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Theorem subcn2 10700
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 7682 . . 3  |-  ( C  e.  CC  ->  -u C  e.  CC )
2 addcn2 10699 . . 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 1209 . 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 7682 . . . . . . . . 9  |-  ( v  e.  CC  ->  -u v  e.  CC )
5 oveq1 5659 . . . . . . . . . . . . . 14  |-  ( w  =  -u v  ->  (
w  -  -u C
)  =  ( -u v  -  -u C ) )
65fveq2d 5309 . . . . . . . . . . . . 13  |-  ( w  =  -u v  ->  ( abs `  ( w  -  -u C ) )  =  ( abs `  ( -u v  -  -u C
) ) )
76breq1d 3855 . . . . . . . . . . . 12  |-  ( w  =  -u v  ->  (
( abs `  (
w  -  -u C
) )  <  z  <->  ( abs `  ( -u v  -  -u C ) )  <  z ) )
87anbi2d 452 . . . . . . . . . . 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 5660 . . . . . . . . . . . . . 14  |-  ( w  =  -u v  ->  (
u  +  w )  =  ( u  +  -u v ) )
109oveq1d 5667 . . . . . . . . . . . . 13  |-  ( w  =  -u v  ->  (
( u  +  w
)  -  ( B  +  -u C ) )  =  ( ( u  +  -u v )  -  ( B  +  -u C
) ) )
1110fveq2d 5309 . . . . . . . . . . . 12  |-  ( w  =  -u v  ->  ( abs `  ( ( u  +  w )  -  ( B  +  -u C
) ) )  =  ( abs `  (
( u  +  -u v )  -  ( B  +  -u C ) ) ) )
1211breq1d 3855 . . . . . . . . . . 11  |-  ( w  =  -u v  ->  (
( abs `  (
( u  +  w
)  -  ( B  +  -u C ) ) )  <  A  <->  ( abs `  ( ( u  +  -u v )  -  ( B  +  -u C ) ) )  <  A
) )
138, 12imbi12d 232 . . . . . . . . . 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 2718 . . . . . . . . 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 14 . . . . . . . 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 271 . . . . . . 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 108 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  v  e.  CC )
18 simpll3 984 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  C  e.  CC )
1917, 18neg2subd 7810 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  ( -u v  -  -u C )  =  ( C  -  v
) )
2019fveq2d 5309 . . . . . . . . . . 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 10626 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  ( abs `  ( C  -  v )
)  =  ( abs `  ( v  -  C
) ) )
2220, 21eqtrd 2120 . . . . . . . . . 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 3855 . . . . . . . . 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 452 . . . . . . . 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 7730 . . . . . . . . . . . 12  |-  ( ( u  e.  CC  /\  v  e.  CC )  ->  ( u  +  -u v )  =  ( u  -  v ) )
2625adantll 460 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  ( u  +  -u v )  =  ( u  -  v ) )
27 simpll2 983 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  B  e.  CC )
2827, 18negsubd 7799 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  ( B  +  -u C )  =  ( B  -  C ) )
2926, 28oveq12d 5670 . . . . . . . . . 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 5309 . . . . . . . . 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 3855 . . . . . . . 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 232 . . . . . . 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 147 . . . . . 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 2453 . . . . 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 2441 . . . 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 2474 . . 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 2474 . 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 13 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 102    /\ w3a 924    = wceq 1289    e. wcel 1438   A.wral 2359   E.wrex 2360   class class class wbr 3845   ` cfv 5015  (class class class)co 5652   CCcc 7348    + caddc 7353    < clt 7522    - cmin 7653   -ucneg 7654   RR+crp 9134   abscabs 10430
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-mulrcl 7444  ax-addcom 7445  ax-mulcom 7446  ax-addass 7447  ax-mulass 7448  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-1rid 7452  ax-0id 7453  ax-rnegex 7454  ax-precex 7455  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-apti 7460  ax-pre-ltadd 7461  ax-pre-mulgt0 7462  ax-pre-mulext 7463  ax-arch 7464  ax-caucvg 7465
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-frec 6156  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-reap 8052  df-ap 8059  df-div 8140  df-inn 8423  df-2 8481  df-3 8482  df-4 8483  df-n0 8674  df-z 8751  df-uz 9020  df-rp 9135  df-iseq 9853  df-seq3 9854  df-exp 9955  df-cj 10276  df-re 10277  df-im 10278  df-rsqrt 10431  df-abs 10432
This theorem is referenced by:  climsub  10716
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