Theorem subcn2 11252

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 8098	. . 3 |- ( C e. CC -> -u C e. CC )
2		addcn2 11251	. . 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 1263	. 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 8098	. . . . . . . . 9 |- ( v e. CC -> -u v e. CC )
5		oveq1 5849	. . . . . . . . . . . . . 14 |- ( w = -u v -> ( w - -u C ) = ( -u v - -u C ) )
6	5	fveq2d 5490	. . . . . . . . . . . . 13 |- ( w = -u v -> ( abs ` ( w - -u C ) ) = ( abs ` ( -u v - -u C ) ) )
7	6	breq1d 3992	. . . . . . . . . . . 12 |- ( w = -u v -> ( ( abs ` ( w - -u C ) ) < z <-> ( abs ` ( -u v - -u C ) ) < z ) )
8	7	anbi2d 460	. . . . . . . . . . 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 5850	. . . . . . . . . . . . . 14 |- ( w = -u v -> ( u + w ) = ( u + -u v ) )
10	9	oveq1d 5857	. . . . . . . . . . . . 13 |- ( w = -u v -> ( ( u + w ) - ( B + -u C ) ) = ( ( u + -u v ) - ( B + -u C ) ) )
11	10	fveq2d 5490	. . . . . . . . . . . 12 |- ( w = -u v -> ( abs ` ( ( u + w ) - ( B + -u C ) ) ) = ( abs ` ( ( u + -u v ) - ( B + -u C ) ) ) )
12	11	breq1d 3992	. . . . . . . . . . 11 |- ( w = -u v -> ( ( abs ` ( ( u + w ) - ( B + -u C ) ) ) < A <-> ( abs ` ( ( u + -u v ) - ( B + -u C ) ) ) < A ) )
13	8, 12	imbi12d 233	. . . . . . . . . 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 2826	. . . . . . . . 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 275	. . . . . . 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 109	. . . . . . . . . . . . 13 |- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> v e. CC )
18		simpll3 1028	. . . . . . . . . . . . 13 |- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> C e. CC )
19	17, 18	neg2subd 8226	. . . . . . . . . . . 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 5490	. . . . . . . . . . 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 11135	. . . . . . . . . . 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 2198	. . . . . . . . . 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 3992	. . . . . . . . 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 460	. . . . . . . 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 8146	. . . . . . . . . . . 12 |- ( ( u e. CC /\ v e. CC ) -> ( u + -u v ) = ( u - v ) )
26	25	adantll 468	. . . . . . . . . . 11 |- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> ( u + -u v ) = ( u - v ) )
27		simpll2 1027	. . . . . . . . . . . 12 |- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> B e. CC )
28	27, 18	negsubd 8215	. . . . . . . . . . 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 5860	. . . . . . . . . 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 5490	. . . . . . . . 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 3992	. . . . . . . 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 233	. . . . . . 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 148	. . . . . 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 2546	. . . . 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 2533	. . . 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 2567	. . 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 2567	. 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 103   /\ w3a 968   = wceq 1343   e. wcel 2136   A.wral 2444   E.wrex 2445   class class class wbr 3982   ` cfv 5188  (class class class)co 5842   CCcc 7751   + caddc 7756   < clt 7933   - cmin 8069   -ucneg 8070   RR+crp 9589   abscabs 10939

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-rp 9590  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941

This theorem is referenced by:  climsub  11269  subcncntop  13193