Theorem lmconst 15007

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 1025	. 2 |- ( ( J e. (TopOn ` X ) /\ P e. X /\ M e. ZZ ) -> P e. X )
2		simp3 1026	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ P e. X /\ M e. ZZ ) -> M e. ZZ )
3		uzid 9813	. . . . . 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 2325	. . . 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 2613	. . . 4 |- ( ( J e. (TopOn ` X ) /\ P e. X /\ M e. ZZ ) -> ( P e. u -> A. k e. ( ZZ>= ` M ) P e. u ) )
9		fveq2 5648	. . . . . 6 |- ( j = M -> ( ZZ>= ` j ) = ( ZZ>= ` M ) )
10	9	raleqdv 2737	. . . . 5 |- ( j = M -> ( A. k e. ( ZZ>= ` j ) P e. u <-> A. k e. ( ZZ>= ` M ) P e. u ) )
11	10	rspcev 2911	. . . 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 1479	. . 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 2607	. 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 1024	. . 3 |- ( ( J e. (TopOn ` X ) /\ P e. X /\ M e. ZZ ) -> J e. (TopOn ` X ) )
15		fconst6g 5544	. . . 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 5876	. . . 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 15006	. 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 953	1 |- ( ( J e. (TopOn ` X ) /\ P e. X /\ M e. ZZ ) -> ( Z X. { P } ) ( ~~> t ` J ) P )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2202   A.wral 2511   E.wrex 2512   {csn 3673   class class class wbr 4093   X. cxp 4729   -->wf 5329   ` cfv 5333   ZZcz 9522   ZZ>=cuz 9798  TopOnctopon 14801   ~~> tclm 14978

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-pm 6863  df-pnf 8259  df-mnf 8260  df-xr 8261  df-ltxr 8262  df-le 8263  df-sub 8395  df-neg 8396  df-inn 9187  df-n0 9446  df-z 9523  df-uz 9799  df-top 14789  df-topon 14802  df-lm 14981

This theorem is referenced by: (None)