Theorem subcn2 11454

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 8219	. . 3 |- ( C e. CC -> -u C e. CC )
2		addcn2 11453	. . 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 8219	. . . . . . . . 9 |- ( v e. CC -> -u v e. CC )
5		oveq1 5925	. . . . . . . . . . . . . 14 |- ( w = -u v -> ( w - -u C ) = ( -u v - -u C ) )
6	5	fveq2d 5558	. . . . . . . . . . . . 13 |- ( w = -u v -> ( abs ` ( w - -u C ) ) = ( abs ` ( -u v - -u C ) ) )
7	6	breq1d 4039	. . . . . . . . . . . 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 5926	. . . . . . . . . . . . . 14 |- ( w = -u v -> ( u + w ) = ( u + -u v ) )
10	9	oveq1d 5933	. . . . . . . . . . . . 13 |- ( w = -u v -> ( ( u + w ) - ( B + -u C ) ) = ( ( u + -u v ) - ( B + -u C ) ) )
11	10	fveq2d 5558	. . . . . . . . . . . 12 |- ( w = -u v -> ( abs ` ( ( u + w ) - ( B + -u C ) ) ) = ( abs ` ( ( u + -u v ) - ( B + -u C ) ) ) )
12	11	breq1d 4039	. . . . . . . . . . 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 2860	. . . . . . . . 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 8347	. . . . . . . . . . . 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 5558	. . . . . . . . . . 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 11337	. . . . . . . . . . 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 2226	. . . . . . . . . 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 4039	. . . . . . . . 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 8267	. . . . . . . . . . . 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 8336	. . . . . . . . . . 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 5936	. . . . . . . . . 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 5558	. . . . . . . . 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 4039	. . . . . . . 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 2574	. . . . 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 2561	. . . 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 2595	. . 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 2595	. 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 2164   A.wral 2472   E.wrex 2473   class class class wbr 4029   ` cfv 5254  (class class class)co 5918   CCcc 7870   + caddc 7875   < clt 8054   - cmin 8190   -ucneg 8191   RR+crp 9719   abscabs 11141

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 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-rp 9720  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143

This theorem is referenced by:  climsub  11471  subcncntop  14721