Theorem lmconst 12856

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 988	. 2 |- ( ( J e. (TopOn ` X ) /\ P e. X /\ M e. ZZ ) -> P e. X )
2		simp3 989	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ P e. X /\ M e. ZZ ) -> M e. ZZ )
3		uzid 9480	. . . . . 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 2260	. . . 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 2546	. . . 4 |- ( ( J e. (TopOn ` X ) /\ P e. X /\ M e. ZZ ) -> ( P e. u -> A. k e. ( ZZ>= ` M ) P e. u ) )
9		fveq2 5486	. . . . . 6 |- ( j = M -> ( ZZ>= ` j ) = ( ZZ>= ` M ) )
10	9	raleqdv 2667	. . . . 5 |- ( j = M -> ( A. k e. ( ZZ>= ` j ) P e. u <-> A. k e. ( ZZ>= ` M ) P e. u ) )
11	10	rspcev 2830	. . . 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 1422	. . 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 2540	. 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 987	. . 3 |- ( ( J e. (TopOn ` X ) /\ P e. X /\ M e. ZZ ) -> J e. (TopOn ` X ) )
15		fconst6g 5386	. . . 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 5699	. . . 4 |- ( ( P e. X /\ k e. Z ) -> ( ( Z X. { P } ) ` k ) = P )
18	1, 17	sylan 281	. . 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 12855	. 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 934	1 |- ( ( J e. (TopOn ` X ) /\ P e. X /\ M e. ZZ ) -> ( Z X. { P } ) ( ~~> t ` J ) P )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 968   = wceq 1343   e. wcel 2136   A.wral 2444   E.wrex 2445   {csn 3576   class class class wbr 3982   X. cxp 4602   -->wf 5184   ` cfv 5188   ZZcz 9191   ZZ>=cuz 9466  TopOnctopon 12648   ~~> tclm 12827

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-pm 6617  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-inn 8858  df-n0 9115  df-z 9192  df-uz 9467  df-top 12636  df-topon 12649  df-lm 12830

This theorem is referenced by: (None)