Theorem addcn2 11816

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 and df-cncf are not yet available to us. See addcncntop 15230 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

Step	Hyp	Ref	Expression
1		rphalfcl 9873	. . 3 |- ( A e. RR+ -> ( A / 2 ) e. RR+ )
2	1	3ad2ant1 1042	. 2 |- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> ( A / 2 ) e. RR+ )
3		simprl 529	. . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> u e. CC )
4		simpl2 1025	. . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> B e. CC )
5		simprr 531	. . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> v e. CC )
6	3, 4, 5	pnpcan2d 8491	. . . . . . 7 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> ( ( u + v ) - ( B + v ) ) = ( u - B ) )
7	6	fveq2d 5630	. . . . . 6 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> ( abs ` ( ( u + v ) - ( B + v ) ) ) = ( abs ` ( u - B ) ) )
8	7	breq1d 4092	. . . . 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 1026	. . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> C e. CC )
10	4, 5, 9	pnpcand 8490	. . . . . . 7 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> ( ( B + v ) - ( B + C ) ) = ( v - C ) )
11	10	fveq2d 5630	. . . . . 6 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> ( abs ` ( ( B + v ) - ( B + C ) ) ) = ( abs ` ( v - C ) ) )
12	11	breq1d 4092	. . . . 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 ) ) )
13	8, 12	anbi12d 473	. . . 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 8120	. . . . . 6 |- ( ( u e. CC /\ v e. CC ) -> ( u + v ) e. CC )
15	14	adantl 277	. . . . 5 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> ( u + v ) e. CC )
16	4, 9	addcld 8162	. . . . 5 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> ( B + C ) e. CC )
17	4, 5	addcld 8162	. . . . 5 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> ( B + v ) e. CC )
18		simpl1 1024	. . . . . 6 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> A e. RR+ )
19	18	rpred 9888	. . . . 5 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> A e. RR )
20		abs3lem 11617	. . . . 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 ) )
21	15, 16, 17, 19, 20	syl22anc 1272	. . . 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 ) )
22	13, 21	sylbird 170	. . 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 ) )
23	22	ralrimivva 2612	. 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 4086	. . . . . 6 |- ( y = ( A / 2 ) -> ( ( abs ` ( u - B ) ) < y <-> ( abs ` ( u - B ) ) < ( A / 2 ) ) )
25	24	anbi1d 465	. . . . 5 |- ( y = ( A / 2 ) -> ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( v - C ) ) < z ) <-> ( ( abs ` ( u - B ) ) < ( A / 2 ) /\ ( abs ` ( v - C ) ) < z ) ) )
26	25	imbi1d 231	. . . 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 ) ) )
27	26	2ralbidv 2554	. . 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 4086	. . . . . 6 |- ( z = ( A / 2 ) -> ( ( abs ` ( v - C ) ) < z <-> ( abs ` ( v - C ) ) < ( A / 2 ) ) )
29	28	anbi2d 464	. . . . 5 |- ( z = ( A / 2 ) -> ( ( ( abs ` ( u - B ) ) < ( A / 2 ) /\ ( abs ` ( v - C ) ) < z ) <-> ( ( abs ` ( u - B ) ) < ( A / 2 ) /\ ( abs ` ( v - C ) ) < ( A / 2 ) ) ) )
30	29	imbi1d 231	. . . 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 ) ) )
31	30	2ralbidv 2554	. . 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 ) ) )
32	27, 31	rspc2ev 2922	. 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 ) )
33	2, 2, 23, 32	syl3anc 1271	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 ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   E.wrex 2509   class class class wbr 4082   ` cfv 5317  (class class class)co 6000   CCcc 7993   RRcr 7994   + caddc 7998   < clt 8177   - cmin 8313   / cdiv 8815   2c2 9157   RR+crp 9845   abscabs 11503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113  ax-arch 8114  ax-caucvg 8115

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-2 9165  df-3 9166  df-4 9167  df-n0 9366  df-z 9443  df-uz 9719  df-rp 9846  df-seqfrec 10665  df-exp 10756  df-cj 11348  df-re 11349  df-im 11350  df-rsqrt 11504  df-abs 11505

This theorem is referenced by:  subcn2  11817  climadd  11832  addcncntop  15230