Theorem ulmi 24140

Description: The uniform limit property. (Contributed by Mario Carneiro, 27-Feb-2015.)

Hypotheses

Ref	Expression
ulm2.z	|- Z = ( ZZ>= ` M )
ulm2.m	|- ( ph -> M e. ZZ )
ulm2.f	|- ( ph -> F : Z --> ( CC ^m S ) )
ulm2.b	|- ( ( ph /\ ( k e. Z /\ z e. S ) ) -> ( ( F ` k ) ` z ) = B )
ulm2.a	|- ( ( ph /\ z e. S ) -> ( G ` z ) = A )
ulmi.u	|- ( ph -> F ( ~~> u ` S ) G )
ulmi.c	|- ( ph -> C e. RR+ )

Assertion

Ref	Expression
ulmi	|- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) A. z e. S ( abs ` ( B - A ) ) < C )

Distinct variable groups:   j, k, z, F   j, G, k, z   j, M, k, z   ph, j, k, z   A, j, k   C, j, k, z   S, j, k, z   j, Z

Allowed substitution hints:   A( z)   B( z, j, k)   Z( z, k)

Proof of Theorem ulmi

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ulmi.c	. 2 |- ( ph -> C e. RR+ )
2		ulmi.u	. . 3 |- ( ph -> F ( ~~> u ` S ) G )
3		ulm2.z	. . . 4 |- Z = ( ZZ>= ` M )
4		ulm2.m	. . . 4 |- ( ph -> M e. ZZ )
5		ulm2.f	. . . 4 |- ( ph -> F : Z --> ( CC ^m S ) )
6		ulm2.b	. . . 4 |- ( ( ph /\ ( k e. Z /\ z e. S ) ) -> ( ( F ` k ) ` z ) = B )
7		ulm2.a	. . . 4 |- ( ( ph /\ z e. S ) -> ( G ` z ) = A )
8		ulmcl 24135	. . . . 5 |- ( F ( ~~> u ` S ) G -> G : S --> CC )
9	2, 8	syl 17	. . . 4 |- ( ph -> G : S --> CC )
10		ulmscl 24133	. . . . 5 |- ( F ( ~~> u ` S ) G -> S e. _V )
11	2, 10	syl 17	. . . 4 |- ( ph -> S e. _V )
12	3, 4, 5, 6, 7, 9, 11	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 ` ( B - A ) ) < x ) )
13	2, 12	mpbid 222	. 2 |- ( ph -> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) A. z e. S ( abs ` ( B - A ) ) < x )
14		breq2 4657	. . . . 5 |- ( x = C -> ( ( abs ` ( B - A ) ) < x <-> ( abs ` ( B - A ) ) < C ) )
15	14	ralbidv 2986	. . . 4 |- ( x = C -> ( A. z e. S ( abs ` ( B - A ) ) < x <-> A. z e. S ( abs ` ( B - A ) ) < C ) )
16	15	rexralbidv 3058	. . 3 |- ( x = C -> ( E. j e. Z A. k e. ( ZZ>= ` j ) A. z e. S ( abs ` ( B - A ) ) < x <-> E. j e. Z A. k e. ( ZZ>= ` j ) A. z e. S ( abs ` ( B - A ) ) < C ) )
17	16	rspcv 3305	. 2 |- ( C e. RR+ -> ( A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) A. z e. S ( abs ` ( B - A ) ) < x -> E. j e. Z A. k e. ( ZZ>= ` j ) A. z e. S ( abs ` ( B - A ) ) < C ) )
18	1, 13, 17	sylc 65	1 |- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) A. z e. S ( abs ` ( B - A ) ) < C )

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   ~~> 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-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-ulm 24131

This theorem is referenced by:  ulmshftlem  24143  ulmcau  24149  ulmbdd  24152  ulmcn  24153  iblulm  24161  itgulm  24162