Theorem subcn2 11346

Description: Complex number subtraction is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by Mario Carneiro, 31-Jan-2014.)

Assertion

Ref	Expression
subcn2	|- ( ( 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 subcn2

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		negcl 8182	. . 3 |- ( C e. CC -> -u C e. CC )
2		addcn2 11345	. . 3 |- ( ( A e. RR+ /\ B e. CC /\ -u C e. CC ) -> E. y e. RR+ E. z e. RR+ A. u e. CC A. w e. CC ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( w - -u C ) ) < z ) -> ( abs ` ( ( u + w ) - ( B + -u C ) ) ) < A ) )
3	1, 2	syl3an3 1284	. 2 |- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> E. y e. RR+ E. z e. RR+ A. u e. CC A. w e. CC ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( w - -u C ) ) < z ) -> ( abs ` ( ( u + w ) - ( B + -u C ) ) ) < A ) )
4		negcl 8182	. . . . . . . . 9 |- ( v e. CC -> -u v e. CC )
5		oveq1 5899	. . . . . . . . . . . . . 14 |- ( w = -u v -> ( w - -u C ) = ( -u v - -u C ) )
6	5	fveq2d 5535	. . . . . . . . . . . . 13 |- ( w = -u v -> ( abs ` ( w - -u C ) ) = ( abs ` ( -u v - -u C ) ) )
7	6	breq1d 4028	. . . . . . . . . . . 12 |- ( w = -u v -> ( ( abs ` ( w - -u C ) ) < z <-> ( abs ` ( -u v - -u C ) ) < z ) )
8	7	anbi2d 464	. . . . . . . . . . 11 |- ( w = -u v -> ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( w - -u C ) ) < z ) <-> ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( -u v - -u C ) ) < z ) ) )
9		oveq2 5900	. . . . . . . . . . . . . 14 |- ( w = -u v -> ( u + w ) = ( u + -u v ) )
10	9	oveq1d 5907	. . . . . . . . . . . . 13 |- ( w = -u v -> ( ( u + w ) - ( B + -u C ) ) = ( ( u + -u v ) - ( B + -u C ) ) )
11	10	fveq2d 5535	. . . . . . . . . . . 12 |- ( w = -u v -> ( abs ` ( ( u + w ) - ( B + -u C ) ) ) = ( abs ` ( ( u + -u v ) - ( B + -u C ) ) ) )
12	11	breq1d 4028	. . . . . . . . . . 11 |- ( w = -u v -> ( ( abs ` ( ( u + w ) - ( B + -u C ) ) ) < A <-> ( abs ` ( ( u + -u v ) - ( B + -u C ) ) ) < A ) )
13	8, 12	imbi12d 234	. . . . . . . . . 10 |- ( w = -u v -> ( ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( w - -u C ) ) < z ) -> ( abs ` ( ( u + w ) - ( B + -u C ) ) ) < A ) <-> ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( -u v - -u C ) ) < z ) -> ( abs ` ( ( u + -u v ) - ( B + -u C ) ) ) < A ) ) )
14	13	rspcv 2852	. . . . . . . . 9 |- ( -u v e. CC -> ( A. w e. CC ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( w - -u C ) ) < z ) -> ( abs ` ( ( u + w ) - ( B + -u C ) ) ) < A ) -> ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( -u v - -u C ) ) < z ) -> ( abs ` ( ( u + -u v ) - ( B + -u C ) ) ) < A ) ) )
15	4, 14	syl 14	. . . . . . . 8 |- ( v e. CC -> ( A. w e. CC ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( w - -u C ) ) < z ) -> ( abs ` ( ( u + w ) - ( B + -u C ) ) ) < A ) -> ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( -u v - -u C ) ) < z ) -> ( abs ` ( ( u + -u v ) - ( B + -u C ) ) ) < A ) ) )
16	15	adantl 277	. . . . . . 7 |- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> ( A. w e. CC ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( w - -u C ) ) < z ) -> ( abs ` ( ( u + w ) - ( B + -u C ) ) ) < A ) -> ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( -u v - -u C ) ) < z ) -> ( abs ` ( ( u + -u v ) - ( B + -u C ) ) ) < A ) ) )
17		simpr 110	. . . . . . . . . . . . 13 |- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> v e. CC )
18		simpll3 1040	. . . . . . . . . . . . 13 |- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> C e. CC )
19	17, 18	neg2subd 8310	. . . . . . . . . . . 12 |- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> ( -u v - -u C ) = ( C - v ) )
20	19	fveq2d 5535	. . . . . . . . . . 11 |- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> ( abs ` ( -u v - -u C ) ) = ( abs ` ( C - v ) ) )
21	18, 17	abssubd 11229	. . . . . . . . . . 11 |- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> ( abs ` ( C - v ) ) = ( abs ` ( v - C ) ) )
22	20, 21	eqtrd 2222	. . . . . . . . . 10 |- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> ( abs ` ( -u v - -u C ) ) = ( abs ` ( v - C ) ) )
23	22	breq1d 4028	. . . . . . . . 9 |- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> ( ( abs ` ( -u v - -u C ) ) < z <-> ( abs ` ( v - C ) ) < z ) )
24	23	anbi2d 464	. . . . . . . 8 |- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( -u v - -u C ) ) < z ) <-> ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( v - C ) ) < z ) ) )
25		negsub 8230	. . . . . . . . . . . 12 |- ( ( u e. CC /\ v e. CC ) -> ( u + -u v ) = ( u - v ) )
26	25	adantll 476	. . . . . . . . . . 11 |- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> ( u + -u v ) = ( u - v ) )
27		simpll2 1039	. . . . . . . . . . . 12 |- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> B e. CC )
28	27, 18	negsubd 8299	. . . . . . . . . . 11 |- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> ( B + -u C ) = ( B - C ) )
29	26, 28	oveq12d 5910	. . . . . . . . . 10 |- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> ( ( u + -u v ) - ( B + -u C ) ) = ( ( u - v ) - ( B - C ) ) )
30	29	fveq2d 5535	. . . . . . . . 9 |- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> ( abs ` ( ( u + -u v ) - ( B + -u C ) ) ) = ( abs ` ( ( u - v ) - ( B - C ) ) ) )
31	30	breq1d 4028	. . . . . . . 8 |- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> ( ( abs ` ( ( u + -u v ) - ( B + -u C ) ) ) < A <-> ( abs ` ( ( u - v ) - ( B - C ) ) ) < A ) )
32	24, 31	imbi12d 234	. . . . . . 7 |- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> ( ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( -u v - -u C ) ) < z ) -> ( abs ` ( ( u + -u v ) - ( B + -u C ) ) ) < A ) <-> ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( v - C ) ) < z ) -> ( abs ` ( ( u - v ) - ( B - C ) ) ) < A ) ) )
33	16, 32	sylibd 149	. . . . . 6 |- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> ( A. w e. CC ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( w - -u C ) ) < z ) -> ( abs ` ( ( u + w ) - ( B + -u C ) ) ) < A ) -> ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( v - C ) ) < z ) -> ( abs ` ( ( u - v ) - ( B - C ) ) ) < A ) ) )
34	33	ralrimdva 2570	. . . . 5 |- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) -> ( A. w e. CC ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( w - -u C ) ) < z ) -> ( abs ` ( ( u + w ) - ( B + -u C ) ) ) < A ) -> A. v e. CC ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( v - C ) ) < z ) -> ( abs ` ( ( u - v ) - ( B - C ) ) ) < A ) ) )
35	34	ralimdva 2557	. . . 4 |- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> ( A. u e. CC A. w e. CC ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( w - -u C ) ) < z ) -> ( abs ` ( ( u + w ) - ( B + -u C ) ) ) < A ) -> A. u e. CC A. v e. CC ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( v - C ) ) < z ) -> ( abs ` ( ( u - v ) - ( B - C ) ) ) < A ) ) )
36	35	reximdv 2591	. . 3 |- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> ( E. z e. RR+ A. u e. CC A. w e. CC ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( w - -u C ) ) < z ) -> ( abs ` ( ( u + w ) - ( B + -u C ) ) ) < A ) -> 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 ) ) )
37	36	reximdv 2591	. 2 |- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> ( E. y e. RR+ E. z e. RR+ A. u e. CC A. w e. CC ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( w - -u C ) ) < z ) -> ( abs ` ( ( u + w ) - ( B + -u 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 ) ) )
38	3, 37	mpd 13	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 980   = wceq 1364   e. wcel 2160   A.wral 2468   E.wrex 2469   class class class wbr 4018   ` cfv 5232  (class class class)co 5892   CCcc 7834   + caddc 7839   < clt 8017   - cmin 8153   -ucneg 8154   RR+crp 9678   abscabs 11033

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7927  ax-resscn 7928  ax-1cn 7929  ax-1re 7930  ax-icn 7931  ax-addcl 7932  ax-addrcl 7933  ax-mulcl 7934  ax-mulrcl 7935  ax-addcom 7936  ax-mulcom 7937  ax-addass 7938  ax-mulass 7939  ax-distr 7940  ax-i2m1 7941  ax-0lt1 7942  ax-1rid 7943  ax-0id 7944  ax-rnegex 7945  ax-precex 7946  ax-cnre 7947  ax-pre-ltirr 7948  ax-pre-ltwlin 7949  ax-pre-lttrn 7950  ax-pre-apti 7951  ax-pre-ltadd 7952  ax-pre-mulgt0 7953  ax-pre-mulext 7954  ax-arch 7955  ax-caucvg 7956

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-riota 5848  df-ov 5895  df-oprab 5896  df-mpo 5897  df-1st 6160  df-2nd 6161  df-recs 6325  df-frec 6411  df-pnf 8019  df-mnf 8020  df-xr 8021  df-ltxr 8022  df-le 8023  df-sub 8155  df-neg 8156  df-reap 8557  df-ap 8564  df-div 8655  df-inn 8945  df-2 9003  df-3 9004  df-4 9005  df-n0 9202  df-z 9279  df-uz 9554  df-rp 9679  df-seqfrec 10472  df-exp 10546  df-cj 10878  df-re 10879  df-im 10880  df-rsqrt 11034  df-abs 11035

This theorem is referenced by:  climsub  11363  subcncntop  14490