Theorem addccncf 15014

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 3212	. 2 |- CC C_ CC
2		addcl 8049	. . . . 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 5733	. . 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 5950	. . . . . . . . 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 8091	. . . . . . . . 9 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( y + A ) e. CC )
12	4, 8, 9, 11	fvmptd3 5672	. . . . . . . 8 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( F ` y ) = ( y + A ) )
13		oveq1 5950	. . . . . . . . 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 8091	. . . . . . . . 9 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( z + A ) e. CC )
16	4, 13, 14, 15	fvmptd3 5672	. . . . . . . 8 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( F ` z ) = ( z + A ) )
17	12, 16	oveq12d 5961	. . . . . . 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 8420	. . . . . . 7 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( ( y + A ) - ( z + A ) ) = ( y - z ) )
19	17, 18	eqtrd 2237	. . . . . 6 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( ( F ` y ) - ( F ` z ) ) = ( y - z ) )
20	19	fveq2d 5579	. . . . 5 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( abs ` ( ( F ` y ) - ( F ` z ) ) ) = ( abs ` ( y - z ) ) )
21	20	breq1d 4053	. . . 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 14993	. 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 1372   e. wcel 2175   C_ wss 3165   class class class wbr 4043   |-> cmpt 4104   ` cfv 5270  (class class class)co 5943   CCcc 7922   + caddc 7927   < clt 8106   - cmin 8242   RR+crp 9774   abscabs 11250   -cn->ccncf 14984

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-addass 8026  ax-distr 8028  ax-i2m1 8029  ax-0id 8032  ax-rnegex 8033  ax-cnre 8035

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-map 6736  df-sub 8244  df-cncf 14985

This theorem is referenced by: (None)