Theorem addcn2 11706

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 15119 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 9833	. . 3 |- ( A e. RR+ -> ( A / 2 ) e. RR+ )
2	1	3ad2ant1 1021	. 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 1004	. . . . . . . 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 8451	. . . . . . 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 5598	. . . . . 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 4064	. . . . 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 1005	. . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> C e. CC )
10	4, 5, 9	pnpcand 8450	. . . . . . 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 5598	. . . . . 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 4064	. . . . 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 8080	. . . . . 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 8122	. . . . 5 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> ( B + C ) e. CC )
17	4, 5	addcld 8122	. . . . 5 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> ( B + v ) e. CC )
18		simpl1 1003	. . . . . 6 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> A e. RR+ )
19	18	rpred 9848	. . . . 5 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> A e. RR )
20		abs3lem 11507	. . . . 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 1251	. . . 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 2589	. 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 4058	. . . . . 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 2531	. . 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 4058	. . . . . 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 2531	. . 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 2896	. 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 1250	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 981   = wceq 1373   e. wcel 2177   A.wral 2485   E.wrex 2486   class class class wbr 4054   ` cfv 5285  (class class class)co 5962   CCcc 7953   RRcr 7954   + caddc 7958   < clt 8137   - cmin 8273   / cdiv 8775   2c2 9117   RR+crp 9805   abscabs 11393

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4170  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-iinf 4649  ax-cnex 8046  ax-resscn 8047  ax-1cn 8048  ax-1re 8049  ax-icn 8050  ax-addcl 8051  ax-addrcl 8052  ax-mulcl 8053  ax-mulrcl 8054  ax-addcom 8055  ax-mulcom 8056  ax-addass 8057  ax-mulass 8058  ax-distr 8059  ax-i2m1 8060  ax-0lt1 8061  ax-1rid 8062  ax-0id 8063  ax-rnegex 8064  ax-precex 8065  ax-cnre 8066  ax-pre-ltirr 8067  ax-pre-ltwlin 8068  ax-pre-lttrn 8069  ax-pre-apti 8070  ax-pre-ltadd 8071  ax-pre-mulgt0 8072  ax-pre-mulext 8073  ax-arch 8074  ax-caucvg 8075

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-tr 4154  df-id 4353  df-po 4356  df-iso 4357  df-iord 4426  df-on 4428  df-ilim 4429  df-suc 4431  df-iom 4652  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-riota 5917  df-ov 5965  df-oprab 5966  df-mpo 5967  df-1st 6244  df-2nd 6245  df-recs 6409  df-frec 6495  df-pnf 8139  df-mnf 8140  df-xr 8141  df-ltxr 8142  df-le 8143  df-sub 8275  df-neg 8276  df-reap 8678  df-ap 8685  df-div 8776  df-inn 9067  df-2 9125  df-3 9126  df-4 9127  df-n0 9326  df-z 9403  df-uz 9679  df-rp 9806  df-seqfrec 10625  df-exp 10716  df-cj 11238  df-re 11239  df-im 11240  df-rsqrt 11394  df-abs 11395

This theorem is referenced by:  subcn2  11707  climadd  11722  addcncntop  15119