Theorem addccncf 15314

Description: Adding a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.)

Hypothesis

Ref	Expression
addccncf.1	|- F = ( x e. CC |-> ( x + A ) )

Assertion

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

Distinct variable group:   x, A

Allowed substitution hint:   F( x)

Proof of Theorem addccncf

Dummy variables w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssid 3245	. 2 |- CC C_ CC
2		addcl 8147	. . . . 5 |- ( ( x e. CC /\ A e. CC ) -> ( x + A ) e. CC )
3	2	ancoms 268	. . . 4 |- ( ( A e. CC /\ x e. CC ) -> ( x + A ) e. CC )
4		addccncf.1	. . . 4 |- F = ( x e. CC |-> ( x + A ) )
5	3, 4	fmptd 5797	. . 3 |- ( A e. CC -> F : CC --> CC )
6		simpr 110	. . . 4 |- ( ( y e. CC /\ w e. RR+ ) -> w e. RR+ )
7	6	a1i 9	. . 3 |- ( A e. CC -> ( ( y e. CC /\ w e. RR+ ) -> w e. RR+ ) )
8		oveq1 6020	. . . . . . . . 9 |- ( x = y -> ( x + A ) = ( y + A ) )
9		simprll 537	. . . . . . . . 9 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> y e. CC )
10		simpl 109	. . . . . . . . . 10 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> A e. CC )
11	9, 10	addcld 8189	. . . . . . . . 9 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( y + A ) e. CC )
12	4, 8, 9, 11	fvmptd3 5736	. . . . . . . 8 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( F ` y ) = ( y + A ) )
13		oveq1 6020	. . . . . . . . 9 |- ( x = z -> ( x + A ) = ( z + A ) )
14		simprlr 538	. . . . . . . . 9 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> z e. CC )
15	14, 10	addcld 8189	. . . . . . . . 9 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( z + A ) e. CC )
16	4, 13, 14, 15	fvmptd3 5736	. . . . . . . 8 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( F ` z ) = ( z + A ) )
17	12, 16	oveq12d 6031	. . . . . . 7 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( ( F ` y ) - ( F ` z ) ) = ( ( y + A ) - ( z + A ) ) )
18	9, 14, 10	pnpcan2d 8518	. . . . . . 7 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( ( y + A ) - ( z + A ) ) = ( y - z ) )
19	17, 18	eqtrd 2262	. . . . . 6 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( ( F ` y ) - ( F ` z ) ) = ( y - z ) )
20	19	fveq2d 5639	. . . . 5 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( abs ` ( ( F ` y ) - ( F ` z ) ) ) = ( abs ` ( y - z ) ) )
21	20	breq1d 4096	. . . 4 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( ( abs ` ( ( F ` y ) - ( F ` z ) ) ) < w <-> ( abs ` ( y - z ) ) < w ) )
22	21	exbiri 382	. . 3 |- ( A e. CC -> ( ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) -> ( ( abs ` ( y - z ) ) < w -> ( abs ` ( ( F ` y ) - ( F ` z ) ) ) < w ) ) )
23	5, 7, 22	elcncf1di 15293	. 2 |- ( A e. CC -> ( ( CC C_ CC /\ CC C_ CC ) -> F e. ( CC -cn-> CC ) ) )
24	1, 1, 23	mp2ani 432	1 |- ( A e. CC -> F e. ( CC -cn-> CC ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   C_ wss 3198   class class class wbr 4086   |-> cmpt 4148   ` cfv 5324  (class class class)co 6013   CCcc 8020   + caddc 8025   < clt 8204   - cmin 8340   RR+crp 9878   abscabs 11548   -cn->ccncf 15284

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-distr 8126  ax-i2m1 8127  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-map 6814  df-sub 8342  df-cncf 15285

This theorem is referenced by: (None)