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Theorem rlimadd 12428
Description: Limit of the sum of two converging functions. Proposition 12-2.1(a) of [Gleason] p. 168. (Contributed by Mario Carneiro, 22-Sep-2014.)
Hypotheses
Ref Expression
rlimadd.3  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  V )
rlimadd.4  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  V )
rlimadd.5  |-  ( ph  ->  ( x  e.  A  |->  B )  ~~> r  D
)
rlimadd.6  |-  ( ph  ->  ( x  e.  A  |->  C )  ~~> r  E
)
Assertion
Ref Expression
rlimadd  |-  ( ph  ->  ( x  e.  A  |->  ( B  +  C
) )  ~~> r  ( D  +  E ) )
Distinct variable groups:    x, A    x, D    ph, x    x, E
Allowed substitution hints:    B( x)    C( x)    V( x)

Proof of Theorem rlimadd
Dummy variables  w  v  y  z  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rlimadd.3 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  V )
2 rlimadd.5 . . 3  |-  ( ph  ->  ( x  e.  A  |->  B )  ~~> r  D
)
31, 2rlimmptrcl 12393 . 2  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  CC )
4 rlimadd.4 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  V )
5 rlimadd.6 . . 3  |-  ( ph  ->  ( x  e.  A  |->  C )  ~~> r  E
)
64, 5rlimmptrcl 12393 . 2  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  CC )
7 rlimcl 12289 . . 3  |-  ( ( x  e.  A  |->  B )  ~~> r  D  ->  D  e.  CC )
82, 7syl 16 . 2  |-  ( ph  ->  D  e.  CC )
9 rlimcl 12289 . . 3  |-  ( ( x  e.  A  |->  C )  ~~> r  E  ->  E  e.  CC )
105, 9syl 16 . 2  |-  ( ph  ->  E  e.  CC )
11 ax-addf 9061 . . 3  |-  +  :
( CC  X.  CC )
--> CC
1211a1i 11 . 2  |-  ( ph  ->  +  : ( CC 
X.  CC ) --> CC )
13 simpr 448 . . 3  |-  ( (
ph  /\  y  e.  RR+ )  ->  y  e.  RR+ )
148adantr 452 . . 3  |-  ( (
ph  /\  y  e.  RR+ )  ->  D  e.  CC )
1510adantr 452 . . 3  |-  ( (
ph  /\  y  e.  RR+ )  ->  E  e.  CC )
16 addcn2 12379 . . 3  |-  ( ( y  e.  RR+  /\  D  e.  CC  /\  E  e.  CC )  ->  E. z  e.  RR+  E. w  e.  RR+  A. u  e.  CC  A. v  e.  CC  (
( ( abs `  (
u  -  D ) )  <  z  /\  ( abs `  ( v  -  E ) )  <  w )  -> 
( abs `  (
( u  +  v )  -  ( D  +  E ) ) )  <  y ) )
1713, 14, 15, 16syl3anc 1184 . 2  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. z  e.  RR+  E. w  e.  RR+  A. u  e.  CC  A. v  e.  CC  (
( ( abs `  (
u  -  D ) )  <  z  /\  ( abs `  ( v  -  E ) )  <  w )  -> 
( abs `  (
( u  +  v )  -  ( D  +  E ) ) )  <  y ) )
183, 6, 8, 10, 2, 5, 12, 17rlimcn2 12376 1  |-  ( ph  ->  ( x  e.  A  |->  ( B  +  C
) )  ~~> r  ( D  +  E ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725   A.wral 2697   E.wrex 2698   class class class wbr 4204    e. cmpt 4258    X. cxp 4868   -->wf 5442   ` cfv 5446  (class class class)co 6073   CCcc 8980    + caddc 8985    < clt 9112    - cmin 9283   RR+crp 10604   abscabs 12031    ~~> r crli 12271
This theorem is referenced by:  caucvgr  12461  fsumrlim  12582  logfacrlim  21000  logexprlim  21001  chpchtlim  21165  selberglem2  21232
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-rlim 12275
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