Theorem ulmclm 24141

Description: A uniform limit of functions converges pointwise. (Contributed by Mario Carneiro, 27-Feb-2015.)

Hypotheses

Ref	Expression
ulmclm.z	|- Z = ( ZZ>= ` M )
ulmclm.m	|- ( ph -> M e. ZZ )
ulmclm.f	|- ( ph -> F : Z --> ( CC ^m S ) )
ulmclm.a	|- ( ph -> A e. S )
ulmclm.h	|- ( ph -> H e. W )
ulmclm.e	|- ( ( ph /\ k e. Z ) -> ( ( F ` k ) ` A ) = ( H ` k ) )
ulmclm.u	|- ( ph -> F ( ~~> u ` S ) G )

Assertion

Ref	Expression
ulmclm	|- ( ph -> H ~~> ( G ` A ) )

Distinct variable groups:   A, k   k, F   k, G   ph, k   k, H   k, M   S, k   k, Z

Allowed substitution hint:   W( k)

Proof of Theorem ulmclm

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

Step	Hyp	Ref	Expression
1		ulmclm.u	. 2 |- ( ph -> F ( ~~> u ` S ) G )
2		ulmclm.a	. . . . . . 7 |- ( ph -> A e. S )
3		fveq2 6191	. . . . . . . . . . 11 |- ( z = A -> ( ( F ` k ) ` z ) = ( ( F ` k ) ` A ) )
4		fveq2 6191	. . . . . . . . . . 11 |- ( z = A -> ( G ` z ) = ( G ` A ) )
5	3, 4	oveq12d 6668	. . . . . . . . . 10 |- ( z = A -> ( ( ( F ` k ) ` z ) - ( G ` z ) ) = ( ( ( F ` k ) ` A ) - ( G ` A ) ) )
6	5	fveq2d 6195	. . . . . . . . 9 |- ( z = A -> ( abs ` ( ( ( F ` k ) ` z ) - ( G ` z ) ) ) = ( abs ` ( ( ( F ` k ) ` A ) - ( G ` A ) ) ) )
7	6	breq1d 4663	. . . . . . . 8 |- ( z = A -> ( ( abs ` ( ( ( F ` k ) ` z ) - ( G ` z ) ) ) < x <-> ( abs ` ( ( ( F ` k ) ` A ) - ( G ` A ) ) ) < x ) )
8	7	rspcv 3305	. . . . . . 7 |- ( A e. S -> ( A. z e. S ( abs ` ( ( ( F ` k ) ` z ) - ( G ` z ) ) ) < x -> ( abs ` ( ( ( F ` k ) ` A ) - ( G ` A ) ) ) < x ) )
9	2, 8	syl 17	. . . . . 6 |- ( ph -> ( A. z e. S ( abs ` ( ( ( F ` k ) ` z ) - ( G ` z ) ) ) < x -> ( abs ` ( ( ( F ` k ) ` A ) - ( G ` A ) ) ) < x ) )
10	9	ralimdv 2963	. . . . 5 |- ( ph -> ( A. k e. ( ZZ>= ` j ) A. z e. S ( abs ` ( ( ( F ` k ) ` z ) - ( G ` z ) ) ) < x -> A. k e. ( ZZ>= ` j ) ( abs ` ( ( ( F ` k ) ` A ) - ( G ` A ) ) ) < x ) )
11	10	reximdv 3016	. . . 4 |- ( ph -> ( E. j e. Z A. k e. ( ZZ>= ` j ) A. z e. S ( abs ` ( ( ( F ` k ) ` z ) - ( G ` z ) ) ) < x -> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( ( F ` k ) ` A ) - ( G ` A ) ) ) < x ) )
12	11	ralimdv 2963	. . 3 |- ( ph -> ( A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) A. z e. S ( abs ` ( ( ( F ` k ) ` z ) - ( G ` z ) ) ) < x -> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( ( F ` k ) ` A ) - ( G ` A ) ) ) < x ) )
13		ulmclm.z	. . . 4 |- Z = ( ZZ>= ` M )
14		ulmclm.m	. . . 4 |- ( ph -> M e. ZZ )
15		ulmclm.f	. . . 4 |- ( ph -> F : Z --> ( CC ^m S ) )
16		eqidd 2623	. . . 4 |- ( ( ph /\ ( k e. Z /\ z e. S ) ) -> ( ( F ` k ) ` z ) = ( ( F ` k ) ` z ) )
17		eqidd 2623	. . . 4 |- ( ( ph /\ z e. S ) -> ( G ` z ) = ( G ` z ) )
18		ulmcl 24135	. . . . 5 |- ( F ( ~~> u ` S ) G -> G : S --> CC )
19	1, 18	syl 17	. . . 4 |- ( ph -> G : S --> CC )
20		ulmscl 24133	. . . . 5 |- ( F ( ~~> u ` S ) G -> S e. _V )
21	1, 20	syl 17	. . . 4 |- ( ph -> S e. _V )
22	13, 14, 15, 16, 17, 19, 21	ulm2 24139	. . 3 |- ( ph -> ( F ( ~~> u ` S ) G <-> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) A. z e. S ( abs ` ( ( ( F ` k ) ` z ) - ( G ` z ) ) ) < x ) )
23		ulmclm.h	. . . 4 |- ( ph -> H e. W )
24		ulmclm.e	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( ( F ` k ) ` A ) = ( H ` k ) )
25	24	eqcomd 2628	. . . 4 |- ( ( ph /\ k e. Z ) -> ( H ` k ) = ( ( F ` k ) ` A ) )
26	19, 2	ffvelrnd 6360	. . . 4 |- ( ph -> ( G ` A ) e. CC )
27	15	ffvelrnda 6359	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. ( CC ^m S ) )
28		elmapi 7879	. . . . . 6 |- ( ( F ` k ) e. ( CC ^m S ) -> ( F ` k ) : S --> CC )
29	27, 28	syl 17	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( F ` k ) : S --> CC )
30	2	adantr 481	. . . . 5 |- ( ( ph /\ k e. Z ) -> A e. S )
31	29, 30	ffvelrnd 6360	. . . 4 |- ( ( ph /\ k e. Z ) -> ( ( F ` k ) ` A ) e. CC )
32	13, 14, 23, 25, 26, 31	clim2c 14236	. . 3 |- ( ph -> ( H ~~> ( G ` A ) <-> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( ( F ` k ) ` A ) - ( G ` A ) ) ) < x ) )
33	12, 22, 32	3imtr4d 283	. 2 |- ( ph -> ( F ( ~~> u ` S ) G -> H ~~> ( G ` A ) ) )
34	1, 33	mpd 15	1 |- ( ph -> H ~~> ( G ` A ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   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   ~~> cli 14215   ~~> 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  ax-pre-lttri 10010  ax-pre-lttrn 10011

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-nel 2898  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-po 5035  df-so 5036  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-1st 7168  df-2nd 7169  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-neg 10269  df-z 11378  df-uz 11688  df-clim 14219  df-ulm 24131

This theorem is referenced by:  ulmuni  24146  ulmdvlem3  24156  mbfulm  24160  pserulm  24176  lgamgulm2  24762  lgamcvglem  24766  knoppcnlem9  32491  knoppndvlem4  32506