MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  addcn2 Unicode version

Theorem addcn2 12307
Description: Complex number addition is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (We write out the definition directly because df-cn 17206 and df-cncf 18772 are not yet available to us. See addcn 18759 for the abbreviated version.) (Contributed by Mario Carneiro, 31-Jan-2014.)
Assertion
Ref Expression
addcn2  |-  ( ( 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 addcn2
StepHypRef Expression
1 rphalfcl 10561 . . 3  |-  ( A  e.  RR+  ->  ( A  /  2 )  e.  RR+ )
213ad2ant1 978 . 2  |-  ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  /  2 )  e.  RR+ )
3 simprl 733 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  u  e.  CC )
4 simpl2 961 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  B  e.  CC )
5 simprr 734 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  v  e.  CC )
63, 4, 5pnpcan2d 9374 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  ( (
u  +  v )  -  ( B  +  v ) )  =  ( u  -  B
) )
76fveq2d 5665 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  ( abs `  ( ( u  +  v )  -  ( B  +  v )
) )  =  ( abs `  ( u  -  B ) ) )
87breq1d 4156 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  ( ( abs `  ( ( u  +  v )  -  ( B  +  v
) ) )  < 
( A  /  2
)  <->  ( abs `  (
u  -  B ) )  <  ( A  /  2 ) ) )
9 simpl3 962 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  C  e.  CC )
104, 5, 9pnpcand 9373 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  ( ( B  +  v )  -  ( B  +  C ) )  =  ( v  -  C
) )
1110fveq2d 5665 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  ( abs `  ( ( B  +  v )  -  ( B  +  C )
) )  =  ( abs `  ( v  -  C ) ) )
1211breq1d 4156 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  ( ( abs `  ( ( B  +  v )  -  ( B  +  C
) ) )  < 
( A  /  2
)  <->  ( abs `  (
v  -  C ) )  <  ( A  /  2 ) ) )
138, 12anbi12d 692 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  ( (
( abs `  (
( u  +  v )  -  ( B  +  v ) ) )  <  ( A  /  2 )  /\  ( abs `  ( ( B  +  v )  -  ( B  +  C ) ) )  <  ( A  / 
2 ) )  <->  ( ( abs `  ( u  -  B ) )  < 
( A  /  2
)  /\  ( abs `  ( v  -  C
) )  <  ( A  /  2 ) ) ) )
14 addcl 8998 . . . . . 6  |-  ( ( u  e.  CC  /\  v  e.  CC )  ->  ( u  +  v )  e.  CC )
1514adantl 453 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  ( u  +  v )  e.  CC )
164, 9addcld 9033 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  ( B  +  C )  e.  CC )
174, 5addcld 9033 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  ( B  +  v )  e.  CC )
18 simpl1 960 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  A  e.  RR+ )
1918rpred 10573 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  A  e.  RR )
20 abs3lem 12062 . . . . 5  |-  ( ( ( ( u  +  v )  e.  CC  /\  ( B  +  C
)  e.  CC )  /\  ( ( B  +  v )  e.  CC  /\  A  e.  RR ) )  -> 
( ( ( abs `  ( ( u  +  v )  -  ( B  +  v )
) )  <  ( A  /  2 )  /\  ( abs `  ( ( B  +  v )  -  ( B  +  C ) ) )  <  ( A  / 
2 ) )  -> 
( abs `  (
( u  +  v )  -  ( B  +  C ) ) )  <  A ) )
2115, 16, 17, 19, 20syl22anc 1185 . . . 4  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  ( (
( abs `  (
( u  +  v )  -  ( B  +  v ) ) )  <  ( A  /  2 )  /\  ( abs `  ( ( B  +  v )  -  ( B  +  C ) ) )  <  ( A  / 
2 ) )  -> 
( abs `  (
( u  +  v )  -  ( B  +  C ) ) )  <  A ) )
2213, 21sylbird 227 . . 3  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  ( (
( abs `  (
u  -  B ) )  <  ( A  /  2 )  /\  ( abs `  ( v  -  C ) )  <  ( A  / 
2 ) )  -> 
( abs `  (
( u  +  v )  -  ( B  +  C ) ) )  <  A ) )
2322ralrimivva 2734 . 2  |-  ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  ->  A. u  e.  CC  A. v  e.  CC  ( ( ( abs `  ( u  -  B ) )  <  ( A  / 
2 )  /\  ( abs `  ( v  -  C ) )  < 
( A  /  2
) )  ->  ( abs `  ( ( u  +  v )  -  ( B  +  C
) ) )  < 
A ) )
24 breq2 4150 . . . . . 6  |-  ( y  =  ( A  / 
2 )  ->  (
( abs `  (
u  -  B ) )  <  y  <->  ( abs `  ( u  -  B
) )  <  ( A  /  2 ) ) )
2524anbi1d 686 . . . . 5  |-  ( y  =  ( A  / 
2 )  ->  (
( ( abs `  (
u  -  B ) )  <  y  /\  ( abs `  ( v  -  C ) )  <  z )  <->  ( ( abs `  ( u  -  B ) )  < 
( A  /  2
)  /\  ( abs `  ( v  -  C
) )  <  z
) ) )
2625imbi1d 309 . . . 4  |-  ( y  =  ( A  / 
2 )  ->  (
( ( ( abs `  ( u  -  B
) )  <  y  /\  ( abs `  (
v  -  C ) )  <  z )  ->  ( abs `  (
( u  +  v )  -  ( B  +  C ) ) )  <  A )  <-> 
( ( ( abs `  ( u  -  B
) )  <  ( A  /  2 )  /\  ( abs `  ( v  -  C ) )  <  z )  -> 
( abs `  (
( u  +  v )  -  ( B  +  C ) ) )  <  A ) ) )
27262ralbidv 2684 . . 3  |-  ( y  =  ( A  / 
2 )  ->  ( A. u  e.  CC  A. v  e.  CC  (
( ( abs `  (
u  -  B ) )  <  y  /\  ( abs `  ( v  -  C ) )  <  z )  -> 
( abs `  (
( u  +  v )  -  ( B  +  C ) ) )  <  A )  <->  A. u  e.  CC  A. v  e.  CC  (
( ( abs `  (
u  -  B ) )  <  ( A  /  2 )  /\  ( abs `  ( v  -  C ) )  <  z )  -> 
( abs `  (
( u  +  v )  -  ( B  +  C ) ) )  <  A ) ) )
28 breq2 4150 . . . . . 6  |-  ( z  =  ( A  / 
2 )  ->  (
( abs `  (
v  -  C ) )  <  z  <->  ( abs `  ( v  -  C
) )  <  ( A  /  2 ) ) )
2928anbi2d 685 . . . . 5  |-  ( z  =  ( A  / 
2 )  ->  (
( ( abs `  (
u  -  B ) )  <  ( A  /  2 )  /\  ( abs `  ( v  -  C ) )  <  z )  <->  ( ( abs `  ( u  -  B ) )  < 
( A  /  2
)  /\  ( abs `  ( v  -  C
) )  <  ( A  /  2 ) ) ) )
3029imbi1d 309 . . . 4  |-  ( z  =  ( A  / 
2 )  ->  (
( ( ( abs `  ( u  -  B
) )  <  ( A  /  2 )  /\  ( abs `  ( v  -  C ) )  <  z )  -> 
( abs `  (
( u  +  v )  -  ( B  +  C ) ) )  <  A )  <-> 
( ( ( abs `  ( u  -  B
) )  <  ( A  /  2 )  /\  ( abs `  ( v  -  C ) )  <  ( A  / 
2 ) )  -> 
( abs `  (
( u  +  v )  -  ( B  +  C ) ) )  <  A ) ) )
31302ralbidv 2684 . . 3  |-  ( z  =  ( A  / 
2 )  ->  ( A. u  e.  CC  A. v  e.  CC  (
( ( abs `  (
u  -  B ) )  <  ( A  /  2 )  /\  ( abs `  ( v  -  C ) )  <  z )  -> 
( abs `  (
( u  +  v )  -  ( B  +  C ) ) )  <  A )  <->  A. u  e.  CC  A. v  e.  CC  (
( ( abs `  (
u  -  B ) )  <  ( A  /  2 )  /\  ( abs `  ( v  -  C ) )  <  ( A  / 
2 ) )  -> 
( abs `  (
( u  +  v )  -  ( B  +  C ) ) )  <  A ) ) )
3227, 31rspc2ev 2996 . 2  |-  ( ( ( A  /  2
)  e.  RR+  /\  ( A  /  2 )  e.  RR+  /\  A. u  e.  CC  A. v  e.  CC  ( ( ( abs `  ( u  -  B ) )  <  ( A  / 
2 )  /\  ( abs `  ( v  -  C ) )  < 
( A  /  2
) )  ->  ( abs `  ( ( u  +  v )  -  ( B  +  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 ) )
332, 2, 23, 32syl3anc 1184 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 1717   A.wral 2642   E.wrex 2643   class class class wbr 4146   ` cfv 5387  (class class class)co 6013   CCcc 8914   RRcr 8915    + caddc 8919    < clt 9046    - cmin 9216    / cdiv 9602   2c2 9974   RR+crp 10537   abscabs 11959
This theorem is referenced by:  subcn2  12308  climadd  12345  rlimadd  12356  addcn  18759
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-sup 7374  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-n0 10147  df-z 10208  df-uz 10414  df-rp 10538  df-seq 11244  df-exp 11303  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961
  Copyright terms: Public domain W3C validator