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Theorem addcn2 13092
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 18853 and df-cncf 20476 are not yet available to us. See addcn 20463 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 11036 . . 3  |-  ( A  e.  RR+  ->  ( A  /  2 )  e.  RR+ )
213ad2ant1 1009 . 2  |-  ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  /  2 )  e.  RR+ )
3 simprl 755 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  u  e.  CC )
4 simpl2 992 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  B  e.  CC )
5 simprr 756 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  v  e.  CC )
63, 4, 5pnpcan2d 9778 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  ( (
u  +  v )  -  ( B  +  v ) )  =  ( u  -  B
) )
76fveq2d 5716 . . . . . 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 4323 . . . . 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 993 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  C  e.  CC )
104, 5, 9pnpcand 9777 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  ( ( B  +  v )  -  ( B  +  C ) )  =  ( v  -  C
) )
1110fveq2d 5716 . . . . . 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 4323 . . . . 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 710 . . . 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 9385 . . . . . 6  |-  ( ( u  e.  CC  /\  v  e.  CC )  ->  ( u  +  v )  e.  CC )
1514adantl 466 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  ( u  +  v )  e.  CC )
164, 9addcld 9426 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  ( B  +  C )  e.  CC )
174, 5addcld 9426 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  ( B  +  v )  e.  CC )
18 simpl1 991 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  A  e.  RR+ )
1918rpred 11048 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  A  e.  RR )
20 abs3lem 12847 . . . . 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 1219 . . . 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 235 . . 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 2829 . 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 4317 . . . . . 6  |-  ( y  =  ( A  / 
2 )  ->  (
( abs `  (
u  -  B ) )  <  y  <->  ( abs `  ( u  -  B
) )  <  ( A  /  2 ) ) )
2524anbi1d 704 . . . . 5  |-  ( y  =  ( A  / 
2 )  ->  (
( ( abs `  (
u  -  B ) )  <  y  /\  ( abs `  ( v  -  C ) )  <  z )  <->  ( ( abs `  ( u  -  B ) )  < 
( A  /  2
)  /\  ( abs `  ( v  -  C
) )  <  z
) ) )
2625imbi1d 317 . . . 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 2778 . . 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 4317 . . . . . 6  |-  ( z  =  ( A  / 
2 )  ->  (
( abs `  (
v  -  C ) )  <  z  <->  ( abs `  ( v  -  C
) )  <  ( A  /  2 ) ) )
2928anbi2d 703 . . . . 5  |-  ( z  =  ( A  / 
2 )  ->  (
( ( abs `  (
u  -  B ) )  <  ( A  /  2 )  /\  ( abs `  ( v  -  C ) )  <  z )  <->  ( ( abs `  ( u  -  B ) )  < 
( A  /  2
)  /\  ( abs `  ( v  -  C
) )  <  ( A  /  2 ) ) ) )
3029imbi1d 317 . . . 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 2778 . . 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 3102 . 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 1218 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 965    = wceq 1369    e. wcel 1756   A.wral 2736   E.wrex 2737   class class class wbr 4313   ` cfv 5439  (class class class)co 6112   CCcc 9301   RRcr 9302    + caddc 9306    < clt 9439    - cmin 9616    / cdiv 10014   2c2 10392   RR+crp 11012   abscabs 12744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-2nd 6599  df-recs 6853  df-rdg 6887  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-sup 7712  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-n0 10601  df-z 10668  df-uz 10883  df-rp 11013  df-seq 11828  df-exp 11887  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746
This theorem is referenced by:  subcn2  13093  climadd  13130  rlimadd  13141  addcn  20463
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