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Theorem addcn2 12375
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 17279 and df-cncf 18896 are not yet available to us. See addcn 18883 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 10625 . . 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 9438 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  ( (
u  +  v )  -  ( B  +  v ) )  =  ( u  -  B
) )
76fveq2d 5723 . . . . . 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 4214 . . . . 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 9437 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  ( ( B  +  v )  -  ( B  +  C ) )  =  ( v  -  C
) )
1110fveq2d 5723 . . . . . 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 4214 . . . . 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 9061 . . . . . 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 9096 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  ( B  +  C )  e.  CC )
174, 5addcld 9096 . . . . 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 10637 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  A  e.  RR )
20 abs3lem 12130 . . . . 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 2790 . 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 4208 . . . . . 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 2739 . . 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 4208 . . . . . 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 2739 . . 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 3052 . 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 2697   E.wrex 2698   class class class wbr 4204   ` cfv 5445  (class class class)co 6072   CCcc 8977   RRcr 8978    + caddc 8982    < clt 9109    - cmin 9280    / cdiv 9666   2c2 10038   RR+crp 10601   abscabs 12027
This theorem is referenced by:  subcn2  12376  climadd  12413  rlimadd  12424  addcn  18883
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056  ax-pre-sup 9057
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-sup 7437  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-div 9667  df-nn 9990  df-2 10047  df-3 10048  df-n0 10211  df-z 10272  df-uz 10478  df-rp 10602  df-seq 11312  df-exp 11371  df-cj 11892  df-re 11893  df-im 11894  df-sqr 12028  df-abs 12029
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