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Theorem subcn2 13072
Description: Complex number subtraction is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by Mario Carneiro, 31-Jan-2014.)
Assertion
Ref Expression
subcn2  |-  ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  ->  E. y  e.  RR+  E. z  e.  RR+  A. u  e.  CC  A. v  e.  CC  (
( ( abs `  (
u  -  B ) )  <  y  /\  ( abs `  ( v  -  C ) )  <  z )  -> 
( abs `  (
( u  -  v
)  -  ( B  -  C ) ) )  <  A ) )
Distinct variable groups:    v, u, y, z, A    u, B, v, y, z    u, C, v, y, z

Proof of Theorem subcn2
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 negcl 9610 . . 3  |-  ( C  e.  CC  ->  -u C  e.  CC )
2 addcn2 13071 . . 3  |-  ( ( A  e.  RR+  /\  B  e.  CC  /\  -u C  e.  CC )  ->  E. y  e.  RR+  E. z  e.  RR+  A. u  e.  CC  A. w  e.  CC  (
( ( abs `  (
u  -  B ) )  <  y  /\  ( abs `  ( w  -  -u C ) )  <  z )  -> 
( abs `  (
( u  +  w
)  -  ( B  +  -u C ) ) )  <  A ) )
31, 2syl3an3 1253 . 2  |-  ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  ->  E. y  e.  RR+  E. z  e.  RR+  A. u  e.  CC  A. w  e.  CC  (
( ( abs `  (
u  -  B ) )  <  y  /\  ( abs `  ( w  -  -u C ) )  <  z )  -> 
( abs `  (
( u  +  w
)  -  ( B  +  -u C ) ) )  <  A ) )
4 negcl 9610 . . . . . . . . 9  |-  ( v  e.  CC  ->  -u v  e.  CC )
5 oveq1 6098 . . . . . . . . . . . . . 14  |-  ( w  =  -u v  ->  (
w  -  -u C
)  =  ( -u v  -  -u C ) )
65fveq2d 5695 . . . . . . . . . . . . 13  |-  ( w  =  -u v  ->  ( abs `  ( w  -  -u C ) )  =  ( abs `  ( -u v  -  -u C
) ) )
76breq1d 4302 . . . . . . . . . . . 12  |-  ( w  =  -u v  ->  (
( abs `  (
w  -  -u C
) )  <  z  <->  ( abs `  ( -u v  -  -u C ) )  <  z ) )
87anbi2d 703 . . . . . . . . . . 11  |-  ( w  =  -u v  ->  (
( ( abs `  (
u  -  B ) )  <  y  /\  ( abs `  ( w  -  -u C ) )  <  z )  <->  ( ( abs `  ( u  -  B ) )  < 
y  /\  ( abs `  ( -u v  -  -u C ) )  < 
z ) ) )
9 oveq2 6099 . . . . . . . . . . . . . 14  |-  ( w  =  -u v  ->  (
u  +  w )  =  ( u  +  -u v ) )
109oveq1d 6106 . . . . . . . . . . . . 13  |-  ( w  =  -u v  ->  (
( u  +  w
)  -  ( B  +  -u C ) )  =  ( ( u  +  -u v )  -  ( B  +  -u C
) ) )
1110fveq2d 5695 . . . . . . . . . . . 12  |-  ( w  =  -u v  ->  ( abs `  ( ( u  +  w )  -  ( B  +  -u C
) ) )  =  ( abs `  (
( u  +  -u v )  -  ( B  +  -u C ) ) ) )
1211breq1d 4302 . . . . . . . . . . 11  |-  ( w  =  -u v  ->  (
( abs `  (
( u  +  w
)  -  ( B  +  -u C ) ) )  <  A  <->  ( abs `  ( ( u  +  -u v )  -  ( B  +  -u C ) ) )  <  A
) )
138, 12imbi12d 320 . . . . . . . . . 10  |-  ( w  =  -u v  ->  (
( ( ( abs `  ( u  -  B
) )  <  y  /\  ( abs `  (
w  -  -u C
) )  <  z
)  ->  ( abs `  ( ( u  +  w )  -  ( B  +  -u C ) ) )  <  A
)  <->  ( ( ( abs `  ( u  -  B ) )  <  y  /\  ( abs `  ( -u v  -  -u C ) )  <  z )  -> 
( abs `  (
( u  +  -u v )  -  ( B  +  -u C ) ) )  <  A
) ) )
1413rspcv 3069 . . . . . . . . 9  |-  ( -u v  e.  CC  ->  ( A. w  e.  CC  ( ( ( abs `  ( u  -  B
) )  <  y  /\  ( abs `  (
w  -  -u C
) )  <  z
)  ->  ( abs `  ( ( u  +  w )  -  ( B  +  -u C ) ) )  <  A
)  ->  ( (
( abs `  (
u  -  B ) )  <  y  /\  ( abs `  ( -u v  -  -u C ) )  <  z )  ->  ( abs `  (
( u  +  -u v )  -  ( B  +  -u C ) ) )  <  A
) ) )
154, 14syl 16 . . . . . . . 8  |-  ( v  e.  CC  ->  ( A. w  e.  CC  ( ( ( abs `  ( u  -  B
) )  <  y  /\  ( abs `  (
w  -  -u C
) )  <  z
)  ->  ( abs `  ( ( u  +  w )  -  ( B  +  -u C ) ) )  <  A
)  ->  ( (
( abs `  (
u  -  B ) )  <  y  /\  ( abs `  ( -u v  -  -u C ) )  <  z )  ->  ( abs `  (
( u  +  -u v )  -  ( B  +  -u C ) ) )  <  A
) ) )
1615adantl 466 . . . . . . 7  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  ( A. w  e.  CC  ( ( ( abs `  ( u  -  B ) )  <  y  /\  ( abs `  ( w  -  -u C ) )  < 
z )  ->  ( abs `  ( ( u  +  w )  -  ( B  +  -u C
) ) )  < 
A )  ->  (
( ( abs `  (
u  -  B ) )  <  y  /\  ( abs `  ( -u v  -  -u C ) )  <  z )  ->  ( abs `  (
( u  +  -u v )  -  ( B  +  -u C ) ) )  <  A
) ) )
17 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  v  e.  CC )
18 simpll3 1029 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  C  e.  CC )
1917, 18neg2subd 9736 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  ( -u v  -  -u C )  =  ( C  -  v
) )
2019fveq2d 5695 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  ( abs `  ( -u v  -  -u C
) )  =  ( abs `  ( C  -  v ) ) )
2118, 17abssubd 12939 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  ( abs `  ( C  -  v )
)  =  ( abs `  ( v  -  C
) ) )
2220, 21eqtrd 2475 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  ( abs `  ( -u v  -  -u C
) )  =  ( abs `  ( v  -  C ) ) )
2322breq1d 4302 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  ( ( abs `  ( -u v  -  -u C ) )  < 
z  <->  ( abs `  (
v  -  C ) )  <  z ) )
2423anbi2d 703 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  ( ( ( abs `  ( u  -  B ) )  <  y  /\  ( abs `  ( -u v  -  -u C ) )  <  z )  <->  ( ( abs `  ( u  -  B ) )  < 
y  /\  ( abs `  ( v  -  C
) )  <  z
) ) )
25 negsub 9657 . . . . . . . . . . . 12  |-  ( ( u  e.  CC  /\  v  e.  CC )  ->  ( u  +  -u v )  =  ( u  -  v ) )
2625adantll 713 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  ( u  +  -u v )  =  ( u  -  v ) )
27 simpll2 1028 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  B  e.  CC )
2827, 18negsubd 9725 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  ( B  +  -u C )  =  ( B  -  C ) )
2926, 28oveq12d 6109 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  ( ( u  +  -u v )  -  ( B  +  -u C
) )  =  ( ( u  -  v
)  -  ( B  -  C ) ) )
3029fveq2d 5695 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  ( abs `  (
( u  +  -u v )  -  ( B  +  -u C ) ) )  =  ( abs `  ( ( u  -  v )  -  ( B  -  C ) ) ) )
3130breq1d 4302 . . . . . . . 8  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  ( ( abs `  ( ( u  +  -u v )  -  ( B  +  -u C ) ) )  <  A  <->  ( abs `  ( ( u  -  v )  -  ( B  -  C ) ) )  <  A ) )
3224, 31imbi12d 320 . . . . . . 7  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  ( ( ( ( abs `  (
u  -  B ) )  <  y  /\  ( abs `  ( -u v  -  -u C ) )  <  z )  ->  ( abs `  (
( u  +  -u v )  -  ( B  +  -u C ) ) )  <  A
)  <->  ( ( ( abs `  ( u  -  B ) )  <  y  /\  ( abs `  ( v  -  C ) )  < 
z )  ->  ( abs `  ( ( u  -  v )  -  ( B  -  C
) ) )  < 
A ) ) )
3316, 32sylibd 214 . . . . . 6  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  ( A. w  e.  CC  ( ( ( abs `  ( u  -  B ) )  <  y  /\  ( abs `  ( w  -  -u C ) )  < 
z )  ->  ( abs `  ( ( u  +  w )  -  ( B  +  -u C
) ) )  < 
A )  ->  (
( ( abs `  (
u  -  B ) )  <  y  /\  ( abs `  ( v  -  C ) )  <  z )  -> 
( abs `  (
( u  -  v
)  -  ( B  -  C ) ) )  <  A ) ) )
3433ralrimdva 2806 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  ->  ( A. w  e.  CC  ( ( ( abs `  ( u  -  B ) )  <  y  /\  ( abs `  ( w  -  -u C ) )  < 
z )  ->  ( abs `  ( ( u  +  w )  -  ( B  +  -u C
) ) )  < 
A )  ->  A. v  e.  CC  ( ( ( abs `  ( u  -  B ) )  <  y  /\  ( abs `  ( v  -  C ) )  < 
z )  ->  ( abs `  ( ( u  -  v )  -  ( B  -  C
) ) )  < 
A ) ) )
3534ralimdva 2794 . . . 4  |-  ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A. u  e.  CC  A. w  e.  CC  (
( ( abs `  (
u  -  B ) )  <  y  /\  ( abs `  ( w  -  -u C ) )  <  z )  -> 
( abs `  (
( u  +  w
)  -  ( B  +  -u C ) ) )  <  A )  ->  A. u  e.  CC  A. v  e.  CC  (
( ( abs `  (
u  -  B ) )  <  y  /\  ( abs `  ( v  -  C ) )  <  z )  -> 
( abs `  (
( u  -  v
)  -  ( B  -  C ) ) )  <  A ) ) )
3635reximdv 2827 . . 3  |-  ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  ->  ( E. z  e.  RR+  A. u  e.  CC  A. w  e.  CC  ( ( ( abs `  ( u  -  B ) )  <  y  /\  ( abs `  ( w  -  -u C ) )  < 
z )  ->  ( abs `  ( ( u  +  w )  -  ( B  +  -u C
) ) )  < 
A )  ->  E. z  e.  RR+  A. u  e.  CC  A. v  e.  CC  ( ( ( abs `  ( u  -  B ) )  <  y  /\  ( abs `  ( v  -  C ) )  < 
z )  ->  ( abs `  ( ( u  -  v )  -  ( B  -  C
) ) )  < 
A ) ) )
3736reximdv 2827 . 2  |-  ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  ->  ( E. y  e.  RR+  E. z  e.  RR+  A. u  e.  CC  A. w  e.  CC  ( ( ( abs `  ( u  -  B ) )  <  y  /\  ( abs `  ( w  -  -u C ) )  < 
z )  ->  ( abs `  ( ( u  +  w )  -  ( B  +  -u C
) ) )  < 
A )  ->  E. y  e.  RR+  E. z  e.  RR+  A. u  e.  CC  A. v  e.  CC  (
( ( abs `  (
u  -  B ) )  <  y  /\  ( abs `  ( v  -  C ) )  <  z )  -> 
( abs `  (
( u  -  v
)  -  ( B  -  C ) ) )  <  A ) ) )
383, 37mpd 15 1  |-  ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  ->  E. y  e.  RR+  E. z  e.  RR+  A. u  e.  CC  A. v  e.  CC  (
( ( abs `  (
u  -  B ) )  <  y  /\  ( abs `  ( v  -  C ) )  <  z )  -> 
( abs `  (
( u  -  v
)  -  ( B  -  C ) ) )  <  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2715   E.wrex 2716   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   CCcc 9280    + caddc 9285    < clt 9418    - cmin 9595   -ucneg 9596   RR+crp 10991   abscabs 12723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-seq 11807  df-exp 11866  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725
This theorem is referenced by:  climsub  13111  rlimsub  13121  subcn  20442
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