Theorem ulmcl 24135

Description: Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.)

Assertion

Ref	Expression
ulmcl	|- ( F ( ~~> u ` S ) G -> G : S --> CC )

Proof of Theorem ulmcl

Dummy variables j k n x z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ulmscl 24133	. . . 4 |- ( F ( ~~> u ` S ) G -> S e. _V )
2		ulmval 24134	. . . 4 |- ( S e. _V -> ( F ( ~~> u ` S ) G <-> E. n e. ZZ ( F : ( ZZ>= ` n ) --> ( CC ^m S ) /\ G : S --> CC /\ A. x e. RR+ E. j e. ( ZZ>= ` n ) A. k e. ( ZZ>= ` j ) A. z e. S ( abs ` ( ( ( F ` k ) ` z ) - ( G ` z ) ) ) < x ) ) )
3	1, 2	syl 17	. . 3 |- ( F ( ~~> u ` S ) G -> ( F ( ~~> u ` S ) G <-> E. n e. ZZ ( F : ( ZZ>= ` n ) --> ( CC ^m S ) /\ G : S --> CC /\ A. x e. RR+ E. j e. ( ZZ>= ` n ) A. k e. ( ZZ>= ` j ) A. z e. S ( abs ` ( ( ( F ` k ) ` z ) - ( G ` z ) ) ) < x ) ) )
4	3	ibi 256	. 2 |- ( F ( ~~> u ` S ) G -> E. n e. ZZ ( F : ( ZZ>= ` n ) --> ( CC ^m S ) /\ G : S --> CC /\ A. x e. RR+ E. j e. ( ZZ>= ` n ) A. k e. ( ZZ>= ` j ) A. z e. S ( abs ` ( ( ( F ` k ) ` z ) - ( G ` z ) ) ) < x ) )
5		simp2 1062	. . 3 |- ( ( F : ( ZZ>= ` n ) --> ( CC ^m S ) /\ G : S --> CC /\ A. x e. RR+ E. j e. ( ZZ>= ` n ) A. k e. ( ZZ>= ` j ) A. z e. S ( abs ` ( ( ( F ` k ) ` z ) - ( G ` z ) ) ) < x ) -> G : S --> CC )
6	5	rexlimivw 3029	. 2 |- ( E. n e. ZZ ( F : ( ZZ>= ` n ) --> ( CC ^m S ) /\ G : S --> CC /\ A. x e. RR+ E. j e. ( ZZ>= ` n ) A. k e. ( ZZ>= ` j ) A. z e. S ( abs ` ( ( ( F ` k ) ` z ) - ( G ` z ) ) ) < x ) -> G : S --> CC )
7	4, 6	syl 17	1 |- ( F ( ~~> u ` S ) G -> G : S --> CC )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   class class class wbr 4653   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   CCcc 9934   < clt 10074   - cmin 10266   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832   abscabs 13974   ~~> uculm 24130

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-rep 4771  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-3or 1038  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-reu 2919  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-iun 4522  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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-pm 7860  df-neg 10269  df-z 11378  df-uz 11688  df-ulm 24131

This theorem is referenced by:  ulmi  24140  ulmclm  24141  ulmres  24142  ulmshftlem  24143  ulmuni  24146  ulmcau  24149  ulmss  24151  ulmbdd  24152  ulmcn  24153  ulmdvlem1  24154  ulmdvlem3  24156  ulmdv  24157  mbfulm  24160  iblulm  24161  itgulm  24162  itgulm2  24163  pserulm  24176  lgamgulmlem6  24760  lgamgulm2  24762  knoppcnlem9  32491