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Theorem subcn2 12070
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 9054 . . 3  |-  ( C  e.  CC  ->  -u C  e.  CC )
2 addcn2 12069 . . 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 1217 . 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 9054 . . . . . . . . 9  |-  ( v  e.  CC  ->  -u v  e.  CC )
5 oveq1 5867 . . . . . . . . . . . . . 14  |-  ( w  =  -u v  ->  (
w  -  -u C
)  =  ( -u v  -  -u C ) )
65fveq2d 5531 . . . . . . . . . . . . 13  |-  ( w  =  -u v  ->  ( abs `  ( w  -  -u C ) )  =  ( abs `  ( -u v  -  -u C
) ) )
76breq1d 4035 . . . . . . . . . . . 12  |-  ( w  =  -u v  ->  (
( abs `  (
w  -  -u C
) )  <  z  <->  ( abs `  ( -u v  -  -u C ) )  <  z ) )
87anbi2d 684 . . . . . . . . . . 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 5868 . . . . . . . . . . . . . 14  |-  ( w  =  -u v  ->  (
u  +  w )  =  ( u  +  -u v ) )
109oveq1d 5875 . . . . . . . . . . . . 13  |-  ( w  =  -u v  ->  (
( u  +  w
)  -  ( B  +  -u C ) )  =  ( ( u  +  -u v )  -  ( B  +  -u C
) ) )
1110fveq2d 5531 . . . . . . . . . . . 12  |-  ( w  =  -u v  ->  ( abs `  ( ( u  +  w )  -  ( B  +  -u C
) ) )  =  ( abs `  (
( u  +  -u v )  -  ( B  +  -u C ) ) ) )
1211breq1d 4035 . . . . . . . . . . 11  |-  ( w  =  -u v  ->  (
( abs `  (
( u  +  w
)  -  ( B  +  -u C ) ) )  <  A  <->  ( abs `  ( ( u  +  -u v )  -  ( B  +  -u C ) ) )  <  A
) )
138, 12imbi12d 311 . . . . . . . . . 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 2882 . . . . . . . . 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 15 . . . . . . . 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 452 . . . . . . 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 447 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  v  e.  CC )
18 simpll3 996 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  C  e.  CC )
1917, 18neg2subd 9176 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  ( -u v  -  -u C )  =  ( C  -  v
) )
2019fveq2d 5531 . . . . . . . . . . 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 11937 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  ( abs `  ( C  -  v )
)  =  ( abs `  ( v  -  C
) ) )
2220, 21eqtrd 2317 . . . . . . . . . 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 4035 . . . . . . . . 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 684 . . . . . . . 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 9097 . . . . . . . . . . . 12  |-  ( ( u  e.  CC  /\  v  e.  CC )  ->  ( u  +  -u v )  =  ( u  -  v ) )
2625adantll 694 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  ( u  +  -u v )  =  ( u  -  v ) )
27 simpll2 995 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  B  e.  CC )
2827, 18negsubd 9165 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  ( B  +  -u C )  =  ( B  -  C ) )
2926, 28oveq12d 5878 . . . . . . . . . 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 5531 . . . . . . . . 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 4035 . . . . . . . 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 311 . . . . . . 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 205 . . . . . 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 2635 . . . . 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 2623 . . . 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 2656 . . 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 2656 . 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 14 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 358    /\ w3a 934    = wceq 1625    e. wcel 1686   A.wral 2545   E.wrex 2546   class class class wbr 4025   ` cfv 5257  (class class class)co 5860   CCcc 8737    + caddc 8742    < clt 8869    - cmin 9039   -ucneg 9040   RR+crp 10356   abscabs 11721
This theorem is referenced by:  climsub  12109  rlimsub  12119  subcn  18372
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-sup 7196  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-3 9807  df-n0 9968  df-z 10027  df-uz 10233  df-rp 10357  df-seq 11049  df-exp 11107  df-cj 11586  df-re 11587  df-im 11588  df-sqr 11722  df-abs 11723
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