Theorem climmulc2 8389

Description: Limit of a sequence multiplied by a constant C. Corollary 12-2.2 of [Gleason] p. 171.

Hypotheses

Ref	Expression
climmulc2.1	|- F e. _V
climmulc2.2	|- G e. _V
climmulc2.3	|- A e. _V

Assertion

Ref	Expression
climmulc2	|- (((C e. CC /\ F ~~> A) /\ (M e. ZZ /\ A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` k) = (C x. (F` k))))) -> G ~~> (C x. A))

Distinct variable groups:   C,k   k,F   k,G   k,M

Proof of Theorem climmulc2

Step	Hyp	Ref	Expression
1		opreq1 4889	. . . . . . . . 9 |- (x = C -> (x x. (F` k)) = (C x. (F` k)))
2	1	eqeq2d 1895	. . . . . . . 8 |- (x = C -> ((G` k) = (x x. (F` k)) <-> (G` k) = (C x. (F` k))))
3	2	anbi2d 678	. . . . . . 7 |- (x = C -> (((F` k) e. CC /\ (G` k) = (x x. (F` k))) <-> ((F` k) e. CC /\ (G` k) = (C x. (F` k)))))
4	3	ralbidv 2123	. . . . . 6 |- (x = C -> (A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` k) = (x x. (F` k))) <-> A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` k) = (C x. (F` k)))))
5		opreq1 4889	. . . . . . 7 |- (x = C -> (x x. A) = (C x. A))
6	5	breq2d 3350	. . . . . 6 |- (x = C -> (G ~~> (x x. A) <-> G ~~> (C x. A)))
7	4, 6	imbi12d 688	. . . . 5 |- (x = C -> ((A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` k) = (x x. (F` k))) -> G ~~> (x x. A)) <-> (A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` k) = (C x. (F` k))) -> G ~~> (C x. A))))
8	7	imbi2d 674	. . . 4 |- (x = C -> ((M e. ZZ -> (A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` k) = (x x. (F` k))) -> G ~~> (x x. A))) <-> (M e. ZZ -> (A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` k) = (C x. (F` k))) -> G ~~> (C x. A)))))
9	8	imbi2d 674	. . 3 |- (x = C -> ((F ~~> A -> (M e. ZZ -> (A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` k) = (x x. (F` k))) -> G ~~> (x x. A)))) <-> (F ~~> A -> (M e. ZZ -> (A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` k) = (C x. (F` k))) -> G ~~> (C x. A))))))
10		climconst2 8355	. . . . . . 7 |- (x e. CC -> (ZZ X. {x}) ~~> x)
11	10	anim1i 361	. . . . . 6 |- ((x e. CC /\ F ~~> A) -> ((ZZ X. {x}) ~~> x /\ F ~~> A))
12	11	adantr 425	. . . . 5 |- (((x e. CC /\ F ~~> A) /\ (M e. ZZ /\ A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` k) = (x x. (F` k))))) -> ((ZZ X. {x}) ~~> x /\ F ~~> A))
13		fvconst2g 4820	. . . . . . . . . . . . 13 |- ((x e. CC /\ k e. ZZ) -> ((ZZ X. {x})` k) = x)
14		eluzelz 7592	. . . . . . . . . . . . 13 |- (k e. (ZZ>=` M) -> k e. ZZ)
15	13, 14	sylan2 500	. . . . . . . . . . . 12 |- ((x e. CC /\ k e. (ZZ>=` M)) -> ((ZZ X. {x})` k) = x)
16		simpl 346	. . . . . . . . . . . 12 |- ((x e. CC /\ k e. (ZZ>=` M)) -> x e. CC)
17	15, 16	eqeltrd 1971	. . . . . . . . . . 11 |- ((x e. CC /\ k e. (ZZ>=` M)) -> ((ZZ X. {x})` k) e. CC)
18	17	a1d 15	. . . . . . . . . 10 |- ((x e. CC /\ k e. (ZZ>=` M)) -> (((F` k) e. CC /\ (G` k) = (x x. (F` k))) -> ((ZZ X. {x})` k) e. CC))
19		simpl 346	. . . . . . . . . . 11 |- (((F` k) e. CC /\ (G` k) = (x x. (F` k))) -> (F` k) e. CC)
20	19	a1i 8	. . . . . . . . . 10 |- ((x e. CC /\ k e. (ZZ>=` M)) -> (((F` k) e. CC /\ (G` k) = (x x. (F` k))) -> (F` k) e. CC))
21	15	opreq1d 4897	. . . . . . . . . . . . 13 |- ((x e. CC /\ k e. (ZZ>=` M)) -> (((ZZ X. {x})` k) x. (F` k)) = (x x. (F` k)))
22	21	eqeq2d 1895	. . . . . . . . . . . 12 |- ((x e. CC /\ k e. (ZZ>=` M)) -> ((G` k) = (((ZZ X. {x})` k) x. (F` k)) <-> (G` k) = (x x. (F` k))))
23	22	biimprd 171	. . . . . . . . . . 11 |- ((x e. CC /\ k e. (ZZ>=` M)) -> ((G` k) = (x x. (F` k)) -> (G` k) = (((ZZ X. {x})` k) x. (F` k))))
24	23	adantld 426	. . . . . . . . . 10 |- ((x e. CC /\ k e. (ZZ>=` M)) -> (((F` k) e. CC /\ (G` k) = (x x. (F` k))) -> (G` k) = (((ZZ X. {x})` k) x. (F` k))))
25	18, 20, 24	3jcad 1051	. . . . . . . . 9 |- ((x e. CC /\ k e. (ZZ>=` M)) -> (((F` k) e. CC /\ (G` k) = (x x. (F` k))) -> (((ZZ X. {x})` k) e. CC /\ (F` k) e. CC /\ (G` k) = (((ZZ X. {x})` k) x. (F` k)))))
26	25	ralimdvaa 2171	. . . . . . . 8 |- (x e. CC -> (A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` k) = (x x. (F` k))) -> A.k e. (ZZ>=` M)(((ZZ X. {x})` k) e. CC /\ (F` k) e. CC /\ (G` k) = (((ZZ X. {x})` k) x. (F` k)))))
27	26	anim2d 620	. . . . . . 7 |- (x e. CC -> ((M e. ZZ /\ A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` k) = (x x. (F` k)))) -> (M e. ZZ /\ A.k e. (ZZ>=` M)(((ZZ X. {x})` k) e. CC /\ (F` k) e. CC /\ (G` k) = (((ZZ X. {x})` k) x. (F` k))))))
28	27	imp 377	. . . . . 6 |- ((x e. CC /\ (M e. ZZ /\ A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` k) = (x x. (F` k))))) -> (M e. ZZ /\ A.k e. (ZZ>=` M)(((ZZ X. {x})` k) e. CC /\ (F` k) e. CC /\ (G` k) = (((ZZ X. {x})` k) x. (F` k)))))
29	28	adantlr 429	. . . . 5 |- (((x e. CC /\ F ~~> A) /\ (M e. ZZ /\ A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` k) = (x x. (F` k))))) -> (M e. ZZ /\ A.k e. (ZZ>=` M)(((ZZ X. {x})` k) e. CC /\ (F` k) e. CC /\ (G` k) = (((ZZ X. {x})` k) x. (F` k)))))
30		zex 7353	. . . . . . 7 |- ZZ e. _V
31		snex 3492	. . . . . . 7 |- {x} e. _V
32	30, 31	xpex 4096	. . . . . 6 |- (ZZ X. {x}) e. _V
33		climmulc2.1	. . . . . 6 |- F e. _V
34		climmulc2.2	. . . . . 6 |- G e. _V
35		visset 2295	. . . . . 6 |- x e. _V
36		climmulc2.3	. . . . . 6 |- A e. _V
37	32, 33, 34, 35, 36	climmul 8388	. . . . 5 |- ((((ZZ X. {x}) ~~> x /\ F ~~> A) /\ (M e. ZZ /\ A.k e. (ZZ>=` M)(((ZZ X. {x})` k) e. CC /\ (F` k) e. CC /\ (G` k) = (((ZZ X. {x})` k) x. (F` k))))) -> G ~~> (x x. A))
38	12, 29, 37	syl11anc 524	. . . 4 |- (((x e. CC /\ F ~~> A) /\ (M e. ZZ /\ A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` k) = (x x. (F` k))))) -> G ~~> (x x. A))
39	38	exp43 415	. . 3 |- (x e. CC -> (F ~~> A -> (M e. ZZ -> (A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` k) = (x x. (F` k))) -> G ~~> (x x. A)))))
40	9, 39	vtoclga 2352	. 2 |- (C e. CC -> (F ~~> A -> (M e. ZZ -> (A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` k) = (C x. (F` k))) -> G ~~> (C x. A)))))
41	40	imp43 397	1 |- (((C e. CC /\ F ~~> A) /\ (M e. ZZ /\ A.k e. (ZZ>=` M)((F` k) e. CC /\ (G` k) = (C x. (F` k))))) -> G ~~> (C x. A))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292  {csn 3044   class class class wbr 3338   X. cxp 3984  ` cfv 3998  (class class class)co 4884  CCcc 6384   x. cmul 6391  ZZcz 6451  ZZ>=cuz 7586   ~~> cli 8234

This theorem is referenced by:  climsub 8390  climmulci 8393  iserzmulc1 8396  caucvg3lem 8426  isummulc1iALT 8474  geolimilem 8497  bfplem8 16005

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-n0 7309  df-z 7345  df-uz 7587  df-seq1 7721  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-clim 8235

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