Theorem esumfzf 30131

Description: Formulating a partial extended sum over integers using the recursive sequence builder. (Contributed by Thierry Arnoux, 18-Oct-2017.)

Hypothesis

Ref	Expression
esumfzf.1	|- F/_ k F

Assertion

Ref	Expression
esumfzf	|- ( ( F : NN --> ( 0 [,] +oo ) /\ N e. NN ) -> Σ* k e. ( 1 ... N ) ( F ` k ) = ( seq 1 ( +e , F ) ` N ) )

Distinct variable group:   k, N

Allowed substitution hint:   F( k)

Proof of Theorem esumfzf

Dummy variables i n x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1843	. . . . . 6 |- F/ k i = 1
2		oveq2 6658	. . . . . 6 |- ( i = 1 -> ( 1 ... i ) = ( 1 ... 1 ) )
3	1, 2	esumeq1d 30097	. . . . 5 |- ( i = 1 -> Σ* k e. ( 1 ... i ) ( F ` k ) = Σ* k e. ( 1 ... 1 ) ( F ` k ) )
4		fveq2 6191	. . . . 5 |- ( i = 1 -> ( seq 1 ( +e , F ) ` i ) = ( seq 1 ( +e , F ) ` 1 ) )
5	3, 4	eqeq12d 2637	. . . 4 |- ( i = 1 -> (Σ* k e. ( 1 ... i ) ( F ` k ) = ( seq 1 ( +e , F ) ` i ) <-> Σ* k e. ( 1 ... 1 ) ( F ` k ) = ( seq 1 ( +e , F ) ` 1 ) ) )
6	5	imbi2d 330	. . 3 |- ( i = 1 -> ( ( F : NN --> ( 0 [,] +oo ) -> Σ* k e. ( 1 ... i ) ( F ` k ) = ( seq 1 ( +e , F ) ` i ) ) <-> ( F : NN --> ( 0 [,] +oo ) -> Σ* k e. ( 1 ... 1 ) ( F ` k ) = ( seq 1 ( +e , F ) ` 1 ) ) ) )
7		nfv 1843	. . . . . 6 |- F/ k i = n
8		oveq2 6658	. . . . . 6 |- ( i = n -> ( 1 ... i ) = ( 1 ... n ) )
9	7, 8	esumeq1d 30097	. . . . 5 |- ( i = n -> Σ* k e. ( 1 ... i ) ( F ` k ) = Σ* k e. ( 1 ... n ) ( F ` k ) )
10		fveq2 6191	. . . . 5 |- ( i = n -> ( seq 1 ( +e , F ) ` i ) = ( seq 1 ( +e , F ) ` n ) )
11	9, 10	eqeq12d 2637	. . . 4 |- ( i = n -> (Σ* k e. ( 1 ... i ) ( F ` k ) = ( seq 1 ( +e , F ) ` i ) <-> Σ* k e. ( 1 ... n ) ( F ` k ) = ( seq 1 ( +e , F ) ` n ) ) )
12	11	imbi2d 330	. . 3 |- ( i = n -> ( ( F : NN --> ( 0 [,] +oo ) -> Σ* k e. ( 1 ... i ) ( F ` k ) = ( seq 1 ( +e , F ) ` i ) ) <-> ( F : NN --> ( 0 [,] +oo ) -> Σ* k e. ( 1 ... n ) ( F ` k ) = ( seq 1 ( +e , F ) ` n ) ) ) )
13		nfv 1843	. . . . . 6 |- F/ k i = ( n + 1 )
14		oveq2 6658	. . . . . 6 |- ( i = ( n + 1 ) -> ( 1 ... i ) = ( 1 ... ( n + 1 ) ) )
15	13, 14	esumeq1d 30097	. . . . 5 |- ( i = ( n + 1 ) -> Σ* k e. ( 1 ... i ) ( F ` k ) = Σ* k e. ( 1 ... ( n + 1 ) ) ( F ` k ) )
16		fveq2 6191	. . . . 5 |- ( i = ( n + 1 ) -> ( seq 1 ( +e , F ) ` i ) = ( seq 1 ( +e , F ) ` ( n + 1 ) ) )
17	15, 16	eqeq12d 2637	. . . 4 |- ( i = ( n + 1 ) -> (Σ* k e. ( 1 ... i ) ( F ` k ) = ( seq 1 ( +e , F ) ` i ) <-> Σ* k e. ( 1 ... ( n + 1 ) ) ( F ` k ) = ( seq 1 ( +e , F ) ` ( n + 1 ) ) ) )
18	17	imbi2d 330	. . 3 |- ( i = ( n + 1 ) -> ( ( F : NN --> ( 0 [,] +oo ) -> Σ* k e. ( 1 ... i ) ( F ` k ) = ( seq 1 ( +e , F ) ` i ) ) <-> ( F : NN --> ( 0 [,] +oo ) -> Σ* k e. ( 1 ... ( n + 1 ) ) ( F ` k ) = ( seq 1 ( +e , F ) ` ( n + 1 ) ) ) ) )
19		nfv 1843	. . . . . 6 |- F/ k i = N
20		oveq2 6658	. . . . . 6 |- ( i = N -> ( 1 ... i ) = ( 1 ... N ) )
21	19, 20	esumeq1d 30097	. . . . 5 |- ( i = N -> Σ* k e. ( 1 ... i ) ( F ` k ) = Σ* k e. ( 1 ... N ) ( F ` k ) )
22		fveq2 6191	. . . . 5 |- ( i = N -> ( seq 1 ( +e , F ) ` i ) = ( seq 1 ( +e , F ) ` N ) )
23	21, 22	eqeq12d 2637	. . . 4 |- ( i = N -> (Σ* k e. ( 1 ... i ) ( F ` k ) = ( seq 1 ( +e , F ) ` i ) <-> Σ* k e. ( 1 ... N ) ( F ` k ) = ( seq 1 ( +e , F ) ` N ) ) )
24	23	imbi2d 330	. . 3 |- ( i = N -> ( ( F : NN --> ( 0 [,] +oo ) -> Σ* k e. ( 1 ... i ) ( F ` k ) = ( seq 1 ( +e , F ) ` i ) ) <-> ( F : NN --> ( 0 [,] +oo ) -> Σ* k e. ( 1 ... N ) ( F ` k ) = ( seq 1 ( +e , F ) ` N ) ) ) )
25		fveq2 6191	. . . . . 6 |- ( k = x -> ( F ` k ) = ( F ` x ) )
26		nfcv 2764	. . . . . 6 |- F/_ x { 1 }
27		nfcv 2764	. . . . . 6 |- F/_ k { 1 }
28		nfcv 2764	. . . . . 6 |- F/_ x ( F ` k )
29		esumfzf.1	. . . . . . 7 |- F/_ k F
30		nfcv 2764	. . . . . . 7 |- F/_ k x
31	29, 30	nffv 6198	. . . . . 6 |- F/_ k ( F ` x )
32	25, 26, 27, 28, 31	cbvesum 30104	. . . . 5 |- Σ* k e. { 1 } ( F ` k ) = Σ* x e. { 1 } ( F ` x )
33		simpr 477	. . . . . . 7 |- ( ( F : NN --> ( 0 [,] +oo ) /\ x = 1 ) -> x = 1 )
34	33	fveq2d 6195	. . . . . 6 |- ( ( F : NN --> ( 0 [,] +oo ) /\ x = 1 ) -> ( F ` x ) = ( F ` 1 ) )
35		1z 11407	. . . . . . 7 |- 1 e. ZZ
36	35	a1i 11	. . . . . 6 |- ( F : NN --> ( 0 [,] +oo ) -> 1 e. ZZ )
37		1nn 11031	. . . . . . 7 |- 1 e. NN
38		ffvelrn 6357	. . . . . . 7 |- ( ( F : NN --> ( 0 [,] +oo ) /\ 1 e. NN ) -> ( F ` 1 ) e. ( 0 [,] +oo ) )
39	37, 38	mpan2 707	. . . . . 6 |- ( F : NN --> ( 0 [,] +oo ) -> ( F ` 1 ) e. ( 0 [,] +oo ) )
40	34, 36, 39	esumsn 30127	. . . . 5 |- ( F : NN --> ( 0 [,] +oo ) -> Σ* x e. { 1 } ( F ` x ) = ( F ` 1 ) )
41	32, 40	syl5eq 2668	. . . 4 |- ( F : NN --> ( 0 [,] +oo ) -> Σ* k e. { 1 } ( F ` k ) = ( F ` 1 ) )
42		fzsn 12383	. . . . . 6 |- ( 1 e. ZZ -> ( 1 ... 1 ) = { 1 } )
43	35, 42	ax-mp 5	. . . . 5 |- ( 1 ... 1 ) = { 1 }
44		esumeq1 30096	. . . . 5 |- ( ( 1 ... 1 ) = { 1 } -> Σ* k e. ( 1 ... 1 ) ( F ` k ) = Σ* k e. { 1 } ( F ` k ) )
45	43, 44	ax-mp 5	. . . 4 |- Σ* k e. ( 1 ... 1 ) ( F ` k ) = Σ* k e. { 1 } ( F ` k )
46		seq1 12814	. . . . 5 |- ( 1 e. ZZ -> ( seq 1 ( +e , F ) ` 1 ) = ( F ` 1 ) )
47	35, 46	ax-mp 5	. . . 4 |- ( seq 1 ( +e , F ) ` 1 ) = ( F ` 1 )
48	41, 45, 47	3eqtr4g 2681	. . 3 |- ( F : NN --> ( 0 [,] +oo ) -> Σ* k e. ( 1 ... 1 ) ( F ` k ) = ( seq 1 ( +e , F ) ` 1 ) )
49		simpl 473	. . . . . . . . 9 |- ( ( n e. NN /\ F : NN --> ( 0 [,] +oo ) ) -> n e. NN )
50		nnuz 11723	. . . . . . . . 9 |- NN = ( ZZ>= ` 1 )
51	49, 50	syl6eleq 2711	. . . . . . . 8 |- ( ( n e. NN /\ F : NN --> ( 0 [,] +oo ) ) -> n e. ( ZZ>= ` 1 ) )
52		seqp1 12816	. . . . . . . 8 |- ( n e. ( ZZ>= ` 1 ) -> ( seq 1 ( +e , F ) ` ( n + 1 ) ) = ( ( seq 1 ( +e , F ) ` n ) +e ( F ` ( n + 1 ) ) ) )
53	51, 52	syl 17	. . . . . . 7 |- ( ( n e. NN /\ F : NN --> ( 0 [,] +oo ) ) -> ( seq 1 ( +e , F ) ` ( n + 1 ) ) = ( ( seq 1 ( +e , F ) ` n ) +e ( F ` ( n + 1 ) ) ) )
54	53	adantr 481	. . . . . 6 |- ( ( ( n e. NN /\ F : NN --> ( 0 [,] +oo ) ) /\ Σ* k e. ( 1 ... n ) ( F ` k ) = ( seq 1 ( +e , F ) ` n ) ) -> ( seq 1 ( +e , F ) ` ( n + 1 ) ) = ( ( seq 1 ( +e , F ) ` n ) +e ( F ` ( n + 1 ) ) ) )
55		simpr 477	. . . . . . 7 |- ( ( ( n e. NN /\ F : NN --> ( 0 [,] +oo ) ) /\ Σ* k e. ( 1 ... n ) ( F ` k ) = ( seq 1 ( +e , F ) ` n ) ) -> Σ* k e. ( 1 ... n ) ( F ` k ) = ( seq 1 ( +e , F ) ` n ) )
56	55	oveq1d 6665	. . . . . 6 |- ( ( ( n e. NN /\ F : NN --> ( 0 [,] +oo ) ) /\ Σ* k e. ( 1 ... n ) ( F ` k ) = ( seq 1 ( +e , F ) ` n ) ) -> (Σ* k e. ( 1 ... n ) ( F ` k ) +e ( F ` ( n + 1 ) ) ) = ( ( seq 1 ( +e , F ) ` n ) +e ( F ` ( n + 1 ) ) ) )
57		nfv 1843	. . . . . . . . . 10 |- F/ k n e. NN
58	57	nfci 2754	. . . . . . . . . . 11 |- F/_ k NN
59		nfcv 2764	. . . . . . . . . . 11 |- F/_ k ( 0 [,] +oo )
60	29, 58, 59	nff 6041	. . . . . . . . . 10 |- F/ k F : NN --> ( 0 [,] +oo )
61	57, 60	nfan 1828	. . . . . . . . 9 |- F/ k ( n e. NN /\ F : NN --> ( 0 [,] +oo ) )
62		fzsuc 12388	. . . . . . . . . 10 |- ( n e. ( ZZ>= ` 1 ) -> ( 1 ... ( n + 1 ) ) = ( ( 1 ... n ) u. { ( n + 1 ) } ) )
63	51, 62	syl 17	. . . . . . . . 9 |- ( ( n e. NN /\ F : NN --> ( 0 [,] +oo ) ) -> ( 1 ... ( n + 1 ) ) = ( ( 1 ... n ) u. { ( n + 1 ) } ) )
64	61, 63	esumeq1d 30097	. . . . . . . 8 |- ( ( n e. NN /\ F : NN --> ( 0 [,] +oo ) ) -> Σ* k e. ( 1 ... ( n + 1 ) ) ( F ` k ) = Σ* k e. ( ( 1 ... n ) u. { ( n + 1 ) } ) ( F ` k ) )
65		nfcv 2764	. . . . . . . . 9 |- F/_ k ( 1 ... n )
66		nfcv 2764	. . . . . . . . 9 |- F/_ k { ( n + 1 ) }
67		ovexd 6680	. . . . . . . . 9 |- ( ( n e. NN /\ F : NN --> ( 0 [,] +oo ) ) -> ( 1 ... n ) e. _V )
68		snex 4908	. . . . . . . . . 10 |- { ( n + 1 ) } e. _V
69	68	a1i 11	. . . . . . . . 9 |- ( ( n e. NN /\ F : NN --> ( 0 [,] +oo ) ) -> { ( n + 1 ) } e. _V )
70		fzp1disj 12399	. . . . . . . . . 10 |- ( ( 1 ... n ) i^i { ( n + 1 ) } ) = (/)
71	70	a1i 11	. . . . . . . . 9 |- ( ( n e. NN /\ F : NN --> ( 0 [,] +oo ) ) -> ( ( 1 ... n ) i^i { ( n + 1 ) } ) = (/) )
72		simplr 792	. . . . . . . . . 10 |- ( ( ( n e. NN /\ F : NN --> ( 0 [,] +oo ) ) /\ k e. ( 1 ... n ) ) -> F : NN --> ( 0 [,] +oo ) )
73		fzssnn 12385	. . . . . . . . . . . 12 |- ( 1 e. NN -> ( 1 ... n ) C_ NN )
74	37, 73	ax-mp 5	. . . . . . . . . . 11 |- ( 1 ... n ) C_ NN
75		simpr 477	. . . . . . . . . . 11 |- ( ( ( n e. NN /\ F : NN --> ( 0 [,] +oo ) ) /\ k e. ( 1 ... n ) ) -> k e. ( 1 ... n ) )
76	74, 75	sseldi 3601	. . . . . . . . . 10 |- ( ( ( n e. NN /\ F : NN --> ( 0 [,] +oo ) ) /\ k e. ( 1 ... n ) ) -> k e. NN )
77	72, 76	ffvelrnd 6360	. . . . . . . . 9 |- ( ( ( n e. NN /\ F : NN --> ( 0 [,] +oo ) ) /\ k e. ( 1 ... n ) ) -> ( F ` k ) e. ( 0 [,] +oo ) )
78		simplr 792	. . . . . . . . . 10 |- ( ( ( n e. NN /\ F : NN --> ( 0 [,] +oo ) ) /\ k e. { ( n + 1 ) } ) -> F : NN --> ( 0 [,] +oo ) )
79		simpr 477	. . . . . . . . . . . 12 |- ( ( ( n e. NN /\ F : NN --> ( 0 [,] +oo ) ) /\ k e. { ( n + 1 ) } ) -> k e. { ( n + 1 ) } )
80		velsn 4193	. . . . . . . . . . . 12 |- ( k e. { ( n + 1 ) } <-> k = ( n + 1 ) )
81	79, 80	sylib 208	. . . . . . . . . . 11 |- ( ( ( n e. NN /\ F : NN --> ( 0 [,] +oo ) ) /\ k e. { ( n + 1 ) } ) -> k = ( n + 1 ) )
82		simpll 790	. . . . . . . . . . . 12 |- ( ( ( n e. NN /\ F : NN --> ( 0 [,] +oo ) ) /\ k e. { ( n + 1 ) } ) -> n e. NN )
83	82	peano2nnd 11037	. . . . . . . . . . 11 |- ( ( ( n e. NN /\ F : NN --> ( 0 [,] +oo ) ) /\ k e. { ( n + 1 ) } ) -> ( n + 1 ) e. NN )
84	81, 83	eqeltrd 2701	. . . . . . . . . 10 |- ( ( ( n e. NN /\ F : NN --> ( 0 [,] +oo ) ) /\ k e. { ( n + 1 ) } ) -> k e. NN )
85	78, 84	ffvelrnd 6360	. . . . . . . . 9 |- ( ( ( n e. NN /\ F : NN --> ( 0 [,] +oo ) ) /\ k e. { ( n + 1 ) } ) -> ( F ` k ) e. ( 0 [,] +oo ) )
86	61, 65, 66, 67, 69, 71, 77, 85	esumsplit 30115	. . . . . . . 8 |- ( ( n e. NN /\ F : NN --> ( 0 [,] +oo ) ) -> Σ* k e. ( ( 1 ... n ) u. { ( n + 1 ) } ) ( F ` k ) = (Σ* k e. ( 1 ... n ) ( F ` k ) +eΣ* k e. { ( n + 1 ) } ( F ` k ) ) )
87		nfcv 2764	. . . . . . . . . . 11 |- F/_ x { ( n + 1 ) }
88	25, 87, 66, 28, 31	cbvesum 30104	. . . . . . . . . 10 |- Σ* k e. { ( n + 1 ) } ( F ` k ) = Σ* x e. { ( n + 1 ) } ( F ` x )
89		simpr 477	. . . . . . . . . . . 12 |- ( ( ( n e. NN /\ F : NN --> ( 0 [,] +oo ) ) /\ x = ( n + 1 ) ) -> x = ( n + 1 ) )
90	89	fveq2d 6195	. . . . . . . . . . 11 |- ( ( ( n e. NN /\ F : NN --> ( 0 [,] +oo ) ) /\ x = ( n + 1 ) ) -> ( F ` x ) = ( F ` ( n + 1 ) ) )
91	49	peano2nnd 11037	. . . . . . . . . . 11 |- ( ( n e. NN /\ F : NN --> ( 0 [,] +oo ) ) -> ( n + 1 ) e. NN )
92		simpr 477	. . . . . . . . . . . 12 |- ( ( n e. NN /\ F : NN --> ( 0 [,] +oo ) ) -> F : NN --> ( 0 [,] +oo ) )
93	92, 91	ffvelrnd 6360	. . . . . . . . . . 11 |- ( ( n e. NN /\ F : NN --> ( 0 [,] +oo ) ) -> ( F ` ( n + 1 ) ) e. ( 0 [,] +oo ) )
94	90, 91, 93	esumsn 30127	. . . . . . . . . 10 |- ( ( n e. NN /\ F : NN --> ( 0 [,] +oo ) ) -> Σ* x e. { ( n + 1 ) } ( F ` x ) = ( F ` ( n + 1 ) ) )
95	88, 94	syl5eq 2668	. . . . . . . . 9 |- ( ( n e. NN /\ F : NN --> ( 0 [,] +oo ) ) -> Σ* k e. { ( n + 1 ) } ( F ` k ) = ( F ` ( n + 1 ) ) )
96	95	oveq2d 6666	. . . . . . . 8 |- ( ( n e. NN /\ F : NN --> ( 0 [,] +oo ) ) -> (Σ* k e. ( 1 ... n ) ( F ` k ) +eΣ* k e. { ( n + 1 ) } ( F ` k ) ) = (Σ* k e. ( 1 ... n ) ( F ` k ) +e ( F ` ( n + 1 ) ) ) )
97	64, 86, 96	3eqtrrd 2661	. . . . . . 7 |- ( ( n e. NN /\ F : NN --> ( 0 [,] +oo ) ) -> (Σ* k e. ( 1 ... n ) ( F ` k ) +e ( F ` ( n + 1 ) ) ) = Σ* k e. ( 1 ... ( n + 1 ) ) ( F ` k ) )
98	97	adantr 481	. . . . . 6 |- ( ( ( n e. NN /\ F : NN --> ( 0 [,] +oo ) ) /\ Σ* k e. ( 1 ... n ) ( F ` k ) = ( seq 1 ( +e , F ) ` n ) ) -> (Σ* k e. ( 1 ... n ) ( F ` k ) +e ( F ` ( n + 1 ) ) ) = Σ* k e. ( 1 ... ( n + 1 ) ) ( F ` k ) )
99	54, 56, 98	3eqtr2rd 2663	. . . . 5 |- ( ( ( n e. NN /\ F : NN --> ( 0 [,] +oo ) ) /\ Σ* k e. ( 1 ... n ) ( F ` k ) = ( seq 1 ( +e , F ) ` n ) ) -> Σ* k e. ( 1 ... ( n + 1 ) ) ( F ` k ) = ( seq 1 ( +e , F ) ` ( n + 1 ) ) )
100	99	exp31 630	. . . 4 |- ( n e. NN -> ( F : NN --> ( 0 [,] +oo ) -> (Σ* k e. ( 1 ... n ) ( F ` k ) = ( seq 1 ( +e , F ) ` n ) -> Σ* k e. ( 1 ... ( n + 1 ) ) ( F ` k ) = ( seq 1 ( +e , F ) ` ( n + 1 ) ) ) ) )
101	100	a2d 29	. . 3 |- ( n e. NN -> ( ( F : NN --> ( 0 [,] +oo ) -> Σ* k e. ( 1 ... n ) ( F ` k ) = ( seq 1 ( +e , F ) ` n ) ) -> ( F : NN --> ( 0 [,] +oo ) -> Σ* k e. ( 1 ... ( n + 1 ) ) ( F ` k ) = ( seq 1 ( +e , F ) ` ( n + 1 ) ) ) ) )
102	6, 12, 18, 24, 48, 101	nnind 11038	. 2 |- ( N e. NN -> ( F : NN --> ( 0 [,] +oo ) -> Σ* k e. ( 1 ... N ) ( F ` k ) = ( seq 1 ( +e , F ) ` N ) ) )
103	102	impcom 446	1 |- ( ( F : NN --> ( 0 [,] +oo ) /\ N e. NN ) -> Σ* k e. ( 1 ... N ) ( F ` k ) = ( seq 1 ( +e , F ) ` N ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   F/_wnfc 2751   _Vcvv 3200   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   +oocpnf 10071   NNcn 11020   ZZcz 11377   ZZ>=cuz 11687   +ecxad 11944   [,]cicc 12178   ...cfz 12326   seqcseq 12801  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-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  ax-addf 10015  ax-mulf 10016

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-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  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-cda 8990  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-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  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-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-ordt 16161  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-ps 17200  df-tsr 17201  df-plusf 17241  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-subrg 18778  df-abv 18817  df-lmod 18865  df-scaf 18866  df-sra 19172  df-rgmod 19173  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-tmd 21876  df-tgp 21877  df-tsms 21930  df-trg 21963  df-xms 22125  df-ms 22126  df-tms 22127  df-nm 22387  df-ngp 22388  df-nrg 22390  df-nlm 22391  df-ii 22680  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303  df-esum 30090

This theorem is referenced by:  esumfsup  30132  esumsup  30151