Theorem addccncf 15323

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 3247	. 2 |- CC C_ CC
2		addcl 8156	. . . . 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 5801	. . 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 6024	. . . . . . . . 9 |- ( x = y -> ( x + A ) = ( y + A ) )
9		simprll 539	. . . . . . . . 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 8198	. . . . . . . . 9 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( y + A ) e. CC )
12	4, 8, 9, 11	fvmptd3 5740	. . . . . . . 8 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( F ` y ) = ( y + A ) )
13		oveq1 6024	. . . . . . . . 9 |- ( x = z -> ( x + A ) = ( z + A ) )
14		simprlr 540	. . . . . . . . 9 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> z e. CC )
15	14, 10	addcld 8198	. . . . . . . . 9 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( z + A ) e. CC )
16	4, 13, 14, 15	fvmptd3 5740	. . . . . . . 8 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( F ` z ) = ( z + A ) )
17	12, 16	oveq12d 6035	. . . . . . 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 8527	. . . . . . 7 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( ( y + A ) - ( z + A ) ) = ( y - z ) )
19	17, 18	eqtrd 2264	. . . . . 6 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( ( F ` y ) - ( F ` z ) ) = ( y - z ) )
20	19	fveq2d 5643	. . . . 5 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( abs ` ( ( F ` y ) - ( F ` z ) ) ) = ( abs ` ( y - z ) ) )
21	20	breq1d 4098	. . . 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 15302	. 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 1397   e. wcel 2202   C_ wss 3200   class class class wbr 4088   |-> cmpt 4150   ` cfv 5326  (class class class)co 6017   CCcc 8029   + caddc 8034   < clt 8213   - cmin 8349   RR+crp 9887   abscabs 11557   -cn->ccncf 15293

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-map 6818  df-sub 8351  df-cncf 15294

This theorem is referenced by: (None)