Theorem climsup 14400

Description: A bounded monotonic sequence converges to the supremum of its range. Theorem 12-5.1 of [Gleason] p. 180. (Contributed by NM, 13-Mar-2005.) (Revised by Mario Carneiro, 10-Feb-2014.)

Hypotheses

Ref	Expression
climsup.1	|- Z = ( ZZ>= ` M )
climsup.2	|- ( ph -> M e. ZZ )
climsup.3	|- ( ph -> F : Z --> RR )
climsup.4	|- ( ( ph /\ k e. Z ) -> ( F ` k ) <_ ( F ` ( k + 1 ) ) )
climsup.5	|- ( ph -> E. x e. RR A. k e. Z ( F ` k ) <_ x )

Assertion

Ref	Expression
climsup	|- ( ph -> F ~~> sup ( ran F , RR , < ) )

Distinct variable groups:   x, k, F   ph, k   k, Z, x

Allowed substitution hints:   ph( x)   M( x, k)

Proof of Theorem climsup

Dummy variables j n y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		climsup.3	. . . . . . . . . 10 |- ( ph -> F : Z --> RR )
2		frn 6053	. . . . . . . . . 10 |- ( F : Z --> RR -> ran F C_ RR )
3	1, 2	syl 17	. . . . . . . . 9 |- ( ph -> ran F C_ RR )
4		ffn 6045	. . . . . . . . . . . 12 |- ( F : Z --> RR -> F Fn Z )
5	1, 4	syl 17	. . . . . . . . . . 11 |- ( ph -> F Fn Z )
6		climsup.2	. . . . . . . . . . . . 13 |- ( ph -> M e. ZZ )
7		uzid 11702	. . . . . . . . . . . . 13 |- ( M e. ZZ -> M e. ( ZZ>= ` M ) )
8	6, 7	syl 17	. . . . . . . . . . . 12 |- ( ph -> M e. ( ZZ>= ` M ) )
9		climsup.1	. . . . . . . . . . . 12 |- Z = ( ZZ>= ` M )
10	8, 9	syl6eleqr 2712	. . . . . . . . . . 11 |- ( ph -> M e. Z )
11		fnfvelrn 6356	. . . . . . . . . . 11 |- ( ( F Fn Z /\ M e. Z ) -> ( F ` M ) e. ran F )
12	5, 10, 11	syl2anc 693	. . . . . . . . . 10 |- ( ph -> ( F ` M ) e. ran F )
13		ne0i 3921	. . . . . . . . . 10 |- ( ( F ` M ) e. ran F -> ran F =/= (/) )
14	12, 13	syl 17	. . . . . . . . 9 |- ( ph -> ran F =/= (/) )
15		climsup.5	. . . . . . . . . 10 |- ( ph -> E. x e. RR A. k e. Z ( F ` k ) <_ x )
16		breq1 4656	. . . . . . . . . . . . 13 |- ( y = ( F ` k ) -> ( y <_ x <-> ( F ` k ) <_ x ) )
17	16	ralrn 6362	. . . . . . . . . . . 12 |- ( F Fn Z -> ( A. y e. ran F y <_ x <-> A. k e. Z ( F ` k ) <_ x ) )
18	17	rexbidv 3052	. . . . . . . . . . 11 |- ( F Fn Z -> ( E. x e. RR A. y e. ran F y <_ x <-> E. x e. RR A. k e. Z ( F ` k ) <_ x ) )
19	5, 18	syl 17	. . . . . . . . . 10 |- ( ph -> ( E. x e. RR A. y e. ran F y <_ x <-> E. x e. RR A. k e. Z ( F ` k ) <_ x ) )
20	15, 19	mpbird 247	. . . . . . . . 9 |- ( ph -> E. x e. RR A. y e. ran F y <_ x )
21	3, 14, 20	3jca 1242	. . . . . . . 8 |- ( ph -> ( ran F C_ RR /\ ran F =/= (/) /\ E. x e. RR A. y e. ran F y <_ x ) )
22		suprcl 10983	. . . . . . . 8 |- ( ( ran F C_ RR /\ ran F =/= (/) /\ E. x e. RR A. y e. ran F y <_ x ) -> sup ( ran F , RR , < ) e. RR )
23	21, 22	syl 17	. . . . . . 7 |- ( ph -> sup ( ran F , RR , < ) e. RR )
24		ltsubrp 11866	. . . . . . 7 |- ( ( sup ( ran F , RR , < ) e. RR /\ y e. RR+ ) -> ( sup ( ran F , RR , < ) - y ) < sup ( ran F , RR , < ) )
25	23, 24	sylan 488	. . . . . 6 |- ( ( ph /\ y e. RR+ ) -> ( sup ( ran F , RR , < ) - y ) < sup ( ran F , RR , < ) )
26	21	adantr 481	. . . . . . 7 |- ( ( ph /\ y e. RR+ ) -> ( ran F C_ RR /\ ran F =/= (/) /\ E. x e. RR A. y e. ran F y <_ x ) )
27		rpre 11839	. . . . . . . 8 |- ( y e. RR+ -> y e. RR )
28		resubcl 10345	. . . . . . . 8 |- ( ( sup ( ran F , RR , < ) e. RR /\ y e. RR ) -> ( sup ( ran F , RR , < ) - y ) e. RR )
29	23, 27, 28	syl2an 494	. . . . . . 7 |- ( ( ph /\ y e. RR+ ) -> ( sup ( ran F , RR , < ) - y ) e. RR )
30		suprlub 10987	. . . . . . 7 |- ( ( ( ran F C_ RR /\ ran F =/= (/) /\ E. x e. RR A. y e. ran F y <_ x ) /\ ( sup ( ran F , RR , < ) - y ) e. RR ) -> ( ( sup ( ran F , RR , < ) - y ) < sup ( ran F , RR , < ) <-> E. k e. ran F ( sup ( ran F , RR , < ) - y ) < k ) )
31	26, 29, 30	syl2anc 693	. . . . . 6 |- ( ( ph /\ y e. RR+ ) -> ( ( sup ( ran F , RR , < ) - y ) < sup ( ran F , RR , < ) <-> E. k e. ran F ( sup ( ran F , RR , < ) - y ) < k ) )
32	25, 31	mpbid 222	. . . . 5 |- ( ( ph /\ y e. RR+ ) -> E. k e. ran F ( sup ( ran F , RR , < ) - y ) < k )
33		breq2 4657	. . . . . . . 8 |- ( k = ( F ` j ) -> ( ( sup ( ran F , RR , < ) - y ) < k <-> ( sup ( ran F , RR , < ) - y ) < ( F ` j ) ) )
34	33	rexrn 6361	. . . . . . 7 |- ( F Fn Z -> ( E. k e. ran F ( sup ( ran F , RR , < ) - y ) < k <-> E. j e. Z ( sup ( ran F , RR , < ) - y ) < ( F ` j ) ) )
35	5, 34	syl 17	. . . . . 6 |- ( ph -> ( E. k e. ran F ( sup ( ran F , RR , < ) - y ) < k <-> E. j e. Z ( sup ( ran F , RR , < ) - y ) < ( F ` j ) ) )
36	35	biimpa 501	. . . . 5 |- ( ( ph /\ E. k e. ran F ( sup ( ran F , RR , < ) - y ) < k ) -> E. j e. Z ( sup ( ran F , RR , < ) - y ) < ( F ` j ) )
37	32, 36	syldan 487	. . . 4 |- ( ( ph /\ y e. RR+ ) -> E. j e. Z ( sup ( ran F , RR , < ) - y ) < ( F ` j ) )
38		ffvelrn 6357	. . . . . . . . . . . 12 |- ( ( F : Z --> RR /\ j e. Z ) -> ( F ` j ) e. RR )
39	1, 38	sylan 488	. . . . . . . . . . 11 |- ( ( ph /\ j e. Z ) -> ( F ` j ) e. RR )
40	39	ad2ant2r 783	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( F ` j ) e. RR )
41	1	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ y e. RR+ ) -> F : Z --> RR )
42	9	uztrn2 11705	. . . . . . . . . . 11 |- ( ( j e. Z /\ k e. ( ZZ>= ` j ) ) -> k e. Z )
43		ffvelrn 6357	. . . . . . . . . . 11 |- ( ( F : Z --> RR /\ k e. Z ) -> ( F ` k ) e. RR )
44	41, 42, 43	syl2an 494	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( F ` k ) e. RR )
45	23	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> sup ( ran F , RR , < ) e. RR )
46		simprr 796	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> k e. ( ZZ>= ` j ) )
47		fzssuz 12382	. . . . . . . . . . . . . 14 |- ( j ... k ) C_ ( ZZ>= ` j )
48		uzss 11708	. . . . . . . . . . . . . . . . 17 |- ( j e. ( ZZ>= ` M ) -> ( ZZ>= ` j ) C_ ( ZZ>= ` M ) )
49	48, 9	syl6sseqr 3652	. . . . . . . . . . . . . . . 16 |- ( j e. ( ZZ>= ` M ) -> ( ZZ>= ` j ) C_ Z )
50	49, 9	eleq2s 2719	. . . . . . . . . . . . . . 15 |- ( j e. Z -> ( ZZ>= ` j ) C_ Z )
51	50	ad2antrl 764	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ZZ>= ` j ) C_ Z )
52	47, 51	syl5ss 3614	. . . . . . . . . . . . 13 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( j ... k ) C_ Z )
53		ffvelrn 6357	. . . . . . . . . . . . . . . 16 |- ( ( F : Z --> RR /\ n e. Z ) -> ( F ` n ) e. RR )
54	53	ralrimiva 2966	. . . . . . . . . . . . . . 15 |- ( F : Z --> RR -> A. n e. Z ( F ` n ) e. RR )
55	1, 54	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> A. n e. Z ( F ` n ) e. RR )
56	55	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> A. n e. Z ( F ` n ) e. RR )
57		ssralv 3666	. . . . . . . . . . . . 13 |- ( ( j ... k ) C_ Z -> ( A. n e. Z ( F ` n ) e. RR -> A. n e. ( j ... k ) ( F ` n ) e. RR ) )
58	52, 56, 57	sylc 65	. . . . . . . . . . . 12 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> A. n e. ( j ... k ) ( F ` n ) e. RR )
59	58	r19.21bi 2932	. . . . . . . . . . 11 |- ( ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) /\ n e. ( j ... k ) ) -> ( F ` n ) e. RR )
60		fzssuz 12382	. . . . . . . . . . . . . 14 |- ( j ... ( k - 1 ) ) C_ ( ZZ>= ` j )
61	60, 51	syl5ss 3614	. . . . . . . . . . . . 13 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( j ... ( k - 1 ) ) C_ Z )
62	61	sselda 3603	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) /\ n e. ( j ... ( k - 1 ) ) ) -> n e. Z )
63		climsup.4	. . . . . . . . . . . . . . 15 |- ( ( ph /\ k e. Z ) -> ( F ` k ) <_ ( F ` ( k + 1 ) ) )
64	63	ralrimiva 2966	. . . . . . . . . . . . . 14 |- ( ph -> A. k e. Z ( F ` k ) <_ ( F ` ( k + 1 ) ) )
65	64	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> A. k e. Z ( F ` k ) <_ ( F ` ( k + 1 ) ) )
66		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( k = n -> ( F ` k ) = ( F ` n ) )
67		oveq1 6657	. . . . . . . . . . . . . . . 16 |- ( k = n -> ( k + 1 ) = ( n + 1 ) )
68	67	fveq2d 6195	. . . . . . . . . . . . . . 15 |- ( k = n -> ( F ` ( k + 1 ) ) = ( F ` ( n + 1 ) ) )
69	66, 68	breq12d 4666	. . . . . . . . . . . . . 14 |- ( k = n -> ( ( F ` k ) <_ ( F ` ( k + 1 ) ) <-> ( F ` n ) <_ ( F ` ( n + 1 ) ) ) )
70	69	rspccva 3308	. . . . . . . . . . . . 13 |- ( ( A. k e. Z ( F ` k ) <_ ( F ` ( k + 1 ) ) /\ n e. Z ) -> ( F ` n ) <_ ( F ` ( n + 1 ) ) )
71	65, 70	sylan 488	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) /\ n e. Z ) -> ( F ` n ) <_ ( F ` ( n + 1 ) ) )
72	62, 71	syldan 487	. . . . . . . . . . 11 |- ( ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) /\ n e. ( j ... ( k - 1 ) ) ) -> ( F ` n ) <_ ( F ` ( n + 1 ) ) )
73	46, 59, 72	monoord 12831	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( F ` j ) <_ ( F ` k ) )
74	40, 44, 45, 73	lesub2dd 10644	. . . . . . . . 9 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( sup ( ran F , RR , < ) - ( F ` k ) ) <_ ( sup ( ran F , RR , < ) - ( F ` j ) ) )
75	45, 44	resubcld 10458	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( sup ( ran F , RR , < ) - ( F ` k ) ) e. RR )
76	45, 40	resubcld 10458	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( sup ( ran F , RR , < ) - ( F ` j ) ) e. RR )
77	27	ad2antlr 763	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> y e. RR )
78		lelttr 10128	. . . . . . . . . 10 |- ( ( ( sup ( ran F , RR , < ) - ( F ` k ) ) e. RR /\ ( sup ( ran F , RR , < ) - ( F ` j ) ) e. RR /\ y e. RR ) -> ( ( ( sup ( ran F , RR , < ) - ( F ` k ) ) <_ ( sup ( ran F , RR , < ) - ( F ` j ) ) /\ ( sup ( ran F , RR , < ) - ( F ` j ) ) < y ) -> ( sup ( ran F , RR , < ) - ( F ` k ) ) < y ) )
79	75, 76, 77, 78	syl3anc 1326	. . . . . . . . 9 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ( ( sup ( ran F , RR , < ) - ( F ` k ) ) <_ ( sup ( ran F , RR , < ) - ( F ` j ) ) /\ ( sup ( ran F , RR , < ) - ( F ` j ) ) < y ) -> ( sup ( ran F , RR , < ) - ( F ` k ) ) < y ) )
80	74, 79	mpand 711	. . . . . . . 8 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ( sup ( ran F , RR , < ) - ( F ` j ) ) < y -> ( sup ( ran F , RR , < ) - ( F ` k ) ) < y ) )
81		ltsub23 10508	. . . . . . . . 9 |- ( ( sup ( ran F , RR , < ) e. RR /\ y e. RR /\ ( F ` j ) e. RR ) -> ( ( sup ( ran F , RR , < ) - y ) < ( F ` j ) <-> ( sup ( ran F , RR , < ) - ( F ` j ) ) < y ) )
82	45, 77, 40, 81	syl3anc 1326	. . . . . . . 8 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ( sup ( ran F , RR , < ) - y ) < ( F ` j ) <-> ( sup ( ran F , RR , < ) - ( F ` j ) ) < y ) )
83	21	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ran F C_ RR /\ ran F =/= (/) /\ E. x e. RR A. y e. ran F y <_ x ) )
84	5	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ y e. RR+ ) -> F Fn Z )
85		fnfvelrn 6356	. . . . . . . . . . . 12 |- ( ( F Fn Z /\ k e. Z ) -> ( F ` k ) e. ran F )
86	84, 42, 85	syl2an 494	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( F ` k ) e. ran F )
87		suprub 10984	. . . . . . . . . . 11 |- ( ( ( ran F C_ RR /\ ran F =/= (/) /\ E. x e. RR A. y e. ran F y <_ x ) /\ ( F ` k ) e. ran F ) -> ( F ` k ) <_ sup ( ran F , RR , < ) )
88	83, 86, 87	syl2anc 693	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( F ` k ) <_ sup ( ran F , RR , < ) )
89	44, 45, 88	abssuble0d 14171	. . . . . . . . 9 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( abs ` ( ( F ` k ) - sup ( ran F , RR , < ) ) ) = ( sup ( ran F , RR , < ) - ( F ` k ) ) )
90	89	breq1d 4663	. . . . . . . 8 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ( abs ` ( ( F ` k ) - sup ( ran F , RR , < ) ) ) < y <-> ( sup ( ran F , RR , < ) - ( F ` k ) ) < y ) )
91	80, 82, 90	3imtr4d 283	. . . . . . 7 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ( sup ( ran F , RR , < ) - y ) < ( F ` j ) -> ( abs ` ( ( F ` k ) - sup ( ran F , RR , < ) ) ) < y ) )
92	91	anassrs 680	. . . . . 6 |- ( ( ( ( ph /\ y e. RR+ ) /\ j e. Z ) /\ k e. ( ZZ>= ` j ) ) -> ( ( sup ( ran F , RR , < ) - y ) < ( F ` j ) -> ( abs ` ( ( F ` k ) - sup ( ran F , RR , < ) ) ) < y ) )
93	92	ralrimdva 2969	. . . . 5 |- ( ( ( ph /\ y e. RR+ ) /\ j e. Z ) -> ( ( sup ( ran F , RR , < ) - y ) < ( F ` j ) -> A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - sup ( ran F , RR , < ) ) ) < y ) )
94	93	reximdva 3017	. . . 4 |- ( ( ph /\ y e. RR+ ) -> ( E. j e. Z ( sup ( ran F , RR , < ) - y ) < ( F ` j ) -> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - sup ( ran F , RR , < ) ) ) < y ) )
95	37, 94	mpd 15	. . 3 |- ( ( ph /\ y e. RR+ ) -> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - sup ( ran F , RR , < ) ) ) < y )
96	95	ralrimiva 2966	. 2 |- ( ph -> A. y e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - sup ( ran F , RR , < ) ) ) < y )
97		fvex 6201	. . . . 5 |- ( ZZ>= ` M ) e. _V
98	9, 97	eqeltri 2697	. . . 4 |- Z e. _V
99		fex 6490	. . . 4 |- ( ( F : Z --> RR /\ Z e. _V ) -> F e. _V )
100	1, 98, 99	sylancl 694	. . 3 |- ( ph -> F e. _V )
101		eqidd 2623	. . 3 |- ( ( ph /\ k e. Z ) -> ( F ` k ) = ( F ` k ) )
102	23	recnd 10068	. . 3 |- ( ph -> sup ( ran F , RR , < ) e. CC )
103	1, 43	sylan 488	. . . 4 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR )
104	103	recnd 10068	. . 3 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
105	9, 6, 100, 101, 102, 104	clim2c 14236	. 2 |- ( ph -> ( F ~~> sup ( ran F , RR , < ) <-> A. y e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - sup ( ran F , RR , < ) ) ) < y ) )
106	96, 105	mpbird 247	1 |- ( ph -> F ~~> sup ( ran F , RR , < ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   (/)c0 3915   class class class wbr 4653   ran crn 5115   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   supcsup 8346   RRcr 9935   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   - cmin 10266   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832   ...cfz 12326   abscabs 13974   ~~> cli 14215

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-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-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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  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-fz 12327  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219

This theorem is referenced by:  isumsup2  14578  climcnds  14583  itg1climres  23481  itg2monolem1  23517  itg2i1fseq  23522  itg2i1fseq2  23523  emcllem6  24727  lmdvg  29999  esumpcvgval  30140  meaiuninclem  40697