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Theorem subcn2 12343
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 9262 . . 3  |-  ( C  e.  CC  ->  -u C  e.  CC )
2 addcn2 12342 . . 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 1219 . 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 9262 . . . . . . . . 9  |-  ( v  e.  CC  ->  -u v  e.  CC )
5 oveq1 6047 . . . . . . . . . . . . . 14  |-  ( w  =  -u v  ->  (
w  -  -u C
)  =  ( -u v  -  -u C ) )
65fveq2d 5691 . . . . . . . . . . . . 13  |-  ( w  =  -u v  ->  ( abs `  ( w  -  -u C ) )  =  ( abs `  ( -u v  -  -u C
) ) )
76breq1d 4182 . . . . . . . . . . . 12  |-  ( w  =  -u v  ->  (
( abs `  (
w  -  -u C
) )  <  z  <->  ( abs `  ( -u v  -  -u C ) )  <  z ) )
87anbi2d 685 . . . . . . . . . . 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 6048 . . . . . . . . . . . . . 14  |-  ( w  =  -u v  ->  (
u  +  w )  =  ( u  +  -u v ) )
109oveq1d 6055 . . . . . . . . . . . . 13  |-  ( w  =  -u v  ->  (
( u  +  w
)  -  ( B  +  -u C ) )  =  ( ( u  +  -u v )  -  ( B  +  -u C
) ) )
1110fveq2d 5691 . . . . . . . . . . . 12  |-  ( w  =  -u v  ->  ( abs `  ( ( u  +  w )  -  ( B  +  -u C
) ) )  =  ( abs `  (
( u  +  -u v )  -  ( B  +  -u C ) ) ) )
1211breq1d 4182 . . . . . . . . . . 11  |-  ( w  =  -u v  ->  (
( abs `  (
( u  +  w
)  -  ( B  +  -u C ) ) )  <  A  <->  ( abs `  ( ( u  +  -u v )  -  ( B  +  -u C ) ) )  <  A
) )
138, 12imbi12d 312 . . . . . . . . . 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 3008 . . . . . . . . 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 453 . . . . . . 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 448 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  v  e.  CC )
18 simpll3 998 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  C  e.  CC )
1917, 18neg2subd 9384 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  ( -u v  -  -u C )  =  ( C  -  v
) )
2019fveq2d 5691 . . . . . . . . . . 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 12210 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  ( abs `  ( C  -  v )
)  =  ( abs `  ( v  -  C
) ) )
2220, 21eqtrd 2436 . . . . . . . . . 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 4182 . . . . . . . . 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 685 . . . . . . . 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 9305 . . . . . . . . . . . 12  |-  ( ( u  e.  CC  /\  v  e.  CC )  ->  ( u  +  -u v )  =  ( u  -  v ) )
2625adantll 695 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  ( u  +  -u v )  =  ( u  -  v ) )
27 simpll2 997 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  B  e.  CC )
2827, 18negsubd 9373 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  ( B  +  -u C )  =  ( B  -  C ) )
2926, 28oveq12d 6058 . . . . . . . . . 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 5691 . . . . . . . . 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 4182 . . . . . . . 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 312 . . . . . . 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 206 . . . . . 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 2756 . . . . 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 2744 . . . 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 2777 . . 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 2777 . 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 set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   CCcc 8944    + caddc 8949    < clt 9076    - cmin 9247   -ucneg 9248   RR+crp 10568   abscabs 11994
This theorem is referenced by:  climsub  12382  rlimsub  12392  subcn  18849
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996
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