Theorem negcncf 15596

Description: The negative function is continuous. (Contributed by Mario Carneiro, 30-Dec-2016.)

Hypothesis

Ref	Expression
negcncf.1	|- F = ( x e. A |-> -u x )

Assertion

Ref	Expression
negcncf	|- ( A C_ CC -> F e. ( A -cn-> CC ) )

Distinct variable group:   x, A

Allowed substitution hint:   F( x)

Proof of Theorem negcncf

Dummy variables e u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 19	. 2 |- ( A C_ CC -> A C_ CC )
2		ssidd 3263	. 2 |- ( A C_ CC -> CC C_ CC )
3		ssel2 3237	. . . . 5 |- ( ( A C_ CC /\ x e. A ) -> x e. CC )
4	3	negcld 8587	. . . 4 |- ( ( A C_ CC /\ x e. A ) -> -u x e. CC )
5		negcncf.1	. . . 4 |- F = ( x e. A |-> -u x )
6	4, 5	fmptd 5836	. . 3 |- ( A C_ CC -> F : A --> CC )
7		simpr 110	. . . 4 |- ( ( u e. A /\ e e. RR+ ) -> e e. RR+ )
8	7	a1i 9	. . 3 |- ( A C_ CC -> ( ( u e. A /\ e e. RR+ ) -> e e. RR+ ) )
9		negeq 8482	. . . . . . . . . 10 |- ( x = u -> -u x = -u u )
10		simprll 539	. . . . . . . . . 10 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> u e. A )
11		simpl 109	. . . . . . . . . . . 12 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> A C_ CC )
12	11, 10	sseldd 3243	. . . . . . . . . . 11 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> u e. CC )
13	12	negcld 8587	. . . . . . . . . 10 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> -u u e. CC )
14	5, 9, 10, 13	fvmptd3 5776	. . . . . . . . 9 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> ( F ` u ) = -u u )
15		negeq 8482	. . . . . . . . . 10 |- ( x = v -> -u x = -u v )
16		simprlr 540	. . . . . . . . . 10 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> v e. A )
17	11, 16	sseldd 3243	. . . . . . . . . . 11 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> v e. CC )
18	17	negcld 8587	. . . . . . . . . 10 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> -u v e. CC )
19	5, 15, 16, 18	fvmptd3 5776	. . . . . . . . 9 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> ( F ` v ) = -u v )
20	14, 19	oveq12d 6076	. . . . . . . 8 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> ( ( F ` u ) - ( F ` v ) ) = ( -u u - -u v ) )
21	12, 17	neg2subd 8617	. . . . . . . 8 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> ( -u u - -u v ) = ( v - u ) )
22	20, 21	eqtrd 2267	. . . . . . 7 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> ( ( F ` u ) - ( F ` v ) ) = ( v - u ) )
23	22	fveq2d 5679	. . . . . 6 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> ( abs ` ( ( F ` u ) - ( F ` v ) ) ) = ( abs ` ( v - u ) ) )
24	17, 12	abssubd 11903	. . . . . 6 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> ( abs ` ( v - u ) ) = ( abs ` ( u - v ) ) )
25	23, 24	eqtrd 2267	. . . . 5 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> ( abs ` ( ( F ` u ) - ( F ` v ) ) ) = ( abs ` ( u - v ) ) )
26	25	breq1d 4124	. . . 4 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> ( ( abs ` ( ( F ` u ) - ( F ` v ) ) ) < e <-> ( abs ` ( u - v ) ) < e ) )
27	26	exbiri 382	. . 3 |- ( A C_ CC -> ( ( ( u e. A /\ v e. A ) /\ e e. RR+ ) -> ( ( abs ` ( u - v ) ) < e -> ( abs ` ( ( F ` u ) - ( F ` v ) ) ) < e ) ) )
28	6, 8, 27	elcncf1di 15570	. 2 |- ( A C_ CC -> ( ( A C_ CC /\ CC C_ CC ) -> F e. ( A -cn-> CC ) ) )
29	1, 2, 28	mp2and 433	1 |- ( A C_ CC -> F e. ( A -cn-> CC ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   C_ wss 3214   class class class wbr 4114   |-> cmpt 4176   ` cfv 5357  (class class class)co 6058   CCcc 8141   < clt 8324   - cmin 8460   -ucneg 8461   RR+crp 10004   abscabs 11707   -cn->ccncf 15561

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-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  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

This theorem depends on definitions:  df-bi 117  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-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-po 4422  df-iso 4423  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-map 6897  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-2 9313  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-cncf 15562

This theorem is referenced by:  negfcncf  15597