Theorem climub 11307

Description: The limit of a monotonic sequence is an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 10-Feb-2014.)

Hypotheses

Ref	Expression
clim2iser.1	|- Z = ( ZZ>= ` M )
climub.2	|- ( ph -> N e. Z )
climub.3	|- ( ph -> F ~~> A )
climub.4	|- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR )
climub.5	|- ( ( ph /\ k e. Z ) -> ( F ` k ) <_ ( F ` ( k + 1 ) ) )

Assertion

Ref	Expression
climub	|- ( ph -> ( F ` N ) <_ A )

Distinct variable groups:   A, k   k, F   k, M   k, N   ph, k   k, Z

Proof of Theorem climub

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2170	. 2 |- ( ZZ>= ` N ) = ( ZZ>= ` N )
2		climub.2	. . . 4 |- ( ph -> N e. Z )
3		clim2iser.1	. . . 4 |- Z = ( ZZ>= ` M )
4	2, 3	eleqtrdi 2263	. . 3 |- ( ph -> N e. ( ZZ>= ` M ) )
5		eluzelz 9496	. . 3 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
6	4, 5	syl 14	. 2 |- ( ph -> N e. ZZ )
7		fveq2 5496	. . . . . 6 |- ( k = N -> ( F ` k ) = ( F ` N ) )
8	7	eleq1d 2239	. . . . 5 |- ( k = N -> ( ( F ` k ) e. RR <-> ( F ` N ) e. RR ) )
9	8	imbi2d 229	. . . 4 |- ( k = N -> ( ( ph -> ( F ` k ) e. RR ) <-> ( ph -> ( F ` N ) e. RR ) ) )
10		climub.4	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR )
11	10	expcom 115	. . . 4 |- ( k e. Z -> ( ph -> ( F ` k ) e. RR ) )
12	9, 11	vtoclga 2796	. . 3 |- ( N e. Z -> ( ph -> ( F ` N ) e. RR ) )
13	2, 12	mpcom 36	. 2 |- ( ph -> ( F ` N ) e. RR )
14		climub.3	. 2 |- ( ph -> F ~~> A )
15	3	uztrn2 9504	. . . 4 |- ( ( N e. Z /\ j e. ( ZZ>= ` N ) ) -> j e. Z )
16	2, 15	sylan 281	. . 3 |- ( ( ph /\ j e. ( ZZ>= ` N ) ) -> j e. Z )
17		fveq2 5496	. . . . . . 7 |- ( k = j -> ( F ` k ) = ( F ` j ) )
18	17	eleq1d 2239	. . . . . 6 |- ( k = j -> ( ( F ` k ) e. RR <-> ( F ` j ) e. RR ) )
19	18	imbi2d 229	. . . . 5 |- ( k = j -> ( ( ph -> ( F ` k ) e. RR ) <-> ( ph -> ( F ` j ) e. RR ) ) )
20	19, 11	vtoclga 2796	. . . 4 |- ( j e. Z -> ( ph -> ( F ` j ) e. RR ) )
21	20	impcom 124	. . 3 |- ( ( ph /\ j e. Z ) -> ( F ` j ) e. RR )
22	16, 21	syldan 280	. 2 |- ( ( ph /\ j e. ( ZZ>= ` N ) ) -> ( F ` j ) e. RR )
23		simpr 109	. . 3 |- ( ( ph /\ j e. ( ZZ>= ` N ) ) -> j e. ( ZZ>= ` N ) )
24		elfzuz 9977	. . . . 5 |- ( k e. ( N ... j ) -> k e. ( ZZ>= ` N ) )
25	3	uztrn2 9504	. . . . . . 7 |- ( ( N e. Z /\ k e. ( ZZ>= ` N ) ) -> k e. Z )
26	2, 25	sylan 281	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` N ) ) -> k e. Z )
27	26, 10	syldan 280	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` N ) ) -> ( F ` k ) e. RR )
28	24, 27	sylan2 284	. . . 4 |- ( ( ph /\ k e. ( N ... j ) ) -> ( F ` k ) e. RR )
29	28	adantlr 474	. . 3 |- ( ( ( ph /\ j e. ( ZZ>= ` N ) ) /\ k e. ( N ... j ) ) -> ( F ` k ) e. RR )
30		elfzuz 9977	. . . . 5 |- ( k e. ( N ... ( j - 1 ) ) -> k e. ( ZZ>= ` N ) )
31		climub.5	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( F ` k ) <_ ( F ` ( k + 1 ) ) )
32	26, 31	syldan 280	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` N ) ) -> ( F ` k ) <_ ( F ` ( k + 1 ) ) )
33	30, 32	sylan2 284	. . . 4 |- ( ( ph /\ k e. ( N ... ( j - 1 ) ) ) -> ( F ` k ) <_ ( F ` ( k + 1 ) ) )
34	33	adantlr 474	. . 3 |- ( ( ( ph /\ j e. ( ZZ>= ` N ) ) /\ k e. ( N ... ( j - 1 ) ) ) -> ( F ` k ) <_ ( F ` ( k + 1 ) ) )
35	23, 29, 34	monoord 10432	. 2 |- ( ( ph /\ j e. ( ZZ>= ` N ) ) -> ( F ` N ) <_ ( F ` j ) )
36	1, 6, 13, 14, 22, 35	climlec2 11304	1 |- ( ph -> ( F ` N ) <_ A )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   RRcr 7773   1c1 7775   + caddc 7777   <_ cle 7955   - cmin 8090   ZZcz 9212   ZZ>=cuz 9487   ...cfz 9965   ~~> cli 11241

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-rp 9611  df-fz 9966  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242

This theorem is referenced by:  climserle  11308