Theorem climub 11655

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 2205	. 2 |- ( ZZ>= ` N ) = ( ZZ>= ` N )
2		climub.2	. . . 4 |- ( ph -> N e. Z )
3		clim2iser.1	. . . 4 |- Z = ( ZZ>= ` M )
4	2, 3	eleqtrdi 2298	. . 3 |- ( ph -> N e. ( ZZ>= ` M ) )
5		eluzelz 9657	. . 3 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
6	4, 5	syl 14	. 2 |- ( ph -> N e. ZZ )
7		fveq2 5576	. . . . . 6 |- ( k = N -> ( F ` k ) = ( F ` N ) )
8	7	eleq1d 2274	. . . . 5 |- ( k = N -> ( ( F ` k ) e. RR <-> ( F ` N ) e. RR ) )
9	8	imbi2d 230	. . . 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 116	. . . 4 |- ( k e. Z -> ( ph -> ( F ` k ) e. RR ) )
12	9, 11	vtoclga 2839	. . 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 9666	. . . 4 |- ( ( N e. Z /\ j e. ( ZZ>= ` N ) ) -> j e. Z )
16	2, 15	sylan 283	. . 3 |- ( ( ph /\ j e. ( ZZ>= ` N ) ) -> j e. Z )
17		fveq2 5576	. . . . . . 7 |- ( k = j -> ( F ` k ) = ( F ` j ) )
18	17	eleq1d 2274	. . . . . 6 |- ( k = j -> ( ( F ` k ) e. RR <-> ( F ` j ) e. RR ) )
19	18	imbi2d 230	. . . . 5 |- ( k = j -> ( ( ph -> ( F ` k ) e. RR ) <-> ( ph -> ( F ` j ) e. RR ) ) )
20	19, 11	vtoclga 2839	. . . 4 |- ( j e. Z -> ( ph -> ( F ` j ) e. RR ) )
21	20	impcom 125	. . 3 |- ( ( ph /\ j e. Z ) -> ( F ` j ) e. RR )
22	16, 21	syldan 282	. 2 |- ( ( ph /\ j e. ( ZZ>= ` N ) ) -> ( F ` j ) e. RR )
23		simpr 110	. . 3 |- ( ( ph /\ j e. ( ZZ>= ` N ) ) -> j e. ( ZZ>= ` N ) )
24		elfzuz 10143	. . . . 5 |- ( k e. ( N ... j ) -> k e. ( ZZ>= ` N ) )
25	3	uztrn2 9666	. . . . . . 7 |- ( ( N e. Z /\ k e. ( ZZ>= ` N ) ) -> k e. Z )
26	2, 25	sylan 283	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` N ) ) -> k e. Z )
27	26, 10	syldan 282	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` N ) ) -> ( F ` k ) e. RR )
28	24, 27	sylan2 286	. . . 4 |- ( ( ph /\ k e. ( N ... j ) ) -> ( F ` k ) e. RR )
29	28	adantlr 477	. . 3 |- ( ( ( ph /\ j e. ( ZZ>= ` N ) ) /\ k e. ( N ... j ) ) -> ( F ` k ) e. RR )
30		elfzuz 10143	. . . . 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 282	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` N ) ) -> ( F ` k ) <_ ( F ` ( k + 1 ) ) )
33	30, 32	sylan2 286	. . . 4 |- ( ( ph /\ k e. ( N ... ( j - 1 ) ) ) -> ( F ` k ) <_ ( F ` ( k + 1 ) ) )
34	33	adantlr 477	. . 3 |- ( ( ( ph /\ j e. ( ZZ>= ` N ) ) /\ k e. ( N ... ( j - 1 ) ) ) -> ( F ` k ) <_ ( F ` ( k + 1 ) ) )
35	23, 29, 34	monoord 10630	. 2 |- ( ( ph /\ j e. ( ZZ>= ` N ) ) -> ( F ` N ) <_ ( F ` j ) )
36	1, 6, 13, 14, 22, 35	climlec2 11652	1 |- ( ph -> ( F ` N ) <_ A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   class class class wbr 4044   ` cfv 5271  (class class class)co 5944   RRcr 7924   1c1 7926   + caddc 7928   <_ cle 8108   - cmin 8243   ZZcz 9372   ZZ>=cuz 9648   ...cfz 10130   ~~> cli 11589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043  ax-arch 8044  ax-caucvg 8045

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-frec 6477  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-inn 9037  df-2 9095  df-3 9096  df-4 9097  df-n0 9296  df-z 9373  df-uz 9649  df-rp 9776  df-fz 10131  df-seqfrec 10593  df-exp 10684  df-cj 11153  df-re 11154  df-im 11155  df-rsqrt 11309  df-abs 11310  df-clim 11590

This theorem is referenced by:  climserle  11656