Theorem negcncf 12500

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 3068	. 2 |- ( A C_ CC -> CC C_ CC )
3		ssel2 3042	. . . . 5 |- ( ( A C_ CC /\ x e. A ) -> x e. CC )
4	3	negcld 7931	. . . 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 5506	. . 3 |- ( A C_ CC -> F : A --> CC )
7		simpr 109	. . . 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 7826	. . . . . . . . . 10 |- ( x = u -> -u x = -u u )
10		simprll 507	. . . . . . . . . 10 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> u e. A )
11		simpl 108	. . . . . . . . . . . 12 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> A C_ CC )
12	11, 10	sseldd 3048	. . . . . . . . . . 11 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> u e. CC )
13	12	negcld 7931	. . . . . . . . . 10 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> -u u e. CC )
14	5, 9, 10, 13	fvmptd3 5446	. . . . . . . . 9 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> ( F ` u ) = -u u )
15		negeq 7826	. . . . . . . . . 10 |- ( x = v -> -u x = -u v )
16		simprlr 508	. . . . . . . . . 10 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> v e. A )
17	11, 16	sseldd 3048	. . . . . . . . . . 11 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> v e. CC )
18	17	negcld 7931	. . . . . . . . . 10 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> -u v e. CC )
19	5, 15, 16, 18	fvmptd3 5446	. . . . . . . . 9 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> ( F ` v ) = -u v )
20	14, 19	oveq12d 5724	. . . . . . . 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 7961	. . . . . . . 8 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> ( -u u - -u v ) = ( v - u ) )
22	20, 21	eqtrd 2132	. . . . . . 7 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> ( ( F ` u ) - ( F ` v ) ) = ( v - u ) )
23	22	fveq2d 5357	. . . . . 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 10805	. . . . . 6 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> ( abs ` ( v - u ) ) = ( abs ` ( u - v ) ) )
25	23, 24	eqtrd 2132	. . . . 5 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> ( abs ` ( ( F ` u ) - ( F ` v ) ) ) = ( abs ` ( u - v ) ) )
26	25	breq1d 3885	. . . 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 377	. . 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 12479	. 2 |- ( A C_ CC -> ( ( A C_ CC /\ CC C_ CC ) -> F e. ( A -cn-> CC ) ) )
29	1, 2, 28	mp2and 427	1 |- ( A C_ CC -> F e. ( A -cn-> CC ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1299   e. wcel 1448   C_ wss 3021   class class class wbr 3875   |-> cmpt 3929   ` cfv 5059  (class class class)co 5706   CCcc 7498   < clt 7672   - cmin 7804   -ucneg 7805   RR+crp 9291   abscabs 10609   -cn->ccncf 12470

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-po 4156  df-iso 4157  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-map 6474  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-2 8637  df-cj 10455  df-re 10456  df-im 10457  df-rsqrt 10610  df-abs 10611  df-cncf 12471

This theorem is referenced by:  negfcncf  12501