Theorem negcncf 15632

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 8617	. . . 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 5856	. . 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 8512	. . . . . . . . . 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 8617	. . . . . . . . . 10 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> -u u e. CC )
14	5, 9, 10, 13	fvmptd3 5796	. . . . . . . . 9 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> ( F ` u ) = -u u )
15		negeq 8512	. . . . . . . . . 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 8617	. . . . . . . . . 10 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> -u v e. CC )
19	5, 15, 16, 18	fvmptd3 5796	. . . . . . . . 9 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> ( F ` v ) = -u v )
20	14, 19	oveq12d 6096	. . . . . . . 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 8647	. . . . . . . 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 5697	. . . . . 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 11940	. . . . . 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 4138	. . . 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 15606	. 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 ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   C_ wss 3220   class class class wbr 4128   |-> cmpt 4190   ` cfv 5375  (class class class)co 6078   CCcc 8170   < clt 8353   - cmin 8490   -ucneg 8491   RR+crp 10036   abscabs 11744   -cn->ccncf 15597

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 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 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8263  ax-resscn 8264  ax-1cn 8265  ax-1re 8266  ax-icn 8267  ax-addcl 8268  ax-addrcl 8269  ax-mulcl 8270  ax-mulrcl 8271  ax-addcom 8272  ax-mulcom 8273  ax-addass 8274  ax-mulass 8275  ax-distr 8276  ax-i2m1 8277  ax-0lt1 8278  ax-1rid 8279  ax-0id 8280  ax-rnegex 8281  ax-precex 8282  ax-cnre 8283  ax-pre-ltirr 8284  ax-pre-ltwlin 8285  ax-pre-lttrn 8286  ax-pre-apti 8287  ax-pre-ltadd 8288  ax-pre-mulgt0 8289  ax-pre-mulext 8290

This theorem 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 3714  df-pr 3715  df-op 3717  df-uni 3934  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-po 4439  df-iso 4440  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6031  df-ov 6081  df-oprab 6082  df-mpo 6083  df-map 6917  df-pnf 8355  df-mnf 8356  df-xr 8357  df-ltxr 8358  df-le 8359  df-sub 8492  df-neg 8493  df-reap 8896  df-ap 8903  df-div 8996  df-2 9345  df-cj 11588  df-re 11589  df-im 11590  df-rsqrt 11745  df-abs 11746  df-cncf 15598

This theorem is referenced by:  negfcncf  15633