Theorem climinf 29947

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 5676	. . . . . . . . . . . 12 |- ( F : Z --> RR -> ran F C_ RR )
3	1, 2	syl 16	. . . . . . . . . . 11 |- ( ph -> ran F C_ RR )
4		ffn 5670	. . . . . . . . . . . . . 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 10989	. . . . . . . . . . . . . . 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 2553	. . . . . . . . . . . . 13 |- ( ph -> M e. Z )
11		fnfvelrn 5952	. . . . . . . . . . . . 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 3754	. . . . . . . . . . . 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 4407	. . . . . . . . . . . . . . 15 |- ( y = ( F ` k ) -> ( x <_ y <-> x <_ ( F ` k ) ) )
17	16	ralrn 5958	. . . . . . . . . . . . . 14 |- ( F Fn Z -> ( A. y e. ran F x <_ y <-> A. k e. Z x <_ ( F ` k ) ) )
18	17	rexbidv 2868	. . . . . . . . . . . . 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 1168	. . . . . . . . . 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 10423	. . . . . . . . 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 11170	. . . . . . 7 |- ( ( ph /\ y e. RR+ ) -> sup ( ran F , RR , `' < ) < ( sup ( ran F , RR , `' < ) + y ) )
27		rpre 11111	. . . . . . . . . 10 |- ( y e. RR+ -> y e. RR )
28	27	adantl 466	. . . . . . . . 9 |- ( ( ph /\ y e. RR+ ) -> y e. RR )
29	24, 28	readdcld 9527	. . . . . . . 8 |- ( ( ph /\ y e. RR+ ) -> ( sup ( ran F , RR , `' < ) + y ) e. RR )
30		infrglb 29939	. . . . . . . 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 3467	. . . . . . . . . . 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 9527	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ k e. ran F ) -> ( sup ( ran F , RR , `' < ) + y ) e. RR )
38	34, 37, 36	ltsub1d 10062	. . . . . . . . 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 1219	. . . . . . . . . . . . 13 |- ( ph -> sup ( ran F , RR , `' < ) e. RR )
40	39	recnd 9526	. . . . . . . . . . . 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 9526	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. RR+ ) /\ k e. ran F ) -> y e. CC )
43	41, 42	pncand 9834	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ k e. ran F ) -> ( ( sup ( ran F , RR , `' < ) + y ) - y ) = sup ( ran F , RR , `' < ) )
44	43	breq2d 4415	. . . . . . . . 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 2934	. . . . . 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 6210	. . . . . . . . 9 |- ( k = ( F ` j ) -> ( k - y ) = ( ( F ` j ) - y ) )
50	49	breq1d 4413	. . . . . . . 8 |- ( k = ( F ` j ) -> ( ( k - y ) < sup ( ran F , RR , `' < ) <-> ( ( F ` j ) - y ) < sup ( ran F , RR , `' < ) ) )
51	50	rexrn 5957	. . . . . . 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 10992	. . . . . . . . . . 11 |- ( ( j e. Z /\ k e. ( ZZ>= ` j ) ) -> k e. Z )
57		ffvelrn 5953	. . . . . . . . . . 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 5953	. . . . . . . . . . 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 756	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> k e. ( ZZ>= ` j ) )
64		fzssuz 11619	. . . . . . . . . . . . . 14 |- ( j ... k ) C_ ( ZZ>= ` j )
65		uzss 10995	. . . . . . . . . . . . . . . . 17 |- ( j e. ( ZZ>= ` M ) -> ( ZZ>= ` j ) C_ ( ZZ>= ` M ) )
66	65, 9	syl6sseqr 3514	. . . . . . . . . . . . . . . 16 |- ( j e. ( ZZ>= ` M ) -> ( ZZ>= ` j ) C_ Z )
67	66, 9	eleq2s 2562	. . . . . . . . . . . . . . 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 3478	. . . . . . . . . . . . 13 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( j ... k ) C_ Z )
70		ffvelrn 5953	. . . . . . . . . . . . . . . 16 |- ( ( F : Z --> RR /\ n e. Z ) -> ( F ` n ) e. RR )
71	70	ralrimiva 2830	. . . . . . . . . . . . . . 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 3527	. . . . . . . . . . . . 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 2920	. . . . . . . . . . 11 |- ( ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) /\ n e. ( j ... k ) ) -> ( F ` n ) e. RR )
77		fzssuz 11619	. . . . . . . . . . . . . 14 |- ( j ... ( k - 1 ) ) C_ ( ZZ>= ` j )
78	77, 68	syl5ss 3478	. . . . . . . . . . . . 13 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( j ... ( k - 1 ) ) C_ Z )
79	78	sselda 3467	. . . . . . . . . . . 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 2830	. . . . . . . . . . . . . 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 6210	. . . . . . . . . . . . . . . 16 |- ( k = n -> ( k + 1 ) = ( n + 1 ) )
84	83	fveq2d 5806	. . . . . . . . . . . . . . 15 |- ( k = n -> ( F ` ( k + 1 ) ) = ( F ` ( n + 1 ) ) )
85		fveq2 5802	. . . . . . . . . . . . . . 15 |- ( k = n -> ( F ` k ) = ( F ` n ) )
86	84, 85	breq12d 4416	. . . . . . . . . . . . . 14 |- ( k = n -> ( ( F ` ( k + 1 ) ) <_ ( F ` k ) <-> ( F ` ( n + 1 ) ) <_ ( F ` n ) ) )
87	86	rspccva 3178	. . . . . . . . . . . . 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 11957	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( F ` k ) <_ ( F ` j ) )
91	58, 61, 62, 90	lesub1dd 10069	. . . . . . . . 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 9890	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ( F ` k ) - sup ( ran F , RR , `' < ) ) e. RR )
93	61, 62	resubcld 9890	. . . . . . . . . 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 9579	. . . . . . . . . 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 1219	. . . . . . . . 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 9933	. . . . . . . . 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 1219	. . . . . . . 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 5952	. . . . . . . . . . . 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 3468	. . . . . . . . . 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 10425	. . . . . . . . . . 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 1219	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> sup ( ran F , RR , `' < ) <_ ( F ` k ) )
108	62, 104, 107	abssubge0d 13039	. . . . . . . . 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 4413	. . . . . . . 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 2912	. . . . 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 2934	. . . 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 2830	. 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 5812	. . . . 5 |- ( ZZ>= ` M ) e. _V
117	9, 116	eqeltri 2538	. . . 4 |- Z e. _V
118		fex 6062	. . . 4 |- ( ( F : Z --> RR /\ Z e. _V ) -> F e. _V )
119	1, 117, 118	sylancl 662	. . 3 |- ( ph -> F e. _V )
120		eqidd 2455	. . 3 |- ( ( ph /\ k e. Z ) -> ( F ` k ) = ( F ` k ) )
121	1	ffvelrnda 5955	. . . 4 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR )
122	121	recnd 9526	. . 3 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
123	9, 6, 119, 120, 40, 122	clim2c 13104	. 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 965   = wceq 1370   e. wcel 1758   =/= wne 2648   A.wral 2799   E.wrex 2800   _Vcvv 3078   C_ wss 3439   (/)c0 3748   class class class wbr 4403   `'ccnv 4950   ran crn 4952   Fn wfn 5524   -->wf 5525   ` cfv 5529  (class class class)co 6203   supcsup 7804   CCcc 9394   RRcr 9395   1c1 9397   + caddc 9399   < clt 9532   <_ cle 9533   - cmin 9709   ZZcz 10760   ZZ>=cuz 10975   RR+crp 11105   ...cfz 11557   abscabs 12844   ~~> cli 13083

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-sup 7805  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-2 10494  df-3 10495  df-n0 10694  df-z 10761  df-uz 10976  df-rp 11106  df-fz 11558  df-seq 11927  df-exp 11986  df-cj 12709  df-re 12710  df-im 12711  df-sqr 12845  df-abs 12846  df-clim 13087

This theorem is referenced by:  climinff  29952  stirlinglem13  30049