Theorem addccncf 15147

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 3217	. 2 |- CC C_ CC
2		addcl 8070	. . . . 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 5747	. . 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 5964	. . . . . . . . 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 8112	. . . . . . . . 9 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( y + A ) e. CC )
12	4, 8, 9, 11	fvmptd3 5686	. . . . . . . 8 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( F ` y ) = ( y + A ) )
13		oveq1 5964	. . . . . . . . 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 8112	. . . . . . . . 9 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( z + A ) e. CC )
16	4, 13, 14, 15	fvmptd3 5686	. . . . . . . 8 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( F ` z ) = ( z + A ) )
17	12, 16	oveq12d 5975	. . . . . . 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 8441	. . . . . . 7 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( ( y + A ) - ( z + A ) ) = ( y - z ) )
19	17, 18	eqtrd 2239	. . . . . 6 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( ( F ` y ) - ( F ` z ) ) = ( y - z ) )
20	19	fveq2d 5593	. . . . 5 |- ( ( A e. CC /\ ( ( y e. CC /\ z e. CC ) /\ w e. RR+ ) ) -> ( abs ` ( ( F ` y ) - ( F ` z ) ) ) = ( abs ` ( y - z ) ) )
21	20	breq1d 4061	. . . 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 15126	. 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 1373   e. wcel 2177   C_ wss 3170   class class class wbr 4051   |-> cmpt 4113   ` cfv 5280  (class class class)co 5957   CCcc 7943   + caddc 7948   < clt 8127   - cmin 8263   RR+crp 9795   abscabs 11383   -cn->ccncf 15117

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 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-addcom 8045  ax-addass 8047  ax-distr 8049  ax-i2m1 8050  ax-0id 8053  ax-rnegex 8054  ax-cnre 8056

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 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-map 6750  df-sub 8265  df-cncf 15118

This theorem is referenced by: (None)