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Theorem rlimsub 12357
Description: Limit of the difference of two converging functions. Proposition 12-2.1(b) 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
rlimsub  |-  ( 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 rlimsub
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 12321 . 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 12321 . 2  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  CC )
7 rlimcl 12217 . . 3  |-  ( ( x  e.  A  |->  B )  ~~> r  D  ->  D  e.  CC )
82, 7syl 16 . 2  |-  ( ph  ->  D  e.  CC )
9 rlimcl 12217 . . 3  |-  ( ( x  e.  A  |->  C )  ~~> r  E  ->  E  e.  CC )
105, 9syl 16 . 2  |-  ( ph  ->  E  e.  CC )
11 subf 9232 . . 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 subcn2 12308 . . 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 12304 1  |-  ( ph  ->  ( x  e.  A  |->  ( B  -  C
) )  ~~> r  ( D  -  E ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1717   A.wral 2642   E.wrex 2643   class class class wbr 4146    e. cmpt 4200    X. cxp 4809   -->wf 5383   ` cfv 5387  (class class class)co 6013   CCcc 8914    < clt 9046    - cmin 9216   RR+crp 10537   abscabs 11959    ~~> r crli 12199
This theorem is referenced by:  rlimneg  12360  rlimle  12361  dvfsumrlim2  19776  logexprlim  20869
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-pm 6950  df-en 7039  df-dom 7040  df-sdom 7041  df-sup 7374  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-n0 10147  df-z 10208  df-uz 10414  df-rp 10538  df-seq 11244  df-exp 11303  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-rlim 12203
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