Theorem lmconst 15319

Description: A constant sequence converges to its value. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 14-Nov-2013.)

Hypothesis

Ref	Expression
lmconst.2	|- Z = ( ZZ>= ` M )

Assertion

Ref	Expression
lmconst	|- ( ( J e. (TopOn ` X ) /\ P e. X /\ M e. ZZ ) -> ( Z X. { P } ) ( ~~> t ` J ) P )

Proof of Theorem lmconst

Dummy variables j k u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1029	. 2 |- ( ( J e. (TopOn ` X ) /\ P e. X /\ M e. ZZ ) -> P e. X )
2		simp3 1030	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ P e. X /\ M e. ZZ ) -> M e. ZZ )
3		uzid 9938	. . . . . 6 |- ( M e. ZZ -> M e. ( ZZ>= ` M ) )
4	2, 3	syl 14	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ P e. X /\ M e. ZZ ) -> M e. ( ZZ>= ` M ) )
5		lmconst.2	. . . . 5 |- Z = ( ZZ>= ` M )
6	4, 5	eleqtrrdi 2332	. . . 4 |- ( ( J e. (TopOn ` X ) /\ P e. X /\ M e. ZZ ) -> M e. Z )
7		idd 21	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ P e. X /\ M e. ZZ ) /\ k e. ( ZZ>= ` M ) ) -> ( P e. u -> P e. u ) )
8	7	ralrimdva 2630	. . . 4 |- ( ( J e. (TopOn ` X ) /\ P e. X /\ M e. ZZ ) -> ( P e. u -> A. k e. ( ZZ>= ` M ) P e. u ) )
9		fveq2 5695	. . . . . 6 |- ( j = M -> ( ZZ>= ` j ) = ( ZZ>= ` M ) )
10	9	raleqdv 2755	. . . . 5 |- ( j = M -> ( A. k e. ( ZZ>= ` j ) P e. u <-> A. k e. ( ZZ>= ` M ) P e. u ) )
11	10	rspcev 2929	. . . 4 |- ( ( M e. Z /\ A. k e. ( ZZ>= ` M ) P e. u ) -> E. j e. Z A. k e. ( ZZ>= ` j ) P e. u )
12	6, 8, 11	syl6an 1483	. . 3 |- ( ( J e. (TopOn ` X ) /\ P e. X /\ M e. ZZ ) -> ( P e. u -> E. j e. Z A. k e. ( ZZ>= ` j ) P e. u ) )
13	12	ralrimivw 2624	. 2 |- ( ( J e. (TopOn ` X ) /\ P e. X /\ M e. ZZ ) -> A. u e. J ( P e. u -> E. j e. Z A. k e. ( ZZ>= ` j ) P e. u ) )
14		simp1 1028	. . 3 |- ( ( J e. (TopOn ` X ) /\ P e. X /\ M e. ZZ ) -> J e. (TopOn ` X ) )
15		fconst6g 5591	. . . 4 |- ( P e. X -> ( Z X. { P } ) : Z --> X )
16	1, 15	syl 14	. . 3 |- ( ( J e. (TopOn ` X ) /\ P e. X /\ M e. ZZ ) -> ( Z X. { P } ) : Z --> X )
17		fvconst2g 5929	. . . 4 |- ( ( P e. X /\ k e. Z ) -> ( ( Z X. { P } ) ` k ) = P )
18	1, 17	sylan 283	. . 3 |- ( ( ( J e. (TopOn ` X ) /\ P e. X /\ M e. ZZ ) /\ k e. Z ) -> ( ( Z X. { P } ) ` k ) = P )
19	14, 5, 2, 16, 18	lmbrf 15318	. 2 |- ( ( J e. (TopOn ` X ) /\ P e. X /\ M e. ZZ ) -> ( ( Z X. { P } ) ( ~~> t ` J ) P <-> ( P e. X /\ A. u e. J ( P e. u -> E. j e. Z A. k e. ( ZZ>= ` j ) P e. u ) ) ) )
20	1, 13, 19	mpbir2and 957	1 |- ( ( J e. (TopOn ` X ) /\ P e. X /\ M e. ZZ ) -> ( Z X. { P } ) ( ~~> t ` J ) P )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   {csn 3709   class class class wbr 4130   X. cxp 4772   -->wf 5373   ` cfv 5377   ZZcz 9646   ZZ>=cuz 9923  TopOnctopon 15113   ~~> tclm 15290

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-pm 6925  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9306  df-n0 9566  df-z 9647  df-uz 9924  df-top 15101  df-topon 15114  df-lm 15293

This theorem is used by: (None)