Theorem climinf2mpt 39946

Description: A bounded below, monotonic non increasing sequence converges to the infimum of its range. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
climinf2mpt.p	|- F/ k ph
climinf2mpt.j	|- F/ j ph
climinf2mpt.m	|- ( ph -> M e. ZZ )
climinf2mpt.z	|- Z = ( ZZ>= ` M )
climinf2mpt.b	|- ( ( ph /\ k e. Z ) -> B e. RR )
climinf2mpt.c	|- ( k = j -> B = C )
climinf2mpt.l	|- ( ( ph /\ k e. Z /\ j = ( k + 1 ) ) -> C <_ B )
climinf2mpt.e	|- ( ph -> ( k e. Z |-> B ) e. dom ~~> )

Assertion

Ref	Expression
climinf2mpt	|- ( ph -> ( k e. Z |-> B ) ~~> inf ( ran ( k e. Z |-> B ) , RR* , < ) )

Distinct variable groups:   B, j   C, k   j, Z, k

Allowed substitution hints:   ph( j, k)   B( k)   C( j)   M( j, k)

Proof of Theorem climinf2mpt

Dummy variables i x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1843	. 2 |- F/ i ph
2		nfcv 2764	. 2 |- F/_ i ( k e. Z |-> B )
3		climinf2mpt.z	. 2 |- Z = ( ZZ>= ` M )
4		climinf2mpt.m	. 2 |- ( ph -> M e. ZZ )
5		climinf2mpt.p	. . 3 |- F/ k ph
6		climinf2mpt.b	. . 3 |- ( ( ph /\ k e. Z ) -> B e. RR )
7	5, 6	fmptd2f 39442	. 2 |- ( ph -> ( k e. Z |-> B ) : Z --> RR )
8		nfv 1843	. . . . . . 7 |- F/ k i e. Z
9	5, 8	nfan 1828	. . . . . 6 |- F/ k ( ph /\ i e. Z )
10		nfv 1843	. . . . . 6 |- F/ k [_ ( i + 1 ) / j ]_ C <_ [_ i / j ]_ C
11	9, 10	nfim 1825	. . . . 5 |- F/ k ( ( ph /\ i e. Z ) -> [_ ( i + 1 ) / j ]_ C <_ [_ i / j ]_ C )
12		eleq1 2689	. . . . . . 7 |- ( k = i -> ( k e. Z <-> i e. Z ) )
13	12	anbi2d 740	. . . . . 6 |- ( k = i -> ( ( ph /\ k e. Z ) <-> ( ph /\ i e. Z ) ) )
14		oveq1 6657	. . . . . . . 8 |- ( k = i -> ( k + 1 ) = ( i + 1 ) )
15	14	csbeq1d 3540	. . . . . . 7 |- ( k = i -> [_ ( k + 1 ) / j ]_ C = [_ ( i + 1 ) / j ]_ C )
16		eqidd 2623	. . . . . . . 8 |- ( k = i -> B = B )
17		csbco 3543	. . . . . . . . . . 11 |- [_ k / j ]_ [_ j / k ]_ B = [_ k / k ]_ B
18		csbid 3541	. . . . . . . . . . 11 |- [_ k / k ]_ B = B
19	17, 18	eqtr2i 2645	. . . . . . . . . 10 |- B = [_ k / j ]_ [_ j / k ]_ B
20		nfcv 2764	. . . . . . . . . . . . 13 |- F/_ j B
21		nfcv 2764	. . . . . . . . . . . . 13 |- F/_ k C
22		climinf2mpt.c	. . . . . . . . . . . . 13 |- ( k = j -> B = C )
23	20, 21, 22	cbvcsb 3538	. . . . . . . . . . . 12 |- [_ j / k ]_ B = [_ j / j ]_ C
24		csbid 3541	. . . . . . . . . . . 12 |- [_ j / j ]_ C = C
25	23, 24	eqtri 2644	. . . . . . . . . . 11 |- [_ j / k ]_ B = C
26	25	csbeq2i 3993	. . . . . . . . . 10 |- [_ k / j ]_ [_ j / k ]_ B = [_ k / j ]_ C
27	19, 26	eqtri 2644	. . . . . . . . 9 |- B = [_ k / j ]_ C
28	27	a1i 11	. . . . . . . 8 |- ( k = i -> B = [_ k / j ]_ C )
29		csbeq1 3536	. . . . . . . 8 |- ( k = i -> [_ k / j ]_ C = [_ i / j ]_ C )
30	16, 28, 29	3eqtrd 2660	. . . . . . 7 |- ( k = i -> B = [_ i / j ]_ C )
31	15, 30	breq12d 4666	. . . . . 6 |- ( k = i -> ( [_ ( k + 1 ) / j ]_ C <_ B <-> [_ ( i + 1 ) / j ]_ C <_ [_ i / j ]_ C ) )
32	13, 31	imbi12d 334	. . . . 5 |- ( k = i -> ( ( ( ph /\ k e. Z ) -> [_ ( k + 1 ) / j ]_ C <_ B ) <-> ( ( ph /\ i e. Z ) -> [_ ( i + 1 ) / j ]_ C <_ [_ i / j ]_ C ) ) )
33		simpl 473	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ph )
34		simpr 477	. . . . . 6 |- ( ( ph /\ k e. Z ) -> k e. Z )
35		eqidd 2623	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( k + 1 ) = ( k + 1 ) )
36		climinf2mpt.j	. . . . . . . . 9 |- F/ j ph
37		nfv 1843	. . . . . . . . 9 |- F/ j k e. Z
38		nfv 1843	. . . . . . . . 9 |- F/ j ( k + 1 ) = ( k + 1 )
39	36, 37, 38	nf3an 1831	. . . . . . . 8 |- F/ j ( ph /\ k e. Z /\ ( k + 1 ) = ( k + 1 ) )
40		nfcsb1v 3549	. . . . . . . . 9 |- F/_ j [_ ( k + 1 ) / j ]_ C
41		nfcv 2764	. . . . . . . . 9 |- F/_ j <_
42	40, 41, 20	nfbr 4699	. . . . . . . 8 |- F/ j [_ ( k + 1 ) / j ]_ C <_ B
43	39, 42	nfim 1825	. . . . . . 7 |- F/ j ( ( ph /\ k e. Z /\ ( k + 1 ) = ( k + 1 ) ) -> [_ ( k + 1 ) / j ]_ C <_ B )
44		ovex 6678	. . . . . . 7 |- ( k + 1 ) e. _V
45		eqeq1 2626	. . . . . . . . 9 |- ( j = ( k + 1 ) -> ( j = ( k + 1 ) <-> ( k + 1 ) = ( k + 1 ) ) )
46	45	3anbi3d 1405	. . . . . . . 8 |- ( j = ( k + 1 ) -> ( ( ph /\ k e. Z /\ j = ( k + 1 ) ) <-> ( ph /\ k e. Z /\ ( k + 1 ) = ( k + 1 ) ) ) )
47		csbeq1a 3542	. . . . . . . . 9 |- ( j = ( k + 1 ) -> C = [_ ( k + 1 ) / j ]_ C )
48	47	breq1d 4663	. . . . . . . 8 |- ( j = ( k + 1 ) -> ( C <_ B <-> [_ ( k + 1 ) / j ]_ C <_ B ) )
49	46, 48	imbi12d 334	. . . . . . 7 |- ( j = ( k + 1 ) -> ( ( ( ph /\ k e. Z /\ j = ( k + 1 ) ) -> C <_ B ) <-> ( ( ph /\ k e. Z /\ ( k + 1 ) = ( k + 1 ) ) -> [_ ( k + 1 ) / j ]_ C <_ B ) ) )
50		climinf2mpt.l	. . . . . . 7 |- ( ( ph /\ k e. Z /\ j = ( k + 1 ) ) -> C <_ B )
51	43, 44, 49, 50	vtoclf 3258	. . . . . 6 |- ( ( ph /\ k e. Z /\ ( k + 1 ) = ( k + 1 ) ) -> [_ ( k + 1 ) / j ]_ C <_ B )
52	33, 34, 35, 51	syl3anc 1326	. . . . 5 |- ( ( ph /\ k e. Z ) -> [_ ( k + 1 ) / j ]_ C <_ B )
53	11, 32, 52	chvar 2262	. . . 4 |- ( ( ph /\ i e. Z ) -> [_ ( i + 1 ) / j ]_ C <_ [_ i / j ]_ C )
54		csbco 3543	. . . . . . . 8 |- [_ ( i + 1 ) / j ]_ [_ j / k ]_ B = [_ ( i + 1 ) / k ]_ B
55	54	eqcomi 2631	. . . . . . 7 |- [_ ( i + 1 ) / k ]_ B = [_ ( i + 1 ) / j ]_ [_ j / k ]_ B
56	25	csbeq2i 3993	. . . . . . 7 |- [_ ( i + 1 ) / j ]_ [_ j / k ]_ B = [_ ( i + 1 ) / j ]_ C
57	55, 56	eqtri 2644	. . . . . 6 |- [_ ( i + 1 ) / k ]_ B = [_ ( i + 1 ) / j ]_ C
58	57	a1i 11	. . . . 5 |- ( ( ph /\ i e. Z ) -> [_ ( i + 1 ) / k ]_ B = [_ ( i + 1 ) / j ]_ C )
59		eqidd 2623	. . . . 5 |- ( ( ph /\ i e. Z ) -> [_ i / j ]_ C = [_ i / j ]_ C )
60	58, 59	breq12d 4666	. . . 4 |- ( ( ph /\ i e. Z ) -> ( [_ ( i + 1 ) / k ]_ B <_ [_ i / j ]_ C <-> [_ ( i + 1 ) / j ]_ C <_ [_ i / j ]_ C ) )
61	53, 60	mpbird 247	. . 3 |- ( ( ph /\ i e. Z ) -> [_ ( i + 1 ) / k ]_ B <_ [_ i / j ]_ C )
62	3	peano2uzs 11742	. . . . . 6 |- ( i e. Z -> ( i + 1 ) e. Z )
63	62	adantl 482	. . . . 5 |- ( ( ph /\ i e. Z ) -> ( i + 1 ) e. Z )
64		nfv 1843	. . . . . . . . 9 |- F/ k ( i + 1 ) e. Z
65	5, 64	nfan 1828	. . . . . . . 8 |- F/ k ( ph /\ ( i + 1 ) e. Z )
66		nfcv 2764	. . . . . . . . . 10 |- F/_ k ( i + 1 )
67	66	nfcsb1 3548	. . . . . . . . 9 |- F/_ k [_ ( i + 1 ) / k ]_ B
68	67	nfel1 2779	. . . . . . . 8 |- F/ k [_ ( i + 1 ) / k ]_ B e. RR
69	65, 68	nfim 1825	. . . . . . 7 |- F/ k ( ( ph /\ ( i + 1 ) e. Z ) -> [_ ( i + 1 ) / k ]_ B e. RR )
70		ovex 6678	. . . . . . 7 |- ( i + 1 ) e. _V
71		eleq1 2689	. . . . . . . . 9 |- ( k = ( i + 1 ) -> ( k e. Z <-> ( i + 1 ) e. Z ) )
72	71	anbi2d 740	. . . . . . . 8 |- ( k = ( i + 1 ) -> ( ( ph /\ k e. Z ) <-> ( ph /\ ( i + 1 ) e. Z ) ) )
73		csbeq1a 3542	. . . . . . . . 9 |- ( k = ( i + 1 ) -> B = [_ ( i + 1 ) / k ]_ B )
74	73	eleq1d 2686	. . . . . . . 8 |- ( k = ( i + 1 ) -> ( B e. RR <-> [_ ( i + 1 ) / k ]_ B e. RR ) )
75	72, 74	imbi12d 334	. . . . . . 7 |- ( k = ( i + 1 ) -> ( ( ( ph /\ k e. Z ) -> B e. RR ) <-> ( ( ph /\ ( i + 1 ) e. Z ) -> [_ ( i + 1 ) / k ]_ B e. RR ) ) )
76	69, 70, 75, 6	vtoclf 3258	. . . . . 6 |- ( ( ph /\ ( i + 1 ) e. Z ) -> [_ ( i + 1 ) / k ]_ B e. RR )
77	62, 76	sylan2 491	. . . . 5 |- ( ( ph /\ i e. Z ) -> [_ ( i + 1 ) / k ]_ B e. RR )
78		eqid 2622	. . . . . 6 |- ( k e. Z |-> B ) = ( k e. Z |-> B )
79	66, 67, 73, 78	fvmptf 6301	. . . . 5 |- ( ( ( i + 1 ) e. Z /\ [_ ( i + 1 ) / k ]_ B e. RR ) -> ( ( k e. Z |-> B ) ` ( i + 1 ) ) = [_ ( i + 1 ) / k ]_ B )
80	63, 77, 79	syl2anc 693	. . . 4 |- ( ( ph /\ i e. Z ) -> ( ( k e. Z |-> B ) ` ( i + 1 ) ) = [_ ( i + 1 ) / k ]_ B )
81		simpr 477	. . . . 5 |- ( ( ph /\ i e. Z ) -> i e. Z )
82		nfv 1843	. . . . . . . 8 |- F/ j i e. Z
83	36, 82	nfan 1828	. . . . . . 7 |- F/ j ( ph /\ i e. Z )
84		nfcsb1v 3549	. . . . . . . 8 |- F/_ j [_ i / j ]_ C
85		nfcv 2764	. . . . . . . 8 |- F/_ j RR
86	84, 85	nfel 2777	. . . . . . 7 |- F/ j [_ i / j ]_ C e. RR
87	83, 86	nfim 1825	. . . . . 6 |- F/ j ( ( ph /\ i e. Z ) -> [_ i / j ]_ C e. RR )
88		eleq1 2689	. . . . . . . 8 |- ( j = i -> ( j e. Z <-> i e. Z ) )
89	88	anbi2d 740	. . . . . . 7 |- ( j = i -> ( ( ph /\ j e. Z ) <-> ( ph /\ i e. Z ) ) )
90		csbeq1a 3542	. . . . . . . 8 |- ( j = i -> C = [_ i / j ]_ C )
91	90	eleq1d 2686	. . . . . . 7 |- ( j = i -> ( C e. RR <-> [_ i / j ]_ C e. RR ) )
92	89, 91	imbi12d 334	. . . . . 6 |- ( j = i -> ( ( ( ph /\ j e. Z ) -> C e. RR ) <-> ( ( ph /\ i e. Z ) -> [_ i / j ]_ C e. RR ) ) )
93		nfv 1843	. . . . . . . . 9 |- F/ k j e. Z
94	5, 93	nfan 1828	. . . . . . . 8 |- F/ k ( ph /\ j e. Z )
95		nfv 1843	. . . . . . . 8 |- F/ k C e. RR
96	94, 95	nfim 1825	. . . . . . 7 |- F/ k ( ( ph /\ j e. Z ) -> C e. RR )
97		eleq1 2689	. . . . . . . . 9 |- ( k = j -> ( k e. Z <-> j e. Z ) )
98	97	anbi2d 740	. . . . . . . 8 |- ( k = j -> ( ( ph /\ k e. Z ) <-> ( ph /\ j e. Z ) ) )
99	22	eleq1d 2686	. . . . . . . 8 |- ( k = j -> ( B e. RR <-> C e. RR ) )
100	98, 99	imbi12d 334	. . . . . . 7 |- ( k = j -> ( ( ( ph /\ k e. Z ) -> B e. RR ) <-> ( ( ph /\ j e. Z ) -> C e. RR ) ) )
101	96, 100, 6	chvar 2262	. . . . . 6 |- ( ( ph /\ j e. Z ) -> C e. RR )
102	87, 92, 101	chvar 2262	. . . . 5 |- ( ( ph /\ i e. Z ) -> [_ i / j ]_ C e. RR )
103		nfcv 2764	. . . . . 6 |- F/_ k i
104		nfcv 2764	. . . . . 6 |- F/_ k [_ i / j ]_ C
105	103, 104, 30, 78	fvmptf 6301	. . . . 5 |- ( ( i e. Z /\ [_ i / j ]_ C e. RR ) -> ( ( k e. Z |-> B ) ` i ) = [_ i / j ]_ C )
106	81, 102, 105	syl2anc 693	. . . 4 |- ( ( ph /\ i e. Z ) -> ( ( k e. Z |-> B ) ` i ) = [_ i / j ]_ C )
107	80, 106	breq12d 4666	. . 3 |- ( ( ph /\ i e. Z ) -> ( ( ( k e. Z |-> B ) ` ( i + 1 ) ) <_ ( ( k e. Z |-> B ) ` i ) <-> [_ ( i + 1 ) / k ]_ B <_ [_ i / j ]_ C ) )
108	61, 107	mpbird 247	. 2 |- ( ( ph /\ i e. Z ) -> ( ( k e. Z |-> B ) ` ( i + 1 ) ) <_ ( ( k e. Z |-> B ) ` i ) )
109		climinf2mpt.e	. . . . 5 |- ( ph -> ( k e. Z |-> B ) e. dom ~~> )
110	106, 102	eqeltrd 2701	. . . . . . 7 |- ( ( ph /\ i e. Z ) -> ( ( k e. Z |-> B ) ` i ) e. RR )
111	110	recnd 10068	. . . . . 6 |- ( ( ph /\ i e. Z ) -> ( ( k e. Z |-> B ) ` i ) e. CC )
112	111	ralrimiva 2966	. . . . 5 |- ( ph -> A. i e. Z ( ( k e. Z |-> B ) ` i ) e. CC )
113	2, 3	climbddf 39919	. . . . 5 |- ( ( M e. ZZ /\ ( k e. Z |-> B ) e. dom ~~> /\ A. i e. Z ( ( k e. Z |-> B ) ` i ) e. CC ) -> E. x e. RR A. i e. Z ( abs ` ( ( k e. Z |-> B ) ` i ) ) <_ x )
114	4, 109, 112, 113	syl3anc 1326	. . . 4 |- ( ph -> E. x e. RR A. i e. Z ( abs ` ( ( k e. Z |-> B ) ` i ) ) <_ x )
115	1, 110	rexabsle2 39654	. . . 4 |- ( ph -> ( E. x e. RR A. i e. Z ( abs ` ( ( k e. Z |-> B ) ` i ) ) <_ x <-> ( E. x e. RR A. i e. Z ( ( k e. Z |-> B ) ` i ) <_ x /\ E. x e. RR A. i e. Z x <_ ( ( k e. Z |-> B ) ` i ) ) ) )
116	114, 115	mpbid 222	. . 3 |- ( ph -> ( E. x e. RR A. i e. Z ( ( k e. Z |-> B ) ` i ) <_ x /\ E. x e. RR A. i e. Z x <_ ( ( k e. Z |-> B ) ` i ) ) )
117	116	simprd 479	. 2 |- ( ph -> E. x e. RR A. i e. Z x <_ ( ( k e. Z |-> B ) ` i ) )
118	1, 2, 3, 4, 7, 108, 117	climinf2 39939	1 |- ( ph -> ( k e. Z |-> B ) ~~> inf ( ran ( k e. Z |-> B ) , RR* , < ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   F/wnf 1708   e. wcel 1990   A.wral 2912   E.wrex 2913   [_csb 3533   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ran crn 5115   ` cfv 5888  (class class class)co 6650  infcinf 8347   CCcc 9934   RRcr 9935   1c1 9937   + caddc 9939   RR*cxr 10073   < clt 10074   <_ cle 10075   ZZcz 11377   ZZ>=cuz 11687   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-int 4476  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-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  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:  smflimsuplem4  41029