Theorem addcn2 11111

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 12760 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 9498	. . 3 |- ( A e. RR+ -> ( A / 2 ) e. RR+ )
2	1	3ad2ant1 1003	. 2 |- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> ( A / 2 ) e. RR+ )
3		simprl 521	. . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> u e. CC )
4		simpl2 986	. . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> B e. CC )
5		simprr 522	. . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> v e. CC )
6	3, 4, 5	pnpcan2d 8135	. . . . . . 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 5433	. . . . . 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 3947	. . . . 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 987	. . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> C e. CC )
10	4, 5, 9	pnpcand 8134	. . . . . . 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 5433	. . . . . 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 3947	. . . . 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 465	. . . 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 7769	. . . . . 6 |- ( ( u e. CC /\ v e. CC ) -> ( u + v ) e. CC )
15	14	adantl 275	. . . . 5 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> ( u + v ) e. CC )
16	4, 9	addcld 7809	. . . . 5 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> ( B + C ) e. CC )
17	4, 5	addcld 7809	. . . . 5 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> ( B + v ) e. CC )
18		simpl1 985	. . . . . 6 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> A e. RR+ )
19	18	rpred 9513	. . . . 5 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> A e. RR )
20		abs3lem 10915	. . . . 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 1218	. . . 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 169	. . 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 2517	. 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 3941	. . . . . 6 |- ( y = ( A / 2 ) -> ( ( abs ` ( u - B ) ) < y <-> ( abs ` ( u - B ) ) < ( A / 2 ) ) )
25	24	anbi1d 461	. . . . 5 |- ( y = ( A / 2 ) -> ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( v - C ) ) < z ) <-> ( ( abs ` ( u - B ) ) < ( A / 2 ) /\ ( abs ` ( v - C ) ) < z ) ) )
26	25	imbi1d 230	. . . 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 2462	. . 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 3941	. . . . . 6 |- ( z = ( A / 2 ) -> ( ( abs ` ( v - C ) ) < z <-> ( abs ` ( v - C ) ) < ( A / 2 ) ) )
29	28	anbi2d 460	. . . . 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 230	. . . 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 2462	. . 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 2808	. 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 1217	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 103   /\ w3a 963   = wceq 1332   e. wcel 1481   A.wral 2417   E.wrex 2418   class class class wbr 3937   ` cfv 5131  (class class class)co 5782   CCcc 7642   RRcr 7643   + caddc 7647   < clt 7824   - cmin 7957   / cdiv 8456   2c2 8795   RR+crp 9470   abscabs 10801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762  ax-arch 7763  ax-caucvg 7764

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-frec 6296  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-2 8803  df-3 8804  df-4 8805  df-n0 9002  df-z 9079  df-uz 9351  df-rp 9471  df-seqfrec 10250  df-exp 10324  df-cj 10646  df-re 10647  df-im 10648  df-rsqrt 10802  df-abs 10803

This theorem is referenced by:  subcn2  11112  climadd  11127  addcncntop  12760