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Theorem addcn2 12387
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 17291 and df-cncf 18908 are not yet available to us. See addcn 18895 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 10636 . . 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 9449 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  ( (
u  +  v )  -  ( B  +  v ) )  =  ( u  -  B
) )
76fveq2d 5732 . . . . . 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 4222 . . . . 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 9448 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  ( ( B  +  v )  -  ( B  +  C ) )  =  ( v  -  C
) )
1110fveq2d 5732 . . . . . 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 4222 . . . . 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 9072 . . . . . 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 9107 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  ( B  +  C )  e.  CC )
174, 5addcld 9107 . . . . 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 10648 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  A  e.  RR )
20 abs3lem 12142 . . . . 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 2798 . 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 4216 . . . . . 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 2747 . . 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 4216 . . . . . 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 2747 . . 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 3060 . 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 1652    e. wcel 1725   A.wral 2705   E.wrex 2706   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   CCcc 8988   RRcr 8989    + caddc 8993    < clt 9120    - cmin 9291    / cdiv 9677   2c2 10049   RR+crp 10612   abscabs 12039
This theorem is referenced by:  subcn2  12388  climadd  12425  rlimadd  12436  addcn  18895
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041
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