Theorem addcn2 11870

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 15285 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 9915	. . 3 |- ( A e. RR+ -> ( A / 2 ) e. RR+ )
2	1	3ad2ant1 1044	. 2 |- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> ( A / 2 ) e. RR+ )
3		simprl 531	. . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> u e. CC )
4		simpl2 1027	. . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> B e. CC )
5		simprr 533	. . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> v e. CC )
6	3, 4, 5	pnpcan2d 8527	. . . . . . 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 5643	. . . . . 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 4098	. . . . 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 1028	. . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> C e. CC )
10	4, 5, 9	pnpcand 8526	. . . . . . 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 5643	. . . . . 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 4098	. . . . 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 8156	. . . . . 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 8198	. . . . 5 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> ( B + C ) e. CC )
17	4, 5	addcld 8198	. . . . 5 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> ( B + v ) e. CC )
18		simpl1 1026	. . . . . 6 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> A e. RR+ )
19	18	rpred 9930	. . . . 5 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ ( u e. CC /\ v e. CC ) ) -> A e. RR )
20		abs3lem 11671	. . . . 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 1274	. . . 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 2614	. 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 4092	. . . . . 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 2556	. . 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 4092	. . . . . 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 2556	. . 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 2925	. 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 1273	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 1004   = wceq 1397   e. wcel 2202   A.wral 2510   E.wrex 2511   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   CCcc 8029   RRcr 8030   + caddc 8034   < clt 8213   - cmin 8349   / cdiv 8851   2c2 9193   RR+crp 9887   abscabs 11557

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-z 9479  df-uz 9755  df-rp 9888  df-seqfrec 10709  df-exp 10800  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559

This theorem is referenced by:  subcn2  11871  climadd  11886  addcncntop  15285