Theorem climub 11984

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 2231	. 2 |- ( ZZ>= ` N ) = ( ZZ>= ` N )
2		climub.2	. . . 4 |- ( ph -> N e. Z )
3		clim2iser.1	. . . 4 |- Z = ( ZZ>= ` M )
4	2, 3	eleqtrdi 2324	. . 3 |- ( ph -> N e. ( ZZ>= ` M ) )
5		eluzelz 9826	. . 3 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
6	4, 5	syl 14	. 2 |- ( ph -> N e. ZZ )
7		fveq2 5648	. . . . . 6 |- ( k = N -> ( F ` k ) = ( F ` N ) )
8	7	eleq1d 2300	. . . . 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 2871	. . 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 9835	. . . 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 5648	. . . . . . 7 |- ( k = j -> ( F ` k ) = ( F ` j ) )
18	17	eleq1d 2300	. . . . . 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 2871	. . . 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 10318	. . . . 5 |- ( k e. ( N ... j ) -> k e. ( ZZ>= ` N ) )
25	3	uztrn2 9835	. . . . . . 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 10318	. . . . 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 10810	. 2 |- ( ( ph /\ j e. ( ZZ>= ` N ) ) -> ( F ` N ) <_ ( F ` j ) )
36	1, 6, 13, 14, 22, 35	climlec2 11981	1 |- ( ph -> ( F ` N ) <_ A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   RRcr 8091   1c1 8093   + caddc 8095   <_ cle 8274   - cmin 8409   ZZcz 9540   ZZ>=cuz 9816   ...cfz 10305   ~~> cli 11918

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210  ax-arch 8211  ax-caucvg 8212

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-n0 9462  df-z 9541  df-uz 9817  df-rp 9950  df-fz 10306  df-seqfrec 10773  df-exp 10864  df-cj 11482  df-re 11483  df-im 11484  df-rsqrt 11638  df-abs 11639  df-clim 11919

This theorem is referenced by:  climserle  11985