Theorem subcn2 12021

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 8489	. . 3 |- ( C e. CC -> -u C e. CC )
2		addcn2 12020	. . 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 1309	. 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 8489	. . . . . . . . 9 |- ( v e. CC -> -u v e. CC )
5		oveq1 6065	. . . . . . . . . . . . . 14 |- ( w = -u v -> ( w - -u C ) = ( -u v - -u C ) )
6	5	fveq2d 5679	. . . . . . . . . . . . 13 |- ( w = -u v -> ( abs ` ( w - -u C ) ) = ( abs ` ( -u v - -u C ) ) )
7	6	breq1d 4124	. . . . . . . . . . . 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 6066	. . . . . . . . . . . . . 14 |- ( w = -u v -> ( u + w ) = ( u + -u v ) )
10	9	oveq1d 6073	. . . . . . . . . . . . 13 |- ( w = -u v -> ( ( u + w ) - ( B + -u C ) ) = ( ( u + -u v ) - ( B + -u C ) ) )
11	10	fveq2d 5679	. . . . . . . . . . . 12 |- ( w = -u v -> ( abs ` ( ( u + w ) - ( B + -u C ) ) ) = ( abs ` ( ( u + -u v ) - ( B + -u C ) ) ) )
12	11	breq1d 4124	. . . . . . . . . . 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 2919	. . . . . . . . 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 1065	. . . . . . . . . . . . 13 |- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> C e. CC )
19	17, 18	neg2subd 8617	. . . . . . . . . . . 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 5679	. . . . . . . . . . 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 11903	. . . . . . . . . . 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 2267	. . . . . . . . . 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 4124	. . . . . . . . 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 8537	. . . . . . . . . . . 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 1064	. . . . . . . . . . . 12 |- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> B e. CC )
28	27, 18	negsubd 8606	. . . . . . . . . . 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 6076	. . . . . . . . . 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 5679	. . . . . . . . 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 4124	. . . . . . . 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 2624	. . . . 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 2611	. . . 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 2645	. . 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 2645	. 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 1005   = wceq 1398   e. wcel 2205   A.wral 2522   E.wrex 2523   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   CCcc 8141   + caddc 8146   < clt 8324   - cmin 8460   -ucneg 8461   RR+crp 10004   abscabs 11707

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-rp 10005  df-seqfrec 10834  df-exp 10925  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709

This theorem is referenced by:  climsub  12038  subcncntop  15554