 29685	. . . . 5 |- ( 0 [,] +oo ) = ( Base ` ( RR*s ↾s ( 0 [,] +oo ) ) )
11		xrge0cmn 19788	. . . . . 6 |- ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd
12	11	a1i 11	. . . . 5 |- ( ph -> ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd )
13	4	ex 450	. . . . . 6 |- ( ph -> ( k e. A -> B e. ( 0 [,] +oo ) ) )
14	1, 13	ralrimi 2957	. . . . 5 |- ( ph -> A. k e. A B e. ( 0 [,] +oo ) )
15	10, 12, 3, 14	gsummptcl 18366	. . . 4 |- ( ph -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) e. ( 0 [,] +oo ) )
16	9, 15	sseldi 3601	. . 3 |- ( ph -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) e. RR* )
17		pwidg 4173	. . . . . . 7 |- ( A e. Fin -> A e. ~P A )
18	3, 17	syl 17	. . . . . 6 |- ( ph -> A e. ~P A )
19	18, 3	elind 3798	. . . . 5 |- ( ph -> A e. ( ~P A i^i Fin ) )
20		eqidd 2623	. . . . 5 |- ( ph -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) )
21		mpteq1 4737	. . . . . . . 8 |- ( x = A -> ( k e. x |-> B ) = ( k e. A |-> B ) )
22	21	oveq2d 6666	. . . . . . 7 |- ( x = A -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) )
23	22	eqeq2d 2632	. . . . . 6 |- ( x = A -> ( ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) <-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) ) )
24	23	rspcev 3309	. . . . 5 |- ( ( A e. ( ~P A i^i Fin ) /\ ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) ) -> E. x e. ( ~P A i^i Fin ) ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) )
25	19, 20, 24	syl2anc 693	. . . 4 |- ( ph -> E. x e. ( ~P A i^i Fin ) ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) )
26		eqid 2622	. . . . 5 |- ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) = ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) )
27		ovex 6678	. . . . 5 |- ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) e. _V
28	26, 27	elrnmpti 5376	. . . 4 |- ( ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) e. ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) <-> E. x e. ( ~P A i^i Fin ) ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) )
29	25, 28	sylibr 224	. . 3 |- ( ph -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) e. ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) )
30		simpr 477	. . . . . 6 |- ( ( ph /\ y e. ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) ) -> y e. ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) )
31		mpteq1 4737	. . . . . . . . 9 |- ( x = a -> ( k e. x |-> B ) = ( k e. a |-> B ) )
32	31	oveq2d 6666	. . . . . . . 8 |- ( x = a -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) )
33	32	cbvmptv 4750	. . . . . . 7 |- ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) = ( a e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) )
34		ovex 6678	. . . . . . 7 |- ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) e. _V
35	33, 34	elrnmpti 5376	. . . . . 6 |- ( y e. ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) <-> E. a e. ( ~P A i^i Fin ) y = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) )
36	30, 35	sylib 208	. . . . 5 |- ( ( ph /\ y e. ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) ) -> E. a e. ( ~P A i^i Fin ) y = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) )
37	11	a1i 11	. . . . . . . . . . 11 |- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd )
38		inss2 3834	. . . . . . . . . . . 12 |- ( ~P A i^i Fin ) C_ Fin
39		simpr 477	. . . . . . . . . . . 12 |- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> a e. ( ~P A i^i Fin ) )
40	38, 39	sseldi 3601	. . . . . . . . . . 11 |- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> a e. Fin )
41		nfv 1843	. . . . . . . . . . . . 13 |- F/ k a e. ( ~P A i^i Fin )
42	1, 41	nfan 1828	. . . . . . . . . . . 12 |- F/ k ( ph /\ a e. ( ~P A i^i Fin ) )
43		simpll 790	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ a e. ( ~P A i^i Fin ) ) /\ k e. a ) -> ph )
44		inss1 3833	. . . . . . . . . . . . . . . . . 18 |- ( ~P A i^i Fin ) C_ ~P A
45	44	sseli 3599	. . . . . . . . . . . . . . . . 17 |- ( a e. ( ~P A i^i Fin ) -> a e. ~P A )
46	45	elpwid 4170	. . . . . . . . . . . . . . . 16 |- ( a e. ( ~P A i^i Fin ) -> a C_ A )
47	46	ad2antlr 763	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ a e. ( ~P A i^i Fin ) ) /\ k e. a ) -> a C_ A )
48		simpr 477	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ a e. ( ~P A i^i Fin ) ) /\ k e. a ) -> k e. a )
49	47, 48	sseldd 3604	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ a e. ( ~P A i^i Fin ) ) /\ k e. a ) -> k e. A )
50	43, 49, 4	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ( ph /\ a e. ( ~P A i^i Fin ) ) /\ k e. a ) -> B e. ( 0 [,] +oo ) )
51	50	ex 450	. . . . . . . . . . . 12 |- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> ( k e. a -> B e. ( 0 [,] +oo ) ) )
52	42, 51	ralrimi 2957	. . . . . . . . . . 11 |- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> A. k e. a B e. ( 0 [,] +oo ) )
53	10, 37, 40, 52	gsummptcl 18366	. . . . . . . . . 10 |- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) e. ( 0 [,] +oo ) )
54	9, 53	sseldi 3601	. . . . . . . . 9 |- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) e. RR* )
55		diffi 8192	. . . . . . . . . . . . 13 |- ( A e. Fin -> ( A \ a ) e. Fin )
56	3, 55	syl 17	. . . . . . . . . . . 12 |- ( ph -> ( A \ a ) e. Fin )
57	56	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> ( A \ a ) e. Fin )
58		simpll 790	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ a e. ( ~P A i^i Fin ) ) /\ k e. ( A \ a ) ) -> ph )
59		simpr 477	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ a e. ( ~P A i^i Fin ) ) /\ k e. ( A \ a ) ) -> k e. ( A \ a ) )
60	59	eldifad 3586	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ a e. ( ~P A i^i Fin ) ) /\ k e. ( A \ a ) ) -> k e. A )
61	58, 60, 4	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ( ph /\ a e. ( ~P A i^i Fin ) ) /\ k e. ( A \ a ) ) -> B e. ( 0 [,] +oo ) )
62	61	ex 450	. . . . . . . . . . . 12 |- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> ( k e. ( A \ a ) -> B e. ( 0 [,] +oo ) ) )
63	42, 62	ralrimi 2957	. . . . . . . . . . 11 |- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> A. k e. ( A \ a ) B e. ( 0 [,] +oo ) )
64	10, 37, 57, 63	gsummptcl 18366	. . . . . . . . . 10 |- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. ( A \ a ) |-> B ) ) e. ( 0 [,] +oo ) )
65	9, 64	sseldi 3601	. . . . . . . . 9 |- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. ( A \ a ) |-> B ) ) e. RR* )
66		elxrge0 12281	. . . . . . . . . . 11 |- ( ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. ( A \ a ) |-> B ) ) e. ( 0 [,] +oo ) <-> ( ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. ( A \ a ) |-> B ) ) e. RR* /\ 0 <_ ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. ( A \ a ) |-> B ) ) ) )
67	66	simprbi 480	. . . . . . . . . 10 |- ( ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. ( A \ a ) |-> B ) ) e. ( 0 [,] +oo ) -> 0 <_ ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. ( A \ a ) |-> B ) ) )
68	64, 67	syl 17	. . . . . . . . 9 |- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> 0 <_ ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. ( A \ a ) |-> B ) ) )
69		xraddge02 29521	. . . . . . . . . 10 |- ( ( ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) e. RR* /\ ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. ( A \ a ) |-> B ) ) e. RR* ) -> ( 0 <_ ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. ( A \ a ) |-> B ) ) -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) <_ ( ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) +e ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. ( A \ a ) |-> B ) ) ) ) )
70	69	imp 445	. . . . . . . . 9 |- ( ( ( ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) e. RR* /\ ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. ( A \ a ) |-> B ) ) e. RR* ) /\ 0 <_ ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. ( A \ a ) |-> B ) ) ) -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) <_ ( ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) +e ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. ( A \ a ) |-> B ) ) ) )
71	54, 65, 68, 70	syl21anc 1325	. . . . . . . 8 |- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) <_ ( ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) +e ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. ( A \ a ) |-> B ) ) ) )
72	71	adantlr 751	. . . . . . 7 |- ( ( ( ph /\ y e. ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) ) /\ a e. ( ~P A i^i Fin ) ) -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) <_ ( ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) +e ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. ( A \ a ) |-> B ) ) ) )
73		simpll 790	. . . . . . . 8 |- ( ( ( ph /\ y e. ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) ) /\ a e. ( ~P A i^i Fin ) ) -> ph )
74	46	adantl 482	. . . . . . . 8 |- ( ( ( ph /\ y e. ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) ) /\ a e. ( ~P A i^i Fin ) ) -> a C_ A )
75		xrge00 29686	. . . . . . . . . 10 |- 0 = ( 0g ` ( RR*s ↾s ( 0 [,] +oo ) ) )
76		xrge0plusg 29687	. . . . . . . . . 10 |- +e = ( +g ` ( RR*s ↾s ( 0 [,] +oo ) ) )
77	11	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ a C_ A ) -> ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd )
78	3	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ a C_ A ) -> A e. Fin )
79		eqid 2622	. . . . . . . . . . . 12 |- ( k e. A |-> B ) = ( k e. A |-> B )
80	1, 4, 79	fmptdf 6387	. . . . . . . . . . 11 |- ( ph -> ( k e. A |-> B ) : A --> ( 0 [,] +oo ) )
81	80	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ a C_ A ) -> ( k e. A |-> B ) : A --> ( 0 [,] +oo ) )
82	79	fnmpt 6020	. . . . . . . . . . . . 13 |- ( A. k e. A B e. ( 0 [,] +oo ) -> ( k e. A |-> B ) Fn A )
83	14, 82	syl 17	. . . . . . . . . . . 12 |- ( ph -> ( k e. A |-> B ) Fn A )
84		c0ex 10034	. . . . . . . . . . . . 13 |- 0 e. _V
85	84	a1i 11	. . . . . . . . . . . 12 |- ( ph -> 0 e. _V )
86	83, 3, 85	fndmfifsupp 8288	. . . . . . . . . . 11 |- ( ph -> ( k e. A |-> B ) finSupp 0 )
87	86	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ a C_ A ) -> ( k e. A |-> B ) finSupp 0 )
88		disjdif 4040	. . . . . . . . . . 11 |- ( a i^i ( A \ a ) ) = (/)
89	88	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ a C_ A ) -> ( a i^i ( A \ a ) ) = (/) )
90		undif 4049	. . . . . . . . . . . . 13 |- ( a C_ A <-> ( a u. ( A \ a ) ) = A )
91	90	biimpi 206	. . . . . . . . . . . 12 |- ( a C_ A -> ( a u. ( A \ a ) ) = A )
92	91	eqcomd 2628	. . . . . . . . . . 11 |- ( a C_ A -> A = ( a u. ( A \ a ) ) )
93	92	adantl 482	. . . . . . . . . 10 |- ( ( ph /\ a C_ A ) -> A = ( a u. ( A \ a ) ) )
94	10, 75, 76, 77, 78, 81, 87, 89, 93	gsumsplit 18328	. . . . . . . . 9 |- ( ( ph /\ a C_ A ) -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) = ( ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( ( k e. A |-> B ) |` a ) ) +e ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( ( k e. A |-> B ) |` ( A \ a ) ) ) ) )
95		resmpt 5449	. . . . . . . . . . . 12 |- ( a C_ A -> ( ( k e. A |-> B ) |` a ) = ( k e. a |-> B ) )
96	95	oveq2d 6666	. . . . . . . . . . 11 |- ( a C_ A -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( ( k e. A |-> B ) |` a ) ) = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) )
97	96	adantl 482	. . . . . . . . . 10 |- ( ( ph /\ a C_ A ) -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( ( k e. A |-> B ) |` a ) ) = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) )
98		difss 3737	. . . . . . . . . . . . 13 |- ( A \ a ) C_ A
99		resmpt 5449	. . . . . . . . . . . . 13 |- ( ( A \ a ) C_ A -> ( ( k e. A |-> B ) |` ( A \ a ) ) = ( k e. ( A \ a ) |-> B ) )
100	98, 99	ax-mp 5	. . . . . . . . . . . 12 |- ( ( k e. A |-> B ) |` ( A \ a ) ) = ( k e. ( A \ a ) |-> B )
101	100	oveq2i 6661	. . . . . . . . . . 11 |- ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( ( k e. A |-> B ) |` ( A \ a ) ) ) = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. ( A \ a ) |-> B ) )
102	101	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ a C_ A ) -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( ( k e. A |-> B ) |` ( A \ a ) ) ) = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. ( A \ a ) |-> B ) ) )
103	97, 102	oveq12d 6668	. . . . . . . . 9 |- ( ( ph /\ a C_ A ) -> ( ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( ( k e. A |-> B ) |` a ) ) +e ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( ( k e. A |-> B ) |` ( A \ a ) ) ) ) = ( ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) +e ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. ( A \ a ) |-> B ) ) ) )
104	94, 103	eqtrd 2656	. . . . . . . 8 |- ( ( ph /\ a C_ A ) -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) = ( ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) +e ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. ( A \ a ) |-> B ) ) ) )
105	73, 74, 104	syl2anc 693	. . . . . . 7 |- ( ( ( ph /\ y e. ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) ) /\ a e. ( ~P A i^i Fin ) ) -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) = ( ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) +e ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. ( A \ a ) |-> B ) ) ) )
106	72, 105	breqtrrd 4681	. . . . . 6 |- ( ( ( ph /\ y e. ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) ) /\ a e. ( ~P A i^i Fin ) ) -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) <_ ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) )
107	106	ralrimiva 2966	. . . . 5 |- ( ( ph /\ y e. ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) ) -> A. a e. ( ~P A i^i Fin ) ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) <_ ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) )
108		r19.29r 3073	. . . . . 6 |- ( ( E. a e. ( ~P A i^i Fin ) y = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) /\ A. a e. ( ~P A i^i Fin ) ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) <_ ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) ) -> E. a e. ( ~P A i^i Fin ) ( y = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) /\ ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) <_ ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) ) )
109		breq1 4656	. . . . . . . 8 |- ( y = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) -> ( y <_ ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) <-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) <_ ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) ) )
110	109	biimpar 502	. . . . . . 7 |- ( ( y = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) /\ ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) <_ ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) ) -> y <_ ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) )
111	110	rexlimivw 3029	. . . . . 6 |- ( E. a e. ( ~P A i^i Fin ) ( y = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) /\ ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) <_ ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) ) -> y <_ ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) )
112	108, 111	syl 17	. . . . 5 |- ( ( E. a e. ( ~P A i^i Fin ) y = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) /\ A. a e. ( ~P A i^i Fin ) ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. a |-> B ) ) <_ ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) ) -> y <_ ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) )
113	36, 107, 112	syl2anc 693	. . . 4 |- ( ( ph /\ y e. ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) ) -> y <_ ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) )
114	16	adantr 481	. . . . 5 |- ( ( ph /\ y e. ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) ) -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) e. RR* )
115	11	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ x e. ( ~P A i^i Fin ) ) -> ( RR*s ↾s ( 0 [,] +oo ) ) e. CMnd )
116		simpr 477	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( ~P A i^i Fin ) ) -> x e. ( ~P A i^i Fin ) )
117	38, 116	sseldi 3601	. . . . . . . . . 10 |- ( ( ph /\ x e. ( ~P A i^i Fin ) ) -> x e. Fin )
118		nfv 1843	. . . . . . . . . . . 12 |- F/ k x e. ( ~P A i^i Fin )
119	1, 118	nfan 1828	. . . . . . . . . . 11 |- F/ k ( ph /\ x e. ( ~P A i^i Fin ) )
120		simpll 790	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. ( ~P A i^i Fin ) ) /\ k e. x ) -> ph )
121	44	sseli 3599	. . . . . . . . . . . . . . . 16 |- ( x e. ( ~P A i^i Fin ) -> x e. ~P A )
122	121	ad2antlr 763	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. ( ~P A i^i Fin ) ) /\ k e. x ) -> x e. ~P A )
123	122	elpwid 4170	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( ~P A i^i Fin ) ) /\ k e. x ) -> x C_ A )
124		simpr 477	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( ~P A i^i Fin ) ) /\ k e. x ) -> k e. x )
125	123, 124	sseldd 3604	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. ( ~P A i^i Fin ) ) /\ k e. x ) -> k e. A )
126	120, 125, 4	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. ( ~P A i^i Fin ) ) /\ k e. x ) -> B e. ( 0 [,] +oo ) )
127	126	ex 450	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( ~P A i^i Fin ) ) -> ( k e. x -> B e. ( 0 [,] +oo ) ) )
128	119, 127	ralrimi 2957	. . . . . . . . . 10 |- ( ( ph /\ x e. ( ~P A i^i Fin ) ) -> A. k e. x B e. ( 0 [,] +oo ) )
129	10, 115, 117, 128	gsummptcl 18366	. . . . . . . . 9 |- ( ( ph /\ x e. ( ~P A i^i Fin ) ) -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) e. ( 0 [,] +oo ) )
130	9, 129	sseldi 3601	. . . . . . . 8 |- ( ( ph /\ x e. ( ~P A i^i Fin ) ) -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) e. RR* )
131	130	ralrimiva 2966	. . . . . . 7 |- ( ph -> A. x e. ( ~P A i^i Fin ) ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) e. RR* )
132	26	rnmptss 6392	. . . . . . 7 |- ( A. x e. ( ~P A i^i Fin ) ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) e. RR* -> ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) C_ RR* )
133	131, 132	syl 17	. . . . . 6 |- ( ph -> ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) C_ RR* )
134	133	sselda 3603	. . . . 5 |- ( ( ph /\ y e. ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) ) -> y e. RR* )
135		xrltnle 10105	. . . . . 6 |- ( ( ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) e. RR* /\ y e. RR* ) -> ( ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) < y <-> -. y <_ ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) ) )
136	135	con2bid 344	. . . . 5 |- ( ( ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) e. RR* /\ y e. RR* ) -> ( y <_ ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) <-> -. ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) < y ) )
137	114, 134, 136	syl2anc 693	. . . 4 |- ( ( ph /\ y e. ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) ) -> ( y <_ ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) <-> -. ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) < y ) )
138	113, 137	mpbid 222	. . 3 |- ( ( ph /\ y e. ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) ) -> -. ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) < y )
139	8, 16, 29, 138	supmax 8373	. 2 |- ( ph -> sup ( ran ( x e. ( ~P A i^i Fin ) |-> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. x |-> B ) ) ) , RR* , < ) = ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) )
140	6, 139	eqtr2d 2657	1 |- ( ph -> ( ( RR*s ↾s ( 0 [,] +oo ) ) gsumg ( k e. A |-> B ) ) = Σ* k e. A B )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   Or wor 5034   ran crn 5115   |` cres 5116   Fn wfn 5883   -->wf 5884  (class class class)co 6650   Fincfn 7955   finSupp cfsupp 8275   supcsup 8346   0cc0 9936   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   +ecxad 11944   [,]cicc 12178   ↾_s cress 15858   gsum_g cgsu 16101   RR*scxrs 16160  CMndccmn 18193  Σ^*cesum 30089

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-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-iin 4523  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-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  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-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-q 11789  df-xadd 11947  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  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-tset 15960  df-ple 15961  df-ds 15964  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-ordt 16161  df-xrs 16162  df-mre 16246  df-mrc 16247  df-acs 16249  df-ps 17200  df-tsr 17201  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-cntz 17750  df-cmn 18195  df-fbas 19743  df-fg 19744  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-ntr 20824  df-nei 20902  df-cn 21031  df-haus 21119  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-tsms 21930  df-esum 30090

This theorem is referenced by:  esumlub  30122