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Theorem addcn2 13501
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 19898 and df-cncf 21551 are not yet available to us. See addcn 21538 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 11246 . . 3  |-  ( A  e.  RR+  ->  ( A  /  2 )  e.  RR+ )
213ad2ant1 1015 . 2  |-  ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  /  2 )  e.  RR+ )
3 simprl 754 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  u  e.  CC )
4 simpl2 998 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  B  e.  CC )
5 simprr 755 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  v  e.  CC )
63, 4, 5pnpcan2d 9960 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  ( (
u  +  v )  -  ( B  +  v ) )  =  ( u  -  B
) )
76fveq2d 5852 . . . . . 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 4449 . . . . 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 999 . . . . . . . 8  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  C  e.  CC )
104, 5, 9pnpcand 9959 . . . . . . 7  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  ( ( B  +  v )  -  ( B  +  C ) )  =  ( v  -  C
) )
1110fveq2d 5852 . . . . . 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 4449 . . . . 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 708 . . . 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 9563 . . . . . 6  |-  ( ( u  e.  CC  /\  v  e.  CC )  ->  ( u  +  v )  e.  CC )
1514adantl 464 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  ( u  +  v )  e.  CC )
164, 9addcld 9604 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  ( B  +  C )  e.  CC )
174, 5addcld 9604 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  ( B  +  v )  e.  CC )
18 simpl1 997 . . . . . 6  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  A  e.  RR+ )
1918rpred 11259 . . . . 5  |-  ( ( ( A  e.  RR+  /\  B  e.  CC  /\  C  e.  CC )  /\  ( u  e.  CC  /\  v  e.  CC ) )  ->  A  e.  RR )
20 abs3lem 13256 . . . . 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 1227 . . . 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 2875 . 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 4443 . . . . . 6  |-  ( y  =  ( A  / 
2 )  ->  (
( abs `  (
u  -  B ) )  <  y  <->  ( abs `  ( u  -  B
) )  <  ( A  /  2 ) ) )
2524anbi1d 702 . . . . 5  |-  ( y  =  ( A  / 
2 )  ->  (
( ( abs `  (
u  -  B ) )  <  y  /\  ( abs `  ( v  -  C ) )  <  z )  <->  ( ( abs `  ( u  -  B ) )  < 
( A  /  2
)  /\  ( abs `  ( v  -  C
) )  <  z
) ) )
2625imbi1d 315 . . . 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 2898 . . 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 4443 . . . . . 6  |-  ( z  =  ( A  / 
2 )  ->  (
( abs `  (
v  -  C ) )  <  z  <->  ( abs `  ( v  -  C
) )  <  ( A  /  2 ) ) )
2928anbi2d 701 . . . . 5  |-  ( z  =  ( A  / 
2 )  ->  (
( ( abs `  (
u  -  B ) )  <  ( A  /  2 )  /\  ( abs `  ( v  -  C ) )  <  z )  <->  ( ( abs `  ( u  -  B ) )  < 
( A  /  2
)  /\  ( abs `  ( v  -  C
) )  <  ( A  /  2 ) ) ) )
3029imbi1d 315 . . . 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 2898 . . 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 3218 . 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 1226 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   CCcc 9479   RRcr 9480    + caddc 9484    < clt 9617    - cmin 9796    / cdiv 10202   2c2 10581   RR+crp 11221   abscabs 13152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-seq 12093  df-exp 12152  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154
This theorem is referenced by:  subcn2  13502  climadd  13539  rlimadd  13550  addcn  21538
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