Theorem climle 11974

Description: Comparison of the limits of two sequences. (Contributed by Paul Chapman, 10-Sep-2007.) (Revised by Mario Carneiro, 1-Feb-2014.)

Hypotheses

Ref	Expression
climadd.1	|- Z = ( ZZ>= ` M )
climadd.2	|- ( ph -> M e. ZZ )
climadd.4	|- ( ph -> F ~~> A )
climle.5	|- ( ph -> G ~~> B )
climle.6	|- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR )
climle.7	|- ( ( ph /\ k e. Z ) -> ( G ` k ) e. RR )
climle.8	|- ( ( ph /\ k e. Z ) -> ( F ` k ) <_ ( G ` k ) )

Assertion

Ref	Expression
climle	|- ( ph -> A <_ B )

Distinct variable groups:   B, k   k, F   ph, k   A, k   k, G   k, M   k, Z

Proof of Theorem climle

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		climadd.1	. . 3 |- Z = ( ZZ>= ` M )
2		climadd.2	. . 3 |- ( ph -> M e. ZZ )
3		climle.5	. . . 4 |- ( ph -> G ~~> B )
4		zex 9549	. . . . . . . 8 |- ZZ e. _V
5		uzssz 9837	. . . . . . . 8 |- ( ZZ>= ` M ) C_ ZZ
6	4, 5	ssexi 4232	. . . . . . 7 |- ( ZZ>= ` M ) e. _V
7	1, 6	eqeltri 2304	. . . . . 6 |- Z e. _V
8	7	mptex 5890	. . . . 5 |- ( j e. Z |-> ( ( G ` j ) - ( F ` j ) ) ) e. _V
9	8	a1i 9	. . . 4 |- ( ph -> ( j e. Z |-> ( ( G ` j ) - ( F ` j ) ) ) e. _V )
10		climadd.4	. . . 4 |- ( ph -> F ~~> A )
11		climle.7	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. RR )
12	11	recnd 8267	. . . 4 |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. CC )
13		climle.6	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR )
14	13	recnd 8267	. . . 4 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
15		simpr 110	. . . . 5 |- ( ( ph /\ k e. Z ) -> k e. Z )
16	11, 13	resubcld 8619	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( ( G ` k ) - ( F ` k ) ) e. RR )
17		fveq2 5648	. . . . . . 7 |- ( j = k -> ( G ` j ) = ( G ` k ) )
18		fveq2 5648	. . . . . . 7 |- ( j = k -> ( F ` j ) = ( F ` k ) )
19	17, 18	oveq12d 6046	. . . . . 6 |- ( j = k -> ( ( G ` j ) - ( F ` j ) ) = ( ( G ` k ) - ( F ` k ) ) )
20		eqid 2231	. . . . . 6 |- ( j e. Z |-> ( ( G ` j ) - ( F ` j ) ) ) = ( j e. Z |-> ( ( G ` j ) - ( F ` j ) ) )
21	19, 20	fvmptg 5731	. . . . 5 |- ( ( k e. Z /\ ( ( G ` k ) - ( F ` k ) ) e. RR ) -> ( ( j e. Z |-> ( ( G ` j ) - ( F ` j ) ) ) ` k ) = ( ( G ` k ) - ( F ` k ) ) )
22	15, 16, 21	syl2anc 411	. . . 4 |- ( ( ph /\ k e. Z ) -> ( ( j e. Z |-> ( ( G ` j ) - ( F ` j ) ) ) ` k ) = ( ( G ` k ) - ( F ` k ) ) )
23	1, 2, 3, 9, 10, 12, 14, 22	climsub 11968	. . 3 |- ( ph -> ( j e. Z |-> ( ( G ` j ) - ( F ` j ) ) ) ~~> ( B - A ) )
24	22, 16	eqeltrd 2308	. . 3 |- ( ( ph /\ k e. Z ) -> ( ( j e. Z |-> ( ( G ` j ) - ( F ` j ) ) ) ` k ) e. RR )
25		climle.8	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( F ` k ) <_ ( G ` k ) )
26	11, 13	subge0d 8774	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( 0 <_ ( ( G ` k ) - ( F ` k ) ) <-> ( F ` k ) <_ ( G ` k ) ) )
27	25, 26	mpbird 167	. . . 4 |- ( ( ph /\ k e. Z ) -> 0 <_ ( ( G ` k ) - ( F ` k ) ) )
28	27, 22	breqtrrd 4121	. . 3 |- ( ( ph /\ k e. Z ) -> 0 <_ ( ( j e. Z |-> ( ( G ` j ) - ( F ` j ) ) ) ` k ) )
29	1, 2, 23, 24, 28	climge0 11965	. 2 |- ( ph -> 0 <_ ( B - A ) )
30	1, 2, 3, 11	climrecl 11964	. . 3 |- ( ph -> B e. RR )
31	1, 2, 10, 13	climrecl 11964	. . 3 |- ( ph -> A e. RR )
32	30, 31	subge0d 8774	. 2 |- ( ph -> ( 0 <_ ( B - A ) <-> A <_ B ) )
33	29, 32	mpbid 147	1 |- ( ph -> A <_ B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   _Vcvv 2803   class class class wbr 4093   |-> cmpt 4155   ` cfv 5333  (class class class)co 6028   RRcr 8091   0cc0 8092   <_ cle 8274   - cmin 8409   ZZcz 9540   ZZ>=cuz 9816   ~~> 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-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:  climlec2  11981  iserle  11982  iserabs  12116  cvgcmpub  12117