Theorem addccncf 14754

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 3199	. 2 |- CC C_ CC
2		addcl 7997	. . . . 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 5712	. . 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 5925	. . . . . . . . 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 8039	. . . . . . . . 9 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( y + A ) e. CC )
12	4, 8, 9, 11	fvmptd3 5651	. . . . . . . 8 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( F ` y ) = ( y + A ) )
13		oveq1 5925	. . . . . . . . 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 8039	. . . . . . . . 9 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( z + A ) e. CC )
16	4, 13, 14, 15	fvmptd3 5651	. . . . . . . 8 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( F ` z ) = ( z + A ) )
17	12, 16	oveq12d 5936	. . . . . . 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 8368	. . . . . . 7 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( ( y + A ) - ( z + A ) ) = ( y - z ) )
19	17, 18	eqtrd 2226	. . . . . 6 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( ( F ` y ) - ( F ` z ) ) = ( y - z ) )
20	19	fveq2d 5558	. . . . 5 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( abs ` ( ( F ` y ) - ( F ` z ) ) ) = ( abs ` ( y - z ) ) )
21	20	breq1d 4039	. . . 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 14734	. 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 1364   e. wcel 2164   C_ wss 3153   class class class wbr 4029   |-> cmpt 4090   ` cfv 5254  (class class class)co 5918   CCcc 7870   + caddc 7875   < clt 8054   - cmin 8190   RR+crp 9719   abscabs 11141   -cn->ccncf 14725

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-map 6704  df-sub 8192  df-cncf 14726

This theorem is referenced by: (None)