Theorem climle 12044

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 9603	. . . . . . . 8 |- ZZ e. _V
5		uzssz 9892	. . . . . . . 8 |- ( ZZ>= ` M ) C_ ZZ
6	4, 5	ssexi 4253	. . . . . . 7 |- ( ZZ>= ` M ) e. _V
7	1, 6	eqeltri 2307	. . . . . 6 |- Z e. _V
8	7	mptex 5917	. . . . 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 8318	. . . 4 |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. CC )
13		climle.6	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR )
14	13	recnd 8318	. . . 4 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
15		simpr 110	. . . . 5 |- ( ( ph /\ k e. Z ) -> k e. Z )
16	11, 13	resubcld 8671	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( ( G ` k ) - ( F ` k ) ) e. RR )
17		fveq2 5675	. . . . . . 7 |- ( j = k -> ( G ` j ) = ( G ` k ) )
18		fveq2 5675	. . . . . . 7 |- ( j = k -> ( F ` j ) = ( F ` k ) )
19	17, 18	oveq12d 6076	. . . . . 6 |- ( j = k -> ( ( G ` j ) - ( F ` j ) ) = ( ( G ` k ) - ( F ` k ) ) )
20		eqid 2234	. . . . . 6 |- ( j e. Z |-> ( ( G ` j ) - ( F ` j ) ) ) = ( j e. Z |-> ( ( G ` j ) - ( F ` j ) ) )
21	19, 20	fvmptg 5758	. . . . 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 12038	. . 3 |- ( ph -> ( j e. Z |-> ( ( G ` j ) - ( F ` j ) ) ) ~~> ( B - A ) )
24	22, 16	eqeltrd 2311	. . 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 8826	. . . . 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 4142	. . 3 |- ( ( ph /\ k e. Z ) -> 0 <_ ( ( j e. Z |-> ( ( G ` j ) - ( F ` j ) ) ) ` k ) )
29	1, 2, 23, 24, 28	climge0 12035	. 2 |- ( ph -> 0 <_ ( B - A ) )
30	1, 2, 3, 11	climrecl 12034	. . 3 |- ( ph -> B e. RR )
31	1, 2, 10, 13	climrecl 12034	. . 3 |- ( ph -> A e. RR )
32	30, 31	subge0d 8826	. 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 2205   _Vcvv 2815   class class class wbr 4114   |-> cmpt 4176   ` cfv 5357  (class class class)co 6058   RRcr 8142   0cc0 8143   <_ cle 8325   - cmin 8460   ZZcz 9594   ZZ>=cuz 9871   ~~> cli 11988

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

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 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-rp 10005  df-seqfrec 10834  df-exp 10925  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-clim 11989

This theorem is referenced by:  climlec2  12051  iserle  12052  iserabs  12186  cvgcmpub  12187