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Theorem rlimsub 12364
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 12328 . 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 12328 . 2  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  CC )
7 rlimcl 12224 . . 3  |-  ( ( x  e.  A  |->  B )  ~~> r  D  ->  D  e.  CC )
82, 7syl 16 . 2  |-  ( ph  ->  D  e.  CC )
9 rlimcl 12224 . . 3  |-  ( ( x  e.  A  |->  C )  ~~> r  E  ->  E  e.  CC )
105, 9syl 16 . 2  |-  ( ph  ->  E  e.  CC )
11 subf 9239 . . 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 12315 . . 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 12311 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 2649   E.wrex 2650   class class class wbr 4153    e. cmpt 4207    X. cxp 4816   -->wf 5390   ` cfv 5394  (class class class)co 6020   CCcc 8921    < clt 9053    - cmin 9223   RR+crp 10544   abscabs 11966    ~~> r crli 12206
This theorem is referenced by:  rlimneg  12367  rlimle  12368  dvfsumrlim2  19783  logexprlim  20876
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 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-pm 6957  df-en 7046  df-dom 7047  df-sdom 7048  df-sup 7381  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-n0 10154  df-z 10215  df-uz 10421  df-rp 10545  df-seq 11251  df-exp 11310  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-rlim 12210
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