Theorem climinf 37684

Description: A bounded monotonic non increasing sequence converges to the infimum of its range. (Contributed by Glauco Siliprandi, 29-Jun-2017.) (Revised by AV, 15-Sep-2020.)

Hypotheses

Ref	Expression
climinf.3	|- Z = ( ZZ>= ` M )
climinf.4	|- ( ph -> M e. ZZ )
climinf.5	|- ( ph -> F : Z --> RR )
climinf.6	|- ( ( ph /\ k e. Z ) -> ( F ` ( k + 1 ) ) <_ ( F ` k ) )
climinf.7	|- ( ph -> E. x e. RR A. k e. Z x <_ ( F ` k ) )

Assertion

Ref	Expression
climinf	|- ( ph -> F ~~> inf ( ran F , RR , < ) )

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

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

Proof of Theorem climinf

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

Step	Hyp	Ref	Expression
1		climinf.5	. . . . . . . . . . . 12 |- ( ph -> F : Z --> RR )
2		frn 5735	. . . . . . . . . . . 12 |- ( F : Z --> RR -> ran F C_ RR )
3	1, 2	syl 17	. . . . . . . . . . 11 |- ( ph -> ran F C_ RR )
4		ffn 5728	. . . . . . . . . . . . . 14 |- ( F : Z --> RR -> F Fn Z )
5	1, 4	syl 17	. . . . . . . . . . . . 13 |- ( ph -> F Fn Z )
6		climinf.4	. . . . . . . . . . . . . . 15 |- ( ph -> M e. ZZ )
7		uzid 11173	. . . . . . . . . . . . . . 15 |- ( M e. ZZ -> M e. ( ZZ>= ` M ) )
8	6, 7	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> M e. ( ZZ>= ` M ) )
9		climinf.3	. . . . . . . . . . . . . 14 |- Z = ( ZZ>= ` M )
10	8, 9	syl6eleqr 2540	. . . . . . . . . . . . 13 |- ( ph -> M e. Z )
11		fnfvelrn 6019	. . . . . . . . . . . . 13 |- ( ( F Fn Z /\ M e. Z ) -> ( F ` M ) e. ran F )
12	5, 10, 11	syl2anc 667	. . . . . . . . . . . 12 |- ( ph -> ( F ` M ) e. ran F )
13		ne0i 3737	. . . . . . . . . . . 12 |- ( ( F ` M ) e. ran F -> ran F =/= (/) )
14	12, 13	syl 17	. . . . . . . . . . 11 |- ( ph -> ran F =/= (/) )
15		climinf.7	. . . . . . . . . . . 12 |- ( ph -> E. x e. RR A. k e. Z x <_ ( F ` k ) )
16		breq2 4406	. . . . . . . . . . . . . . 15 |- ( y = ( F ` k ) -> ( x <_ y <-> x <_ ( F ` k ) ) )
17	16	ralrn 6025	. . . . . . . . . . . . . 14 |- ( F Fn Z -> ( A. y e. ran F x <_ y <-> A. k e. Z x <_ ( F ` k ) ) )
18	17	rexbidv 2901	. . . . . . . . . . . . 13 |- ( F Fn Z -> ( E. x e. RR A. y e. ran F x <_ y <-> E. x e. RR A. k e. Z x <_ ( F ` k ) ) )
19	5, 18	syl 17	. . . . . . . . . . . 12 |- ( ph -> ( E. x e. RR A. y e. ran F x <_ y <-> E. x e. RR A. k e. Z x <_ ( F ` k ) ) )
20	15, 19	mpbird 236	. . . . . . . . . . 11 |- ( ph -> E. x e. RR A. y e. ran F x <_ y )
21	3, 14, 20	3jca 1188	. . . . . . . . . 10 |- ( ph -> ( ran F C_ RR /\ ran F =/= (/) /\ E. x e. RR A. y e. ran F x <_ y ) )
22	21	adantr 467	. . . . . . . . 9 |- ( ( ph /\ y e. RR+ ) -> ( ran F C_ RR /\ ran F =/= (/) /\ E. x e. RR A. y e. ran F x <_ y ) )
23		infrecl 10590	. . . . . . . . 9 |- ( ( ran F C_ RR /\ ran F =/= (/) /\ E. x e. RR A. y e. ran F x <_ y ) -> inf ( ran F , RR , < ) e. RR )
24	22, 23	syl 17	. . . . . . . 8 |- ( ( ph /\ y e. RR+ ) -> inf ( ran F , RR , < ) e. RR )
25		simpr 463	. . . . . . . 8 |- ( ( ph /\ y e. RR+ ) -> y e. RR+ )
26	24, 25	ltaddrpd 11371	. . . . . . 7 |- ( ( ph /\ y e. RR+ ) -> inf ( ran F , RR , < ) < (inf ( ran F , RR , < ) + y ) )
27		rpre 11308	. . . . . . . . . 10 |- ( y e. RR+ -> y e. RR )
28	27	adantl 468	. . . . . . . . 9 |- ( ( ph /\ y e. RR+ ) -> y e. RR )
29	24, 28	readdcld 9670	. . . . . . . 8 |- ( ( ph /\ y e. RR+ ) -> (inf ( ran F , RR , < ) + y ) e. RR )
30		infrglb 37668	. . . . . . . 8 |- ( ( ( ran F C_ RR /\ ran F =/= (/) /\ E. x e. RR A. y e. ran F x <_ y ) /\ (inf ( ran F , RR , < ) + y ) e. RR ) -> (inf ( ran F , RR , < ) < (inf ( ran F , RR , < ) + y ) <-> E. k e. ran F k < (inf ( ran F , RR , < ) + y ) ) )
31	22, 29, 30	syl2anc 667	. . . . . . 7 |- ( ( ph /\ y e. RR+ ) -> (inf ( ran F , RR , < ) < (inf ( ran F , RR , < ) + y ) <-> E. k e. ran F k < (inf ( ran F , RR , < ) + y ) ) )
32	26, 31	mpbid 214	. . . . . 6 |- ( ( ph /\ y e. RR+ ) -> E. k e. ran F k < (inf ( ran F , RR , < ) + y ) )
33	3	sselda 3432	. . . . . . . . . . 11 |- ( ( ph /\ k e. ran F ) -> k e. RR )
34	33	adantlr 721	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ k e. ran F ) -> k e. RR )
35	24	adantr 467	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. RR+ ) /\ k e. ran F ) -> inf ( ran F , RR , < ) e. RR )
36	27	ad2antlr 733	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. RR+ ) /\ k e. ran F ) -> y e. RR )
37	35, 36	readdcld 9670	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ k e. ran F ) -> (inf ( ran F , RR , < ) + y ) e. RR )
38	34, 37, 36	ltsub1d 10222	. . . . . . . . 9 |- ( ( ( ph /\ y e. RR+ ) /\ k e. ran F ) -> ( k < (inf ( ran F , RR , < ) + y ) <-> ( k - y ) < ( (inf ( ran F , RR , < ) + y ) - y ) ) )
39	3, 14, 20, 23	syl3anc 1268	. . . . . . . . . . . . 13 |- ( ph -> inf ( ran F , RR , < ) e. RR )
40	39	recnd 9669	. . . . . . . . . . . 12 |- ( ph -> inf ( ran F , RR , < ) e. CC )
41	40	ad2antrr 732	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. RR+ ) /\ k e. ran F ) -> inf ( ran F , RR , < ) e. CC )
42	36	recnd 9669	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. RR+ ) /\ k e. ran F ) -> y e. CC )
43	41, 42	pncand 9987	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ k e. ran F ) -> ( (inf ( ran F , RR , < ) + y ) - y ) = inf ( ran F , RR , < ) )
44	43	breq2d 4414	. . . . . . . . 9 |- ( ( ( ph /\ y e. RR+ ) /\ k e. ran F ) -> ( ( k - y ) < ( (inf ( ran F , RR , < ) + y ) - y ) <-> ( k - y ) < inf ( ran F , RR , < ) ) )
45	38, 44	bitrd 257	. . . . . . . 8 |- ( ( ( ph /\ y e. RR+ ) /\ k e. ran F ) -> ( k < (inf ( ran F , RR , < ) + y ) <-> ( k - y ) < inf ( ran F , RR , < ) ) )
46	45	biimpd 211	. . . . . . 7 |- ( ( ( ph /\ y e. RR+ ) /\ k e. ran F ) -> ( k < (inf ( ran F , RR , < ) + y ) -> ( k - y ) < inf ( ran F , RR , < ) ) )
47	46	reximdva 2862	. . . . . 6 |- ( ( ph /\ y e. RR+ ) -> ( E. k e. ran F k < (inf ( ran F , RR , < ) + y ) -> E. k e. ran F ( k - y ) < inf ( ran F , RR , < ) ) )
48	32, 47	mpd 15	. . . . 5 |- ( ( ph /\ y e. RR+ ) -> E. k e. ran F ( k - y ) < inf ( ran F , RR , < ) )
49		oveq1 6297	. . . . . . . . 9 |- ( k = ( F ` j ) -> ( k - y ) = ( ( F ` j ) - y ) )
50	49	breq1d 4412	. . . . . . . 8 |- ( k = ( F ` j ) -> ( ( k - y ) < inf ( ran F , RR , < ) <-> ( ( F ` j ) - y ) < inf ( ran F , RR , < ) ) )
51	50	rexrn 6024	. . . . . . 7 |- ( F Fn Z -> ( E. k e. ran F ( k - y ) < inf ( ran F , RR , < ) <-> E. j e. Z ( ( F ` j ) - y ) < inf ( ran F , RR , < ) ) )
52	5, 51	syl 17	. . . . . 6 |- ( ph -> ( E. k e. ran F ( k - y ) < inf ( ran F , RR , < ) <-> E. j e. Z ( ( F ` j ) - y ) < inf ( ran F , RR , < ) ) )
53	52	biimpa 487	. . . . 5 |- ( ( ph /\ E. k e. ran F ( k - y ) < inf ( ran F , RR , < ) ) -> E. j e. Z ( ( F ` j ) - y ) < inf ( ran F , RR , < ) )
54	48, 53	syldan 473	. . . 4 |- ( ( ph /\ y e. RR+ ) -> E. j e. Z ( ( F ` j ) - y ) < inf ( ran F , RR , < ) )
55	1	adantr 467	. . . . . . . . . . 11 |- ( ( ph /\ y e. RR+ ) -> F : Z --> RR )
56	9	uztrn2 11176	. . . . . . . . . . 11 |- ( ( j e. Z /\ k e. ( ZZ>= ` j ) ) -> k e. Z )
57		ffvelrn 6020	. . . . . . . . . . 11 |- ( ( F : Z --> RR /\ k e. Z ) -> ( F ` k ) e. RR )
58	55, 56, 57	syl2an 480	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( F ` k ) e. RR )
59		simpl 459	. . . . . . . . . . 11 |- ( ( j e. Z /\ k e. ( ZZ>= ` j ) ) -> j e. Z )
60		ffvelrn 6020	. . . . . . . . . . 11 |- ( ( F : Z --> RR /\ j e. Z ) -> ( F ` j ) e. RR )
61	55, 59, 60	syl2an 480	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( F ` j ) e. RR )
62	39	ad2antrr 732	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> inf ( ran F , RR , < ) e. RR )
63		simprr 766	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> k e. ( ZZ>= ` j ) )
64		fzssuz 11839	. . . . . . . . . . . . . 14 |- ( j ... k ) C_ ( ZZ>= ` j )
65		uzss 11179	. . . . . . . . . . . . . . . . 17 |- ( j e. ( ZZ>= ` M ) -> ( ZZ>= ` j ) C_ ( ZZ>= ` M ) )
66	65, 9	syl6sseqr 3479	. . . . . . . . . . . . . . . 16 |- ( j e. ( ZZ>= ` M ) -> ( ZZ>= ` j ) C_ Z )
67	66, 9	eleq2s 2547	. . . . . . . . . . . . . . 15 |- ( j e. Z -> ( ZZ>= ` j ) C_ Z )
68	67	ad2antrl 734	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ZZ>= ` j ) C_ Z )
69	64, 68	syl5ss 3443	. . . . . . . . . . . . 13 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( j ... k ) C_ Z )
70		ffvelrn 6020	. . . . . . . . . . . . . . . 16 |- ( ( F : Z --> RR /\ n e. Z ) -> ( F ` n ) e. RR )
71	70	ralrimiva 2802	. . . . . . . . . . . . . . 15 |- ( F : Z --> RR -> A. n e. Z ( F ` n ) e. RR )
72	1, 71	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> A. n e. Z ( F ` n ) e. RR )
73	72	ad2antrr 732	. . . . . . . . . . . . 13 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> A. n e. Z ( F ` n ) e. RR )
74		ssralv 3493	. . . . . . . . . . . . 13 |- ( ( j ... k ) C_ Z -> ( A. n e. Z ( F ` n ) e. RR -> A. n e. ( j ... k ) ( F ` n ) e. RR ) )
75	69, 73, 74	sylc 62	. . . . . . . . . . . 12 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> A. n e. ( j ... k ) ( F ` n ) e. RR )
76	75	r19.21bi 2757	. . . . . . . . . . 11 |- ( ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) /\ n e. ( j ... k ) ) -> ( F ` n ) e. RR )
77		fzssuz 11839	. . . . . . . . . . . . . 14 |- ( j ... ( k - 1 ) ) C_ ( ZZ>= ` j )
78	77, 68	syl5ss 3443	. . . . . . . . . . . . 13 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( j ... ( k - 1 ) ) C_ Z )
79	78	sselda 3432	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) /\ n e. ( j ... ( k - 1 ) ) ) -> n e. Z )
80		climinf.6	. . . . . . . . . . . . . . 15 |- ( ( ph /\ k e. Z ) -> ( F ` ( k + 1 ) ) <_ ( F ` k ) )
81	80	ralrimiva 2802	. . . . . . . . . . . . . 14 |- ( ph -> A. k e. Z ( F ` ( k + 1 ) ) <_ ( F ` k ) )
82	81	ad2antrr 732	. . . . . . . . . . . . 13 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> A. k e. Z ( F ` ( k + 1 ) ) <_ ( F ` k ) )
83		oveq1 6297	. . . . . . . . . . . . . . . 16 |- ( k = n -> ( k + 1 ) = ( n + 1 ) )
84	83	fveq2d 5869	. . . . . . . . . . . . . . 15 |- ( k = n -> ( F ` ( k + 1 ) ) = ( F ` ( n + 1 ) ) )
85		fveq2 5865	. . . . . . . . . . . . . . 15 |- ( k = n -> ( F ` k ) = ( F ` n ) )
86	84, 85	breq12d 4415	. . . . . . . . . . . . . 14 |- ( k = n -> ( ( F ` ( k + 1 ) ) <_ ( F ` k ) <-> ( F ` ( n + 1 ) ) <_ ( F ` n ) ) )
87	86	rspccva 3149	. . . . . . . . . . . . 13 |- ( ( A. k e. Z ( F ` ( k + 1 ) ) <_ ( F ` k ) /\ n e. Z ) -> ( F ` ( n + 1 ) ) <_ ( F ` n ) )
88	82, 87	sylan 474	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) /\ n e. Z ) -> ( F ` ( n + 1 ) ) <_ ( F ` n ) )
89	79, 88	syldan 473	. . . . . . . . . . 11 |- ( ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) /\ n e. ( j ... ( k - 1 ) ) ) -> ( F ` ( n + 1 ) ) <_ ( F ` n ) )
90	63, 76, 89	monoord2 12244	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( F ` k ) <_ ( F ` j ) )
91	58, 61, 62, 90	lesub1dd 10229	. . . . . . . . 9 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ( F ` k ) - inf ( ran F , RR , < ) ) <_ ( ( F ` j ) - inf ( ran F , RR , < ) ) )
92	58, 62	resubcld 10047	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ( F ` k ) - inf ( ran F , RR , < ) ) e. RR )
93	61, 62	resubcld 10047	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ( F ` j ) - inf ( ran F , RR , < ) ) e. RR )
94	27	ad2antlr 733	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> y e. RR )
95		lelttr 9724	. . . . . . . . . 10 |- ( ( ( ( F ` k ) - inf ( ran F , RR , < ) ) e. RR /\ ( ( F ` j ) - inf ( ran F , RR , < ) ) e. RR /\ y e. RR ) -> ( ( ( ( F ` k ) - inf ( ran F , RR , < ) ) <_ ( ( F ` j ) - inf ( ran F , RR , < ) ) /\ ( ( F ` j ) - inf ( ran F , RR , < ) ) < y ) -> ( ( F ` k ) - inf ( ran F , RR , < ) ) < y ) )
96	92, 93, 94, 95	syl3anc 1268	. . . . . . . . 9 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ( ( ( F ` k ) - inf ( ran F , RR , < ) ) <_ ( ( F ` j ) - inf ( ran F , RR , < ) ) /\ ( ( F ` j ) - inf ( ran F , RR , < ) ) < y ) -> ( ( F ` k ) - inf ( ran F , RR , < ) ) < y ) )
97	91, 96	mpand 681	. . . . . . . 8 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ( ( F ` j ) - inf ( ran F , RR , < ) ) < y -> ( ( F ` k ) - inf ( ran F , RR , < ) ) < y ) )
98		ltsub23 10094	. . . . . . . . 9 |- ( ( ( F ` j ) e. RR /\ y e. RR /\ inf ( ran F , RR , < ) e. RR ) -> ( ( ( F ` j ) - y ) < inf ( ran F , RR , < ) <-> ( ( F ` j ) - inf ( ran F , RR , < ) ) < y ) )
99	61, 94, 62, 98	syl3anc 1268	. . . . . . . 8 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ( ( F ` j ) - y ) < inf ( ran F , RR , < ) <-> ( ( F ` j ) - inf ( ran F , RR , < ) ) < y ) )
100	3	ad2antrr 732	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ran F C_ RR )
101	5	adantr 467	. . . . . . . . . . . 12 |- ( ( ph /\ y e. RR+ ) -> F Fn Z )
102		fnfvelrn 6019	. . . . . . . . . . . 12 |- ( ( F Fn Z /\ k e. Z ) -> ( F ` k ) e. ran F )
103	101, 56, 102	syl2an 480	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( F ` k ) e. ran F )
104	100, 103	sseldd 3433	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( F ` k ) e. RR )
105	20	ad2antrr 732	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> E. x e. RR A. y e. ran F x <_ y )
106		infrelb 10596	. . . . . . . . . . 11 |- ( ( ran F C_ RR /\ E. x e. RR A. y e. ran F x <_ y /\ ( F ` k ) e. ran F ) -> inf ( ran F , RR , < ) <_ ( F ` k ) )
107	100, 105, 103, 106	syl3anc 1268	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> inf ( ran F , RR , < ) <_ ( F ` k ) )
108	62, 104, 107	abssubge0d 13493	. . . . . . . . 9 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( abs ` ( ( F ` k ) - inf ( ran F , RR , < ) ) ) = ( ( F ` k ) - inf ( ran F , RR , < ) ) )
109	108	breq1d 4412	. . . . . . . 8 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ( abs ` ( ( F ` k ) - inf ( ran F , RR , < ) ) ) < y <-> ( ( F ` k ) - inf ( ran F , RR , < ) ) < y ) )
110	97, 99, 109	3imtr4d 272	. . . . . . 7 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ( ( F ` j ) - y ) < inf ( ran F , RR , < ) -> ( abs ` ( ( F ` k ) - inf ( ran F , RR , < ) ) ) < y ) )
111	110	anassrs 654	. . . . . 6 |- ( ( ( ( ph /\ y e. RR+ ) /\ j e. Z ) /\ k e. ( ZZ>= ` j ) ) -> ( ( ( F ` j ) - y ) < inf ( ran F , RR , < ) -> ( abs ` ( ( F ` k ) - inf ( ran F , RR , < ) ) ) < y ) )
112	111	ralrimdva 2806	. . . . 5 |- ( ( ( ph /\ y e. RR+ ) /\ j e. Z ) -> ( ( ( F ` j ) - y ) < inf ( ran F , RR , < ) -> A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - inf ( ran F , RR , < ) ) ) < y ) )
113	112	reximdva 2862	. . . 4 |- ( ( ph /\ y e. RR+ ) -> ( E. j e. Z ( ( F ` j ) - y ) < inf ( ran F , RR , < ) -> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - inf ( ran F , RR , < ) ) ) < y ) )
114	54, 113	mpd 15	. . 3 |- ( ( ph /\ y e. RR+ ) -> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - inf ( ran F , RR , < ) ) ) < y )
115	114	ralrimiva 2802	. 2 |- ( ph -> A. y e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - inf ( ran F , RR , < ) ) ) < y )
116		fvex 5875	. . . . 5 |- ( ZZ>= ` M ) e. _V
117	9, 116	eqeltri 2525	. . . 4 |- Z e. _V
118		fex 6138	. . . 4 |- ( ( F : Z --> RR /\ Z e. _V ) -> F e. _V )
119	1, 117, 118	sylancl 668	. . 3 |- ( ph -> F e. _V )
120		eqidd 2452	. . 3 |- ( ( ph /\ k e. Z ) -> ( F ` k ) = ( F ` k ) )
121	1	ffvelrnda 6022	. . . 4 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR )
122	121	recnd 9669	. . 3 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
123	9, 6, 119, 120, 40, 122	clim2c 13569	. 2 |- ( ph -> ( F ~~> inf ( ran F , RR , < ) <-> A. y e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - inf ( ran F , RR , < ) ) ) < y ) )
124	115, 123	mpbird 236	1 |- ( ph -> F ~~> inf ( ran F , RR , < ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   =/= wne 2622   A.wral 2737   E.wrex 2738   _Vcvv 3045   C_ wss 3404   (/)c0 3731   class class class wbr 4402   ran crn 4835   Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290  infcinf 7955   CCcc 9537   RRcr 9538   1c1 9540   + caddc 9542   < clt 9675   <_ cle 9676   - cmin 9860   ZZcz 10937   ZZ>=cuz 11159   RR+crp 11302   ...cfz 11784   abscabs 13297   ~~> cli 13548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-sup 7956  df-inf 7957  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11785  df-seq 12214  df-exp 12273  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552

This theorem is referenced by:  climinff  37690  stirlinglem13  37948