Theorem climinf 31815

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

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 ~~> sup ( 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 5743	. . . . . . . . . . . 12 |- ( F : Z --> RR -> ran F C_ RR )
3	1, 2	syl 16	. . . . . . . . . . 11 |- ( ph -> ran F C_ RR )
4		ffn 5737	. . . . . . . . . . . . . 14 |- ( F : Z --> RR -> F Fn Z )
5	1, 4	syl 16	. . . . . . . . . . . . 13 |- ( ph -> F Fn Z )
6		climinf.4	. . . . . . . . . . . . . . 15 |- ( ph -> M e. ZZ )
7		uzid 11120	. . . . . . . . . . . . . . 15 |- ( M e. ZZ -> M e. ( ZZ>= ` M ) )
8	6, 7	syl 16	. . . . . . . . . . . . . 14 |- ( ph -> M e. ( ZZ>= ` M ) )
9		climinf.3	. . . . . . . . . . . . . 14 |- Z = ( ZZ>= ` M )
10	8, 9	syl6eleqr 2556	. . . . . . . . . . . . 13 |- ( ph -> M e. Z )
11		fnfvelrn 6029	. . . . . . . . . . . . 13 |- ( ( F Fn Z /\ M e. Z ) -> ( F ` M ) e. ran F )
12	5, 10, 11	syl2anc 661	. . . . . . . . . . . 12 |- ( ph -> ( F ` M ) e. ran F )
13		ne0i 3799	. . . . . . . . . . . 12 |- ( ( F ` M ) e. ran F -> ran F =/= (/) )
14	12, 13	syl 16	. . . . . . . . . . 11 |- ( ph -> ran F =/= (/) )
15		climinf.7	. . . . . . . . . . . 12 |- ( ph -> E. x e. RR A. k e. Z x <_ ( F ` k ) )
16		breq2 4460	. . . . . . . . . . . . . . 15 |- ( y = ( F ` k ) -> ( x <_ y <-> x <_ ( F ` k ) ) )
17	16	ralrn 6035	. . . . . . . . . . . . . 14 |- ( F Fn Z -> ( A. y e. ran F x <_ y <-> A. k e. Z x <_ ( F ` k ) ) )
18	17	rexbidv 2968	. . . . . . . . . . . . 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 16	. . . . . . . . . . . 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 232	. . . . . . . . . . 11 |- ( ph -> E. x e. RR A. y e. ran F x <_ y )
21	3, 14, 20	3jca 1176	. . . . . . . . . 10 |- ( ph -> ( ran F C_ RR /\ ran F =/= (/) /\ E. x e. RR A. y e. ran F x <_ y ) )
22	21	adantr 465	. . . . . . . . 9 |- ( ( ph /\ y e. RR+ ) -> ( ran F C_ RR /\ ran F =/= (/) /\ E. x e. RR A. y e. ran F x <_ y ) )
23		infmrcl 10542	. . . . . . . . 9 |- ( ( ran F C_ RR /\ ran F =/= (/) /\ E. x e. RR A. y e. ran F x <_ y ) -> sup ( ran F , RR , `' < ) e. RR )
24	22, 23	syl 16	. . . . . . . 8 |- ( ( ph /\ y e. RR+ ) -> sup ( ran F , RR , `' < ) e. RR )
25		simpr 461	. . . . . . . 8 |- ( ( ph /\ y e. RR+ ) -> y e. RR+ )
26	24, 25	ltaddrpd 11310	. . . . . . 7 |- ( ( ph /\ y e. RR+ ) -> sup ( ran F , RR , `' < ) < ( sup ( ran F , RR , `' < ) + y ) )
27		rpre 11251	. . . . . . . . . 10 |- ( y e. RR+ -> y e. RR )
28	27	adantl 466	. . . . . . . . 9 |- ( ( ph /\ y e. RR+ ) -> y e. RR )
29	24, 28	readdcld 9640	. . . . . . . 8 |- ( ( ph /\ y e. RR+ ) -> ( sup ( ran F , RR , `' < ) + y ) e. RR )
30		infrglb 31787	. . . . . . . 8 |- ( ( ( ran F C_ RR /\ ran F =/= (/) /\ E. x e. RR A. y e. ran F x <_ y ) /\ ( sup ( ran F , RR , `' < ) + y ) e. RR ) -> ( sup ( ran F , RR , `' < ) < ( sup ( ran F , RR , `' < ) + y ) <-> E. k e. ran F k < ( sup ( ran F , RR , `' < ) + y ) ) )
31	22, 29, 30	syl2anc 661	. . . . . . 7 |- ( ( ph /\ y e. RR+ ) -> ( sup ( ran F , RR , `' < ) < ( sup ( ran F , RR , `' < ) + y ) <-> E. k e. ran F k < ( sup ( ran F , RR , `' < ) + y ) ) )
32	26, 31	mpbid 210	. . . . . 6 |- ( ( ph /\ y e. RR+ ) -> E. k e. ran F k < ( sup ( ran F , RR , `' < ) + y ) )
33	3	sselda 3499	. . . . . . . . . . 11 |- ( ( ph /\ k e. ran F ) -> k e. RR )
34	33	adantlr 714	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ k e. ran F ) -> k e. RR )
35	24	adantr 465	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. RR+ ) /\ k e. ran F ) -> sup ( ran F , RR , `' < ) e. RR )
36	27	ad2antlr 726	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. RR+ ) /\ k e. ran F ) -> y e. RR )
37	35, 36	readdcld 9640	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ k e. ran F ) -> ( sup ( ran F , RR , `' < ) + y ) e. RR )
38	34, 37, 36	ltsub1d 10182	. . . . . . . . 9 |- ( ( ( ph /\ y e. RR+ ) /\ k e. ran F ) -> ( k < ( sup ( ran F , RR , `' < ) + y ) <-> ( k - y ) < ( ( sup ( ran F , RR , `' < ) + y ) - y ) ) )
39	3, 14, 20, 23	syl3anc 1228	. . . . . . . . . . . . 13 |- ( ph -> sup ( ran F , RR , `' < ) e. RR )
40	39	recnd 9639	. . . . . . . . . . . 12 |- ( ph -> sup ( ran F , RR , `' < ) e. CC )
41	40	ad2antrr 725	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. RR+ ) /\ k e. ran F ) -> sup ( ran F , RR , `' < ) e. CC )
42	36	recnd 9639	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. RR+ ) /\ k e. ran F ) -> y e. CC )
43	41, 42	pncand 9951	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ k e. ran F ) -> ( ( sup ( ran F , RR , `' < ) + y ) - y ) = sup ( ran F , RR , `' < ) )
44	43	breq2d 4468	. . . . . . . . 9 |- ( ( ( ph /\ y e. RR+ ) /\ k e. ran F ) -> ( ( k - y ) < ( ( sup ( ran F , RR , `' < ) + y ) - y ) <-> ( k - y ) < sup ( ran F , RR , `' < ) ) )
45	38, 44	bitrd 253	. . . . . . . 8 |- ( ( ( ph /\ y e. RR+ ) /\ k e. ran F ) -> ( k < ( sup ( ran F , RR , `' < ) + y ) <-> ( k - y ) < sup ( ran F , RR , `' < ) ) )
46	45	biimpd 207	. . . . . . 7 |- ( ( ( ph /\ y e. RR+ ) /\ k e. ran F ) -> ( k < ( sup ( ran F , RR , `' < ) + y ) -> ( k - y ) < sup ( ran F , RR , `' < ) ) )
47	46	reximdva 2932	. . . . . 6 |- ( ( ph /\ y e. RR+ ) -> ( E. k e. ran F k < ( sup ( ran F , RR , `' < ) + y ) -> E. k e. ran F ( k - y ) < sup ( ran F , RR , `' < ) ) )
48	32, 47	mpd 15	. . . . 5 |- ( ( ph /\ y e. RR+ ) -> E. k e. ran F ( k - y ) < sup ( ran F , RR , `' < ) )
49		oveq1 6303	. . . . . . . . 9 |- ( k = ( F ` j ) -> ( k - y ) = ( ( F ` j ) - y ) )
50	49	breq1d 4466	. . . . . . . 8 |- ( k = ( F ` j ) -> ( ( k - y ) < sup ( ran F , RR , `' < ) <-> ( ( F ` j ) - y ) < sup ( ran F , RR , `' < ) ) )
51	50	rexrn 6034	. . . . . . 7 |- ( F Fn Z -> ( E. k e. ran F ( k - y ) < sup ( ran F , RR , `' < ) <-> E. j e. Z ( ( F ` j ) - y ) < sup ( ran F , RR , `' < ) ) )
52	5, 51	syl 16	. . . . . 6 |- ( ph -> ( E. k e. ran F ( k - y ) < sup ( ran F , RR , `' < ) <-> E. j e. Z ( ( F ` j ) - y ) < sup ( ran F , RR , `' < ) ) )
53	52	biimpa 484	. . . . 5 |- ( ( ph /\ E. k e. ran F ( k - y ) < sup ( ran F , RR , `' < ) ) -> E. j e. Z ( ( F ` j ) - y ) < sup ( ran F , RR , `' < ) )
54	48, 53	syldan 470	. . . 4 |- ( ( ph /\ y e. RR+ ) -> E. j e. Z ( ( F ` j ) - y ) < sup ( ran F , RR , `' < ) )
55	1	adantr 465	. . . . . . . . . . 11 |- ( ( ph /\ y e. RR+ ) -> F : Z --> RR )
56	9	uztrn2 11123	. . . . . . . . . . 11 |- ( ( j e. Z /\ k e. ( ZZ>= ` j ) ) -> k e. Z )
57		ffvelrn 6030	. . . . . . . . . . 11 |- ( ( F : Z --> RR /\ k e. Z ) -> ( F ` k ) e. RR )
58	55, 56, 57	syl2an 477	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( F ` k ) e. RR )
59		simpl 457	. . . . . . . . . . 11 |- ( ( j e. Z /\ k e. ( ZZ>= ` j ) ) -> j e. Z )
60		ffvelrn 6030	. . . . . . . . . . 11 |- ( ( F : Z --> RR /\ j e. Z ) -> ( F ` j ) e. RR )
61	55, 59, 60	syl2an 477	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( F ` j ) e. RR )
62	39	ad2antrr 725	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> sup ( ran F , RR , `' < ) e. RR )
63		simprr 757	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> k e. ( ZZ>= ` j ) )
64		fzssuz 11750	. . . . . . . . . . . . . 14 |- ( j ... k ) C_ ( ZZ>= ` j )
65		uzss 11126	. . . . . . . . . . . . . . . . 17 |- ( j e. ( ZZ>= ` M ) -> ( ZZ>= ` j ) C_ ( ZZ>= ` M ) )
66	65, 9	syl6sseqr 3546	. . . . . . . . . . . . . . . 16 |- ( j e. ( ZZ>= ` M ) -> ( ZZ>= ` j ) C_ Z )
67	66, 9	eleq2s 2565	. . . . . . . . . . . . . . 15 |- ( j e. Z -> ( ZZ>= ` j ) C_ Z )
68	67	ad2antrl 727	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ZZ>= ` j ) C_ Z )
69	64, 68	syl5ss 3510	. . . . . . . . . . . . 13 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( j ... k ) C_ Z )
70		ffvelrn 6030	. . . . . . . . . . . . . . . 16 |- ( ( F : Z --> RR /\ n e. Z ) -> ( F ` n ) e. RR )
71	70	ralrimiva 2871	. . . . . . . . . . . . . . 15 |- ( F : Z --> RR -> A. n e. Z ( F ` n ) e. RR )
72	1, 71	syl 16	. . . . . . . . . . . . . 14 |- ( ph -> A. n e. Z ( F ` n ) e. RR )
73	72	ad2antrr 725	. . . . . . . . . . . . 13 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> A. n e. Z ( F ` n ) e. RR )
74		ssralv 3560	. . . . . . . . . . . . 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 60	. . . . . . . . . . . 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 2826	. . . . . . . . . . 11 |- ( ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) /\ n e. ( j ... k ) ) -> ( F ` n ) e. RR )
77		fzssuz 11750	. . . . . . . . . . . . . 14 |- ( j ... ( k - 1 ) ) C_ ( ZZ>= ` j )
78	77, 68	syl5ss 3510	. . . . . . . . . . . . 13 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( j ... ( k - 1 ) ) C_ Z )
79	78	sselda 3499	. . . . . . . . . . . 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 2871	. . . . . . . . . . . . . 14 |- ( ph -> A. k e. Z ( F ` ( k + 1 ) ) <_ ( F ` k ) )
82	81	ad2antrr 725	. . . . . . . . . . . . 13 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> A. k e. Z ( F ` ( k + 1 ) ) <_ ( F ` k ) )
83		oveq1 6303	. . . . . . . . . . . . . . . 16 |- ( k = n -> ( k + 1 ) = ( n + 1 ) )
84	83	fveq2d 5876	. . . . . . . . . . . . . . 15 |- ( k = n -> ( F ` ( k + 1 ) ) = ( F ` ( n + 1 ) ) )
85		fveq2 5872	. . . . . . . . . . . . . . 15 |- ( k = n -> ( F ` k ) = ( F ` n ) )
86	84, 85	breq12d 4469	. . . . . . . . . . . . . 14 |- ( k = n -> ( ( F ` ( k + 1 ) ) <_ ( F ` k ) <-> ( F ` ( n + 1 ) ) <_ ( F ` n ) ) )
87	86	rspccva 3209	. . . . . . . . . . . . 13 |- ( ( A. k e. Z ( F ` ( k + 1 ) ) <_ ( F ` k ) /\ n e. Z ) -> ( F ` ( n + 1 ) ) <_ ( F ` n ) )
88	82, 87	sylan 471	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) /\ n e. Z ) -> ( F ` ( n + 1 ) ) <_ ( F ` n ) )
89	79, 88	syldan 470	. . . . . . . . . . 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 12141	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( F ` k ) <_ ( F ` j ) )
91	58, 61, 62, 90	lesub1dd 10189	. . . . . . . . 9 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ( F ` k ) - sup ( ran F , RR , `' < ) ) <_ ( ( F ` j ) - sup ( ran F , RR , `' < ) ) )
92	58, 62	resubcld 10008	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ( F ` k ) - sup ( ran F , RR , `' < ) ) e. RR )
93	61, 62	resubcld 10008	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ( F ` j ) - sup ( ran F , RR , `' < ) ) e. RR )
94	27	ad2antlr 726	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> y e. RR )
95		lelttr 9692	. . . . . . . . . 10 |- ( ( ( ( F ` k ) - sup ( ran F , RR , `' < ) ) e. RR /\ ( ( F ` j ) - sup ( ran F , RR , `' < ) ) e. RR /\ y e. RR ) -> ( ( ( ( F ` k ) - sup ( ran F , RR , `' < ) ) <_ ( ( F ` j ) - sup ( ran F , RR , `' < ) ) /\ ( ( F ` j ) - sup ( ran F , RR , `' < ) ) < y ) -> ( ( F ` k ) - sup ( ran F , RR , `' < ) ) < y ) )
96	92, 93, 94, 95	syl3anc 1228	. . . . . . . . 9 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ( ( ( F ` k ) - sup ( ran F , RR , `' < ) ) <_ ( ( F ` j ) - sup ( ran F , RR , `' < ) ) /\ ( ( F ` j ) - sup ( ran F , RR , `' < ) ) < y ) -> ( ( F ` k ) - sup ( ran F , RR , `' < ) ) < y ) )
97	91, 96	mpand 675	. . . . . . . 8 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ( ( F ` j ) - sup ( ran F , RR , `' < ) ) < y -> ( ( F ` k ) - sup ( ran F , RR , `' < ) ) < y ) )
98		ltsub23 10053	. . . . . . . . 9 |- ( ( ( F ` j ) e. RR /\ y e. RR /\ sup ( ran F , RR , `' < ) e. RR ) -> ( ( ( F ` j ) - y ) < sup ( ran F , RR , `' < ) <-> ( ( F ` j ) - sup ( ran F , RR , `' < ) ) < y ) )
99	61, 94, 62, 98	syl3anc 1228	. . . . . . . 8 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ( ( F ` j ) - y ) < sup ( ran F , RR , `' < ) <-> ( ( F ` j ) - sup ( ran F , RR , `' < ) ) < y ) )
100	3	ad2antrr 725	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ran F C_ RR )
101	5	adantr 465	. . . . . . . . . . . 12 |- ( ( ph /\ y e. RR+ ) -> F Fn Z )
102		fnfvelrn 6029	. . . . . . . . . . . 12 |- ( ( F Fn Z /\ k e. Z ) -> ( F ` k ) e. ran F )
103	101, 56, 102	syl2an 477	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( F ` k ) e. ran F )
104	100, 103	sseldd 3500	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( F ` k ) e. RR )
105	20	ad2antrr 725	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> E. x e. RR A. y e. ran F x <_ y )
106		infmrlb 10544	. . . . . . . . . . 11 |- ( ( ran F C_ RR /\ E. x e. RR A. y e. ran F x <_ y /\ ( F ` k ) e. ran F ) -> sup ( ran F , RR , `' < ) <_ ( F ` k ) )
107	100, 105, 103, 106	syl3anc 1228	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> sup ( ran F , RR , `' < ) <_ ( F ` k ) )
108	62, 104, 107	abssubge0d 13275	. . . . . . . . 9 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( abs ` ( ( F ` k ) - sup ( ran F , RR , `' < ) ) ) = ( ( F ` k ) - sup ( ran F , RR , `' < ) ) )
109	108	breq1d 4466	. . . . . . . 8 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ( abs ` ( ( F ` k ) - sup ( ran F , RR , `' < ) ) ) < y <-> ( ( F ` k ) - sup ( ran F , RR , `' < ) ) < y ) )
110	97, 99, 109	3imtr4d 268	. . . . . . 7 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ( ( F ` j ) - y ) < sup ( ran F , RR , `' < ) -> ( abs ` ( ( F ` k ) - sup ( ran F , RR , `' < ) ) ) < y ) )
111	110	anassrs 648	. . . . . 6 |- ( ( ( ( ph /\ y e. RR+ ) /\ j e. Z ) /\ k e. ( ZZ>= ` j ) ) -> ( ( ( F ` j ) - y ) < sup ( ran F , RR , `' < ) -> ( abs ` ( ( F ` k ) - sup ( ran F , RR , `' < ) ) ) < y ) )
112	111	ralrimdva 2875	. . . . 5 |- ( ( ( ph /\ y e. RR+ ) /\ j e. Z ) -> ( ( ( F ` j ) - y ) < sup ( ran F , RR , `' < ) -> A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - sup ( ran F , RR , `' < ) ) ) < y ) )
113	112	reximdva 2932	. . . 4 |- ( ( ph /\ y e. RR+ ) -> ( E. j e. Z ( ( F ` j ) - y ) < sup ( ran F , RR , `' < ) -> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - sup ( 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 ) - sup ( ran F , RR , `' < ) ) ) < y )
115	114	ralrimiva 2871	. 2 |- ( ph -> A. y e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - sup ( ran F , RR , `' < ) ) ) < y )
116		fvex 5882	. . . . 5 |- ( ZZ>= ` M ) e. _V
117	9, 116	eqeltri 2541	. . . 4 |- Z e. _V
118		fex 6146	. . . 4 |- ( ( F : Z --> RR /\ Z e. _V ) -> F e. _V )
119	1, 117, 118	sylancl 662	. . 3 |- ( ph -> F e. _V )
120		eqidd 2458	. . 3 |- ( ( ph /\ k e. Z ) -> ( F ` k ) = ( F ` k ) )
121	1	ffvelrnda 6032	. . . 4 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR )
122	121	recnd 9639	. . 3 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
123	9, 6, 119, 120, 40, 122	clim2c 13340	. 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 ) )
124	115, 123	mpbird 232	1 |- ( ph -> F ~~> sup ( ran F , RR , `' < ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1395   e. wcel 1819   =/= wne 2652   A.wral 2807   E.wrex 2808   _Vcvv 3109   C_ wss 3471   (/)c0 3793   class class class wbr 4456   `'ccnv 5007   ran crn 5009   Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296   supcsup 7918   CCcc 9507   RRcr 9508   1c1 9510   + caddc 9512   < clt 9645   <_ cle 9646   - cmin 9824   ZZcz 10885   ZZ>=cuz 11106   RR+crp 11245   ...cfz 11697   abscabs 13079   ~~> cli 13319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-fz 11698  df-seq 12111  df-exp 12170  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-clim 13323

This theorem is referenced by:  climinff  31820  stirlinglem13  32071