Theorem negcncf 15192

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 3222	. 2 |- ( A C_ CC -> CC C_ CC )
3		ssel2 3196	. . . . 5 |- ( ( A C_ CC /\ x e. A ) -> x e. CC )
4	3	negcld 8405	. . . 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 5757	. . 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 8300	. . . . . . . . . 10 |- ( x = u -> -u x = -u u )
10		simprll 537	. . . . . . . . . 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 3202	. . . . . . . . . . 11 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> u e. CC )
13	12	negcld 8405	. . . . . . . . . 10 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> -u u e. CC )
14	5, 9, 10, 13	fvmptd3 5696	. . . . . . . . 9 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> ( F ` u ) = -u u )
15		negeq 8300	. . . . . . . . . 10 |- ( x = v -> -u x = -u v )
16		simprlr 538	. . . . . . . . . 10 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> v e. A )
17	11, 16	sseldd 3202	. . . . . . . . . . 11 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> v e. CC )
18	17	negcld 8405	. . . . . . . . . 10 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> -u v e. CC )
19	5, 15, 16, 18	fvmptd3 5696	. . . . . . . . 9 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> ( F ` v ) = -u v )
20	14, 19	oveq12d 5985	. . . . . . . 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 8435	. . . . . . . 8 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> ( -u u - -u v ) = ( v - u ) )
22	20, 21	eqtrd 2240	. . . . . . 7 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> ( ( F ` u ) - ( F ` v ) ) = ( v - u ) )
23	22	fveq2d 5603	. . . . . 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 11619	. . . . . 6 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> ( abs ` ( v - u ) ) = ( abs ` ( u - v ) ) )
25	23, 24	eqtrd 2240	. . . . 5 |- ( ( A C_ CC /\ ( ( u e. A /\ v e. A ) /\ e e. RR+ ) ) -> ( abs ` ( ( F ` u ) - ( F ` v ) ) ) = ( abs ` ( u - v ) ) )
26	25	breq1d 4069	. . . 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 15166	. 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 1373   e. wcel 2178   C_ wss 3174   class class class wbr 4059   |-> cmpt 4121   ` cfv 5290  (class class class)co 5967   CCcc 7958   < clt 8142   - cmin 8278   -ucneg 8279   RR+crp 9810   abscabs 11423   -cn->ccncf 15157

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-po 4361  df-iso 4362  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-map 6760  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-2 9130  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425  df-cncf 15158

This theorem is referenced by:  negfcncf  15193