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Theorem subcn2 13428
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 9839 . . 3  |-  ( C  e.  CC  ->  -u C  e.  CC )
2 addcn2 13427 . . 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 1263 . 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 9839 . . . . . . . . 9  |-  ( v  e.  CC  ->  -u v  e.  CC )
5 oveq1 6303 . . . . . . . . . . . . . 14  |-  ( w  =  -u v  ->  (
w  -  -u C
)  =  ( -u v  -  -u C ) )
65fveq2d 5876 . . . . . . . . . . . . 13  |-  ( w  =  -u v  ->  ( abs `  ( w  -  -u C ) )  =  ( abs `  ( -u v  -  -u C
) ) )
76breq1d 4466 . . . . . . . . . . . 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 6304 . . . . . . . . . . . . . 14  |-  ( w  =  -u v  ->  (
u  +  w )  =  ( u  +  -u v ) )
109oveq1d 6311 . . . . . . . . . . . . 13  |-  ( w  =  -u v  ->  (
( u  +  w
)  -  ( B  +  -u C ) )  =  ( ( u  +  -u v )  -  ( B  +  -u C
) ) )
1110fveq2d 5876 . . . . . . . . . . . 12  |-  ( w  =  -u v  ->  ( abs `  ( ( u  +  w )  -  ( B  +  -u C
) ) )  =  ( abs `  (
( u  +  -u v )  -  ( B  +  -u C ) ) ) )
1211breq1d 4466 . . . . . . . . . . 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 3206 . . . . . . . . 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 1037 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  C  e.  CC )
1917, 18neg2subd 9967 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  ( -u v  -  -u C )  =  ( C  -  v
) )
2019fveq2d 5876 . . . . . . . . . . 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 13295 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  ( abs `  ( C  -  v )
)  =  ( abs `  ( v  -  C
) ) )
2220, 21eqtrd 2498 . . . . . . . . . 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 4466 . . . . . . . . 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 9886 . . . . . . . . . . . 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 1036 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  B  e.  CC )
2827, 18negsubd 9956 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  u  e.  CC )  /\  v  e.  CC )  ->  ( B  +  -u C )  =  ( B  -  C ) )
2926, 28oveq12d 6314 . . . . . . . . . 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 5876 . . . . . . . . 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 4466 . . . . . . . 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 2875 . . . . 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 2865 . . . 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 2931 . . 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 2931 . 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 973    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   CCcc 9507    + caddc 9512    < clt 9645    - cmin 9824   -ucneg 9825   RR+crp 11245   abscabs 13078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-seq 12110  df-exp 12169  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080
This theorem is referenced by:  climsub  13467  rlimsub  13477  subcn  21495
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