Theorem rlimi 14244

Description: Convergence at infinity of a function on the reals. (Contributed by Mario Carneiro, 28-Feb-2015.)

Hypotheses

Ref	Expression
rlimi.1	|- ( ph -> A. z e. A B e. V )
rlimi.2	|- ( ph -> R e. RR+ )
rlimi.3	|- ( ph -> ( z e. A |-> B ) ~~> r C )

Assertion

Ref	Expression
rlimi	|- ( ph -> E. y e. RR A. z e. A ( y <_ z -> ( abs ` ( B - C ) ) < R ) )

Distinct variable groups:   y, z, A   y, B   y, C, z   ph, y   y, R, z   z, V

Allowed substitution hints:   ph( z)   B( z)   V( y)

Proof of Theorem rlimi

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rlimi.2	. 2 |- ( ph -> R e. RR+ )
2		rlimi.3	. . 3 |- ( ph -> ( z e. A |-> B ) ~~> r C )
3		rlimf 14232	. . . . . . 7 |- ( ( z e. A |-> B ) ~~> r C -> ( z e. A |-> B ) : dom ( z e. A |-> B ) --> CC )
4	2, 3	syl 17	. . . . . 6 |- ( ph -> ( z e. A |-> B ) : dom ( z e. A |-> B ) --> CC )
5		rlimi.1	. . . . . . . . 9 |- ( ph -> A. z e. A B e. V )
6		eqid 2622	. . . . . . . . . 10 |- ( z e. A |-> B ) = ( z e. A |-> B )
7	6	fmpt 6381	. . . . . . . . 9 |- ( A. z e. A B e. V <-> ( z e. A |-> B ) : A --> V )
8	5, 7	sylib 208	. . . . . . . 8 |- ( ph -> ( z e. A |-> B ) : A --> V )
9		fdm 6051	. . . . . . . 8 |- ( ( z e. A |-> B ) : A --> V -> dom ( z e. A |-> B ) = A )
10	8, 9	syl 17	. . . . . . 7 |- ( ph -> dom ( z e. A |-> B ) = A )
11	10	feq2d 6031	. . . . . 6 |- ( ph -> ( ( z e. A |-> B ) : dom ( z e. A |-> B ) --> CC <-> ( z e. A |-> B ) : A --> CC ) )
12	4, 11	mpbid 222	. . . . 5 |- ( ph -> ( z e. A |-> B ) : A --> CC )
13	6	fmpt 6381	. . . . 5 |- ( A. z e. A B e. CC <-> ( z e. A |-> B ) : A --> CC )
14	12, 13	sylibr 224	. . . 4 |- ( ph -> A. z e. A B e. CC )
15		rlimss 14233	. . . . . 6 |- ( ( z e. A |-> B ) ~~> r C -> dom ( z e. A |-> B ) C_ RR )
16	2, 15	syl 17	. . . . 5 |- ( ph -> dom ( z e. A |-> B ) C_ RR )
17	10, 16	eqsstr3d 3640	. . . 4 |- ( ph -> A C_ RR )
18		rlimcl 14234	. . . . 5 |- ( ( z e. A |-> B ) ~~> r C -> C e. CC )
19	2, 18	syl 17	. . . 4 |- ( ph -> C e. CC )
20	14, 17, 19	rlim2 14227	. . 3 |- ( ph -> ( ( z e. A |-> B ) ~~> r C <-> A. x e. RR+ E. y e. RR A. z e. A ( y <_ z -> ( abs ` ( B - C ) ) < x ) ) )
21	2, 20	mpbid 222	. 2 |- ( ph -> A. x e. RR+ E. y e. RR A. z e. A ( y <_ z -> ( abs ` ( B - C ) ) < x ) )
22		breq2 4657	. . . . 5 |- ( x = R -> ( ( abs ` ( B - C ) ) < x <-> ( abs ` ( B - C ) ) < R ) )
23	22	imbi2d 330	. . . 4 |- ( x = R -> ( ( y <_ z -> ( abs ` ( B - C ) ) < x ) <-> ( y <_ z -> ( abs ` ( B - C ) ) < R ) ) )
24	23	rexralbidv 3058	. . 3 |- ( x = R -> ( E. y e. RR A. z e. A ( y <_ z -> ( abs ` ( B - C ) ) < x ) <-> E. y e. RR A. z e. A ( y <_ z -> ( abs ` ( B - C ) ) < R ) ) )
25	24	rspcv 3305	. 2 |- ( R e. RR+ -> ( A. x e. RR+ E. y e. RR A. z e. A ( y <_ z -> ( abs ` ( B - C ) ) < x ) -> E. y e. RR A. z e. A ( y <_ z -> ( abs ` ( B - C ) ) < R ) ) )
26	1, 21, 25	sylc 65	1 |- ( ph -> E. y e. RR A. z e. A ( y <_ z -> ( abs ` ( B - C ) ) < R ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   < clt 10074   <_ cle 10075   - cmin 10266   RR+crp 11832   abscabs 13974   ~~> r crli 14216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-pm 7860  df-rlim 14220

This theorem is referenced by:  rlimi2  14245  rlimclim1  14276  rlimuni  14281  rlimcld2  14309  rlimcn1  14319  rlimcn2  14321  rlimo1  14347  o1rlimmul  14349  rlimno1  14384  xrlimcnp  24695  rlimcxp  24700  chtppilimlem2  25163  dchrisumlem3  25180