Theorem negcncf 15706

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 3269	. 2 |- ( A C_ CC -> CC C_ CC )
3		ssel2 3243	. . . . 5 |- ( ( A C_ CC /\ x e. A ) -> x e. CC )
4	3	negcld 8624	. . . 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 5862	. . 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 8519	. . . . . . . . . 10 |- ( x = u -> -u x = -u u )
10		simprll 543	. . . . . . . . . 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 3249	. . . . . . . . . . 11 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> u e. CC )
13	12	negcld 8624	. . . . . . . . . 10 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> -u u e. CC )
14	5, 9, 10, 13	fvmptd3 5799	. . . . . . . . 9 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> ( F ` u ) = -u u )
15		negeq 8519	. . . . . . . . . 10 |- ( x = v -> -u x = -u v )
16		simprlr 544	. . . . . . . . . 10 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> v e. A )
17	11, 16	sseldd 3249	. . . . . . . . . . 11 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> v e. CC )
18	17	negcld 8624	. . . . . . . . . 10 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> -u v e. CC )
19	5, 15, 16, 18	fvmptd3 5799	. . . . . . . . 9 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> ( F ` v ) = -u v )
20	14, 19	oveq12d 6103	. . . . . . . 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 8654	. . . . . . . 8 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> ( -u u - -u v ) = ( v - u ) )
22	20, 21	eqtrd 2271	. . . . . . 7 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> ( ( F ` u ) - ( F ` v ) ) = ( v - u ) )
23	22	fveq2d 5699	. . . . . 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 11959	. . . . . 6 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> ( abs ` ( v - u ) ) = ( abs ` ( u - v ) ) )
25	23, 24	eqtrd 2271	. . . . 5 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> ( abs ` ( ( F ` u ) - ( F ` v ) ) ) = ( abs ` ( u - v ) ) )
26	25	breq1d 4140	. . . 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 15680	. 2 |- ( A C_ CC -> ( ( A C_ CC /\ CC C_ CC ) -> F e. ( A -cn-> CC ) ) )
29	1, 2, 28	mp2and 437	1 |- ( A C_ CC -> F e. ( A -cn-> CC ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   C_ wss 3220   class class class wbr 4130   |-> cmpt 4192   ` cfv 5377  (class class class)co 6085   CCcc 8177   < clt 8360   - cmin 8497   -ucneg 8498   RR+crp 10054   abscabs 11763   -cn->ccncf 15671

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-po 4441  df-iso 4442  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-map 6924  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-2 9363  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-cncf 15672

This theorem is used by:  negfcncf  15707