Theorem climinf 31375

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 5737	. . . . . . . . . . . 12 |- ( F : Z --> RR -> ran F C_ RR )
3	1, 2	syl 16	. . . . . . . . . . 11 |- ( ph -> ran F C_ RR )
4		ffn 5731	. . . . . . . . . . . . . 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 11097	. . . . . . . . . . . . . . 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 2566	. . . . . . . . . . . . 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 661	. . . . . . . . . . . 12 |- ( ph -> ( F ` M ) e. ran F )
13		ne0i 3791	. . . . . . . . . . . 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 4451	. . . . . . . . . . . . . . 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 2973	. . . . . . . . . . . . 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 10523	. . . . . . . . 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 11286	. . . . . . 7 |- ( ( ph /\ y e. RR+ ) -> sup ( ran F , RR , `' < ) < ( sup ( ran F , RR , `' < ) + y ) )
27		rpre 11227	. . . . . . . . . 10 |- ( y e. RR+ -> y e. RR )
28	27	adantl 466	. . . . . . . . 9 |- ( ( ph /\ y e. RR+ ) -> y e. RR )
29	24, 28	readdcld 9624	. . . . . . . 8 |- ( ( ph /\ y e. RR+ ) -> ( sup ( ran F , RR , `' < ) + y ) e. RR )
30		infrglb 31367	. . . . . . . 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 3504	. . . . . . . . . . 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 9624	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ k e. ran F ) -> ( sup ( ran F , RR , `' < ) + y ) e. RR )
38	34, 37, 36	ltsub1d 10162	. . . . . . . . 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 9623	. . . . . . . . . . . 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 9623	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. RR+ ) /\ k e. ran F ) -> y e. CC )
43	41, 42	pncand 9932	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ k e. ran F ) -> ( ( sup ( ran F , RR , `' < ) + y ) - y ) = sup ( ran F , RR , `' < ) )
44	43	breq2d 4459	. . . . . . . . 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 2938	. . . . . 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 6292	. . . . . . . . 9 |- ( k = ( F ` j ) -> ( k - y ) = ( ( F ` j ) - y ) )
50	49	breq1d 4457	. . . . . . . 8 |- ( k = ( F ` j ) -> ( ( k - y ) < sup ( ran F , RR , `' < ) <-> ( ( F ` j ) - y ) < sup ( ran F , RR , `' < ) ) )
51	50	rexrn 6024	. . . . . . 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 11100	. . . . . . . . . . 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 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 6020	. . . . . . . . . . 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 11725	. . . . . . . . . . . . . 14 |- ( j ... k ) C_ ( ZZ>= ` j )
65		uzss 11103	. . . . . . . . . . . . . . . . 17 |- ( j e. ( ZZ>= ` M ) -> ( ZZ>= ` j ) C_ ( ZZ>= ` M ) )
66	65, 9	syl6sseqr 3551	. . . . . . . . . . . . . . . 16 |- ( j e. ( ZZ>= ` M ) -> ( ZZ>= ` j ) C_ Z )
67	66, 9	eleq2s 2575	. . . . . . . . . . . . . . 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 3515	. . . . . . . . . . . . 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 2878	. . . . . . . . . . . . . . 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 3564	. . . . . . . . . . . . 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 2833	. . . . . . . . . . 11 |- ( ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) /\ n e. ( j ... k ) ) -> ( F ` n ) e. RR )
77		fzssuz 11725	. . . . . . . . . . . . . 14 |- ( j ... ( k - 1 ) ) C_ ( ZZ>= ` j )
78	77, 68	syl5ss 3515	. . . . . . . . . . . . 13 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( j ... ( k - 1 ) ) C_ Z )
79	78	sselda 3504	. . . . . . . . . . . 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 2878	. . . . . . . . . . . . . 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 6292	. . . . . . . . . . . . . . . 16 |- ( k = n -> ( k + 1 ) = ( n + 1 ) )
84	83	fveq2d 5870	. . . . . . . . . . . . . . 15 |- ( k = n -> ( F ` ( k + 1 ) ) = ( F ` ( n + 1 ) ) )
85		fveq2 5866	. . . . . . . . . . . . . . 15 |- ( k = n -> ( F ` k ) = ( F ` n ) )
86	84, 85	breq12d 4460	. . . . . . . . . . . . . 14 |- ( k = n -> ( ( F ` ( k + 1 ) ) <_ ( F ` k ) <-> ( F ` ( n + 1 ) ) <_ ( F ` n ) ) )
87	86	rspccva 3213	. . . . . . . . . . . . 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 12107	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( F ` k ) <_ ( F ` j ) )
91	58, 61, 62, 90	lesub1dd 10169	. . . . . . . . 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 9988	. . . . . . . . . 10 |- ( ( ( ph /\ y e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ( F ` k ) - sup ( ran F , RR , `' < ) ) e. RR )
93	61, 62	resubcld 9988	. . . . . . . . . 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 9676	. . . . . . . . . 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 10033	. . . . . . . . 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 6019	. . . . . . . . . . . 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 3505	. . . . . . . . . 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 10525	. . . . . . . . . . 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 13229	. . . . . . . . 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 4457	. . . . . . . 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 2882	. . . . 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 2938	. . . 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 2878	. 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 5876	. . . . 5 |- ( ZZ>= ` M ) e. _V
117	9, 116	eqeltri 2551	. . . 4 |- Z e. _V
118		fex 6134	. . . 4 |- ( ( F : Z --> RR /\ Z e. _V ) -> F e. _V )
119	1, 117, 118	sylancl 662	. . 3 |- ( ph -> F e. _V )
120		eqidd 2468	. . 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 9623	. . 3 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
123	9, 6, 119, 120, 40, 122	clim2c 13294	. 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 1379   e. wcel 1767   =/= wne 2662   A.wral 2814   E.wrex 2815   _Vcvv 3113   C_ wss 3476   (/)c0 3785   class class class wbr 4447   `'ccnv 4998   ran crn 5000   Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6285   supcsup 7901   CCcc 9491   RRcr 9492   1c1 9494   + caddc 9496   < clt 9629   <_ cle 9630   - cmin 9806   ZZcz 10865   ZZ>=cuz 11083   RR+crp 11221   ...cfz 11673   abscabs 13033   ~~> cli 13273

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-sup 7902  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11084  df-rp 11222  df-fz 11674  df-seq 12077  df-exp 12136  df-cj 12898  df-re 12899  df-im 12900  df-sqrt 13034  df-abs 13035  df-clim 13277

This theorem is referenced by:  climinff  31380  stirlinglem13  31613