Theorem climub 11285

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 2165	. 2 |- ( ZZ>= ` N ) = ( ZZ>= ` N )
2		climub.2	. . . 4 |- ( ph -> N e. Z )
3		clim2iser.1	. . . 4 |- Z = ( ZZ>= ` M )
4	2, 3	eleqtrdi 2259	. . 3 |- ( ph -> N e. ( ZZ>= ` M ) )
5		eluzelz 9475	. . 3 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
6	4, 5	syl 14	. 2 |- ( ph -> N e. ZZ )
7		fveq2 5486	. . . . . 6 |- ( k = N -> ( F ` k ) = ( F ` N ) )
8	7	eleq1d 2235	. . . . 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 2792	. . 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 9483	. . . 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 5486	. . . . . . 7 |- ( k = j -> ( F ` k ) = ( F ` j ) )
18	17	eleq1d 2235	. . . . . 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 2792	. . . 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 9956	. . . . 5 |- ( k e. ( N ... j ) -> k e. ( ZZ>= ` N ) )
25	3	uztrn2 9483	. . . . . . 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 469	. . 3 |- ( ( ( ph /\ j e. ( ZZ>= ` N ) ) /\ k e. ( N ... j ) ) -> ( F ` k ) e. RR )
30		elfzuz 9956	. . . . 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 469	. . 3 |- ( ( ( ph /\ j e. ( ZZ>= ` N ) ) /\ k e. ( N ... ( j - 1 ) ) ) -> ( F ` k ) <_ ( F ` ( k + 1 ) ) )
35	23, 29, 34	monoord 10411	. 2 |- ( ( ph /\ j e. ( ZZ>= ` N ) ) -> ( F ` N ) <_ ( F ` j ) )
36	1, 6, 13, 14, 22, 35	climlec2 11282	1 |- ( ph -> ( F ` N ) <_ A )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   class class class wbr 3982   ` cfv 5188  (class class class)co 5842   RRcr 7752   1c1 7754   + caddc 7756   <_ cle 7934   - cmin 8069   ZZcz 9191   ZZ>=cuz 9466   ...cfz 9944   ~~> cli 11219

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-rp 9590  df-fz 9945  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-clim 11220

This theorem is referenced by:  climserle  11286