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Theorem climadd 13601
Description: Limit of the sum of two converging sequences. Proposition 12-2.1(a) of [Gleason] p. 168. (Contributed by NM, 24-Sep-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)
Hypotheses
Ref Expression
climadd.1  |-  Z  =  ( ZZ>= `  M )
climadd.2  |-  ( ph  ->  M  e.  ZZ )
climadd.4  |-  ( ph  ->  F  ~~>  A )
climadd.6  |-  ( ph  ->  H  e.  X )
climadd.7  |-  ( ph  ->  G  ~~>  B )
climadd.8  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
climadd.9  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  CC )
climadd.h  |-  ( (
ph  /\  k  e.  Z )  ->  ( H `  k )  =  ( ( F `
 k )  +  ( G `  k
) ) )
Assertion
Ref Expression
climadd  |-  ( ph  ->  H  ~~>  ( A  +  B ) )
Distinct variable groups:    B, k    k, F    ph, k    A, k   
k, G    k, H    k, M    k, Z
Allowed substitution hint:    X( k)

Proof of Theorem climadd
Dummy variables  u  v  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 climadd.1 . 2  |-  Z  =  ( ZZ>= `  M )
2 climadd.2 . 2  |-  ( ph  ->  M  e.  ZZ )
3 climadd.4 . . 3  |-  ( ph  ->  F  ~~>  A )
4 climcl 13469 . . 3  |-  ( F  ~~>  A  ->  A  e.  CC )
53, 4syl 17 . 2  |-  ( ph  ->  A  e.  CC )
6 climadd.7 . . 3  |-  ( ph  ->  G  ~~>  B )
7 climcl 13469 . . 3  |-  ( G  ~~>  B  ->  B  e.  CC )
86, 7syl 17 . 2  |-  ( ph  ->  B  e.  CC )
9 addcl 9603 . . 3  |-  ( ( u  e.  CC  /\  v  e.  CC )  ->  ( u  +  v )  e.  CC )
109adantl 464 . 2  |-  ( (
ph  /\  ( u  e.  CC  /\  v  e.  CC ) )  -> 
( u  +  v )  e.  CC )
11 climadd.6 . 2  |-  ( ph  ->  H  e.  X )
12 simpr 459 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  x  e.  RR+ )
135adantr 463 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  A  e.  CC )
148adantr 463 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  B  e.  CC )
15 addcn2 13563 . . 3  |-  ( ( x  e.  RR+  /\  A  e.  CC  /\  B  e.  CC )  ->  E. y  e.  RR+  E. z  e.  RR+  A. u  e.  CC  A. v  e.  CC  (
( ( abs `  (
u  -  A ) )  <  y  /\  ( abs `  ( v  -  B ) )  <  z )  -> 
( abs `  (
( u  +  v )  -  ( A  +  B ) ) )  <  x ) )
1612, 13, 14, 15syl3anc 1230 . 2  |-  ( (
ph  /\  x  e.  RR+ )  ->  E. y  e.  RR+  E. z  e.  RR+  A. u  e.  CC  A. v  e.  CC  (
( ( abs `  (
u  -  A ) )  <  y  /\  ( abs `  ( v  -  B ) )  <  z )  -> 
( abs `  (
( u  +  v )  -  ( A  +  B ) ) )  <  x ) )
17 climadd.8 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
18 climadd.9 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  CC )
19 climadd.h . 2  |-  ( (
ph  /\  k  e.  Z )  ->  ( H `  k )  =  ( ( F `
 k )  +  ( G `  k
) ) )
201, 2, 5, 8, 10, 3, 6, 11, 16, 17, 18, 19climcn2 13562 1  |-  ( ph  ->  H  ~~>  ( A  +  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2753   E.wrex 2754   class class class wbr 4394   ` cfv 5568  (class class class)co 6277   CCcc 9519    + caddc 9524    < clt 9657    - cmin 9840   ZZcz 10904   ZZ>=cuz 11126   RR+crp 11264   abscabs 13214    ~~> cli 13454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-sup 7934  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-n0 10836  df-z 10905  df-uz 11127  df-rp 11265  df-seq 12150  df-exp 12209  df-cj 13079  df-re 13080  df-im 13081  df-sqrt 13215  df-abs 13216  df-clim 13458
This theorem is referenced by:  climaddc1  13604  isumadd  13731  ioombl1lem4  22261  itg2addlem  22455  emcllem6  23654  basellem7  23739  basellem9  23741  clim1fr1  36956  climaddf  36970  ioodvbdlimc1lem2  37078  stirlinglem5  37209  fourierdlem103  37341  fourierdlem104  37342
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