Theorem addcn2 14324

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 21031 and df-cncf 22681 are not yet available to us. See addcn 22668 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 11858	. . 3 |- ( A e. RR+ -> ( A / 2 ) e. RR+ )
2	1	3ad2ant1 1082	. 2 |- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> ( A / 2 ) e. RR+ )
3		simprl 794	. . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> u e. CC )
4		simpl2 1065	. . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> B e. CC )
5		simprr 796	. . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> v e. CC )
6	3, 4, 5	pnpcan2d 10430	. . . . . . 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 6195	. . . . . 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 4663	. . . . 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 1066	. . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> C e. CC )
10	4, 5, 9	pnpcand 10429	. . . . . . 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 6195	. . . . . 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 4663	. . . . 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 747	. . . 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 10018	. . . . . 6 |- ( ( u e. CC /\ v e. CC ) -> ( u + v ) e. CC )
15	14	adantl 482	. . . . 5 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> ( u + v ) e. CC )
16	4, 9	addcld 10059	. . . . 5 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> ( B + C ) e. CC )
17	4, 5	addcld 10059	. . . . 5 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> ( B + v ) e. CC )
18		simpl1 1064	. . . . . 6 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> A e. RR+ )
19	18	rpred 11872	. . . . 5 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> A e. RR )
20		abs3lem 14078	. . . . 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 1327	. . . 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 250	. . 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 2971	. 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 4657	. . . . . 6 |- ( y = ( A / 2 ) -> ( ( abs ` ( u - B ) ) < y <-> ( abs ` ( u - B ) ) < ( A / 2 ) ) )
25	24	anbi1d 741	. . . . 5 |- ( y = ( A / 2 ) -> ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( v - C ) ) < z ) <-> ( ( abs ` ( u - B ) ) < ( A / 2 ) /\ ( abs ` ( v - C ) ) < z ) ) )
26	25	imbi1d 331	. . . 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 2989	. . 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 4657	. . . . . 6 |- ( z = ( A / 2 ) -> ( ( abs ` ( v - C ) ) < z <-> ( abs ` ( v - C ) ) < ( A / 2 ) ) )
29	28	anbi2d 740	. . . . 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 331	. . . 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 2989	. . 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 3324	. 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 1326	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 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   + caddc 9939   < clt 10074   - cmin 10266   / cdiv 10684   2c2 11070   RR+crp 11832   abscabs 13974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976

This theorem is referenced by:  subcn2  14325  climadd  14362  rlimadd  14373  addcn  22668