Theorem reclimc 39885

Description: Limit of the reciprocal of a function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
reclimc.f	|- F = ( x e. A |-> B )
reclimc.g	|- G = ( x e. A |-> ( 1 / B ) )
reclimc.b	|- ( ( ph /\ x e. A ) -> B e. ( CC \ { 0 } ) )
reclimc.c	|- ( ph -> C e. ( F lim CC D ) )
reclimc.cne0	|- ( ph -> C =/= 0 )

Assertion

Ref	Expression
reclimc	|- ( ph -> ( 1 / C ) e. ( G lim CC D ) )

Distinct variable groups:   x, A   x, C   x, D   ph, x

Allowed substitution hints:   B( x)   F( x)   G( x)

Proof of Theorem reclimc

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- ( x e. A |-> ( C - B ) ) = ( x e. A |-> ( C - B ) )
2		eqid 2622	. . . 4 |- ( x e. A |-> ( B x. C ) ) = ( x e. A |-> ( B x. C ) )
3		eqid 2622	. . . 4 |- ( x e. A |-> ( ( C - B ) / ( B x. C ) ) ) = ( x e. A |-> ( ( C - B ) / ( B x. C ) ) )
4		limccl 23639	. . . . . . 7 |- ( F lim CC D ) C_ CC
5		reclimc.c	. . . . . . 7 |- ( ph -> C e. ( F lim CC D ) )
6	4, 5	sseldi 3601	. . . . . 6 |- ( ph -> C e. CC )
7	6	adantr 481	. . . . 5 |- ( ( ph /\ x e. A ) -> C e. CC )
8		reclimc.b	. . . . . 6 |- ( ( ph /\ x e. A ) -> B e. ( CC \ { 0 } ) )
9	8	eldifad 3586	. . . . 5 |- ( ( ph /\ x e. A ) -> B e. CC )
10	7, 9	subcld 10392	. . . 4 |- ( ( ph /\ x e. A ) -> ( C - B ) e. CC )
11	9, 7	mulcld 10060	. . . . 5 |- ( ( ph /\ x e. A ) -> ( B x. C ) e. CC )
12		eldifsni 4320	. . . . . . . . 9 |- ( B e. ( CC \ { 0 } ) -> B =/= 0 )
13	8, 12	syl 17	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> B =/= 0 )
14		reclimc.cne0	. . . . . . . . 9 |- ( ph -> C =/= 0 )
15	14	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> C =/= 0 )
16	9, 7, 13, 15	mulne0d 10679	. . . . . . 7 |- ( ( ph /\ x e. A ) -> ( B x. C ) =/= 0 )
17	16	neneqd 2799	. . . . . 6 |- ( ( ph /\ x e. A ) -> -. ( B x. C ) = 0 )
18		elsng 4191	. . . . . . 7 |- ( ( B x. C ) e. CC -> ( ( B x. C ) e. { 0 } <-> ( B x. C ) = 0 ) )
19	11, 18	syl 17	. . . . . 6 |- ( ( ph /\ x e. A ) -> ( ( B x. C ) e. { 0 } <-> ( B x. C ) = 0 ) )
20	17, 19	mtbird 315	. . . . 5 |- ( ( ph /\ x e. A ) -> -. ( B x. C ) e. { 0 } )
21	11, 20	eldifd 3585	. . . 4 |- ( ( ph /\ x e. A ) -> ( B x. C ) e. ( CC \ { 0 } ) )
22		eqid 2622	. . . . . 6 |- ( x e. A |-> C ) = ( x e. A |-> C )
23		eqid 2622	. . . . . 6 |- ( x e. A |-> -u B ) = ( x e. A |-> -u B )
24		eqid 2622	. . . . . 6 |- ( x e. A |-> ( C + -u B ) ) = ( x e. A |-> ( C + -u B ) )
25	9	negcld 10379	. . . . . 6 |- ( ( ph /\ x e. A ) -> -u B e. CC )
26		reclimc.f	. . . . . . . 8 |- F = ( x e. A |-> B )
27	26, 9, 5	limcmptdm 39867	. . . . . . 7 |- ( ph -> A C_ CC )
28		limcrcl 23638	. . . . . . . . 9 |- ( C e. ( F lim CC D ) -> ( F : dom F --> CC /\ dom F C_ CC /\ D e. CC ) )
29	5, 28	syl 17	. . . . . . . 8 |- ( ph -> ( F : dom F --> CC /\ dom F C_ CC /\ D e. CC ) )
30	29	simp3d 1075	. . . . . . 7 |- ( ph -> D e. CC )
31	22, 27, 6, 30	constlimc 39856	. . . . . 6 |- ( ph -> C e. ( ( x e. A |-> C ) lim CC D ) )
32	26, 23, 9, 5	neglimc 39879	. . . . . 6 |- ( ph -> -u C e. ( ( x e. A |-> -u B ) lim CC D ) )
33	22, 23, 24, 7, 25, 31, 32	addlimc 39880	. . . . 5 |- ( ph -> ( C + -u C ) e. ( ( x e. A |-> ( C + -u B ) ) lim CC D ) )
34	6	negidd 10382	. . . . 5 |- ( ph -> ( C + -u C ) = 0 )
35	7, 9	negsubd 10398	. . . . . . 7 |- ( ( ph /\ x e. A ) -> ( C + -u B ) = ( C - B ) )
36	35	mpteq2dva 4744	. . . . . 6 |- ( ph -> ( x e. A |-> ( C + -u B ) ) = ( x e. A |-> ( C - B ) ) )
37	36	oveq1d 6665	. . . . 5 |- ( ph -> ( ( x e. A |-> ( C + -u B ) ) lim CC D ) = ( ( x e. A |-> ( C - B ) ) lim CC D ) )
38	33, 34, 37	3eltr3d 2715	. . . 4 |- ( ph -> 0 e. ( ( x e. A |-> ( C - B ) ) lim CC D ) )
39	26, 22, 2, 9, 7, 5, 31	mullimc 39848	. . . 4 |- ( ph -> ( C x. C ) e. ( ( x e. A |-> ( B x. C ) ) lim CC D ) )
40	6, 6, 14, 14	mulne0d 10679	. . . 4 |- ( ph -> ( C x. C ) =/= 0 )
41	1, 2, 3, 10, 21, 38, 39, 40	0ellimcdiv 39881	. . 3 |- ( ph -> 0 e. ( ( x e. A |-> ( ( C - B ) / ( B x. C ) ) ) lim CC D ) )
42		1cnd 10056	. . . . . . 7 |- ( ( ph /\ x e. A ) -> 1 e. CC )
43	42, 9, 42, 7, 13, 15	divsubdivd 10846	. . . . . 6 |- ( ( ph /\ x e. A ) -> ( ( 1 / B ) - ( 1 / C ) ) = ( ( ( 1 x. C ) - ( 1 x. B ) ) / ( B x. C ) ) )
44	7	mulid2d 10058	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> ( 1 x. C ) = C )
45	9	mulid2d 10058	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> ( 1 x. B ) = B )
46	44, 45	oveq12d 6668	. . . . . . 7 |- ( ( ph /\ x e. A ) -> ( ( 1 x. C ) - ( 1 x. B ) ) = ( C - B ) )
47	46	oveq1d 6665	. . . . . 6 |- ( ( ph /\ x e. A ) -> ( ( ( 1 x. C ) - ( 1 x. B ) ) / ( B x. C ) ) = ( ( C - B ) / ( B x. C ) ) )
48	43, 47	eqtr2d 2657	. . . . 5 |- ( ( ph /\ x e. A ) -> ( ( C - B ) / ( B x. C ) ) = ( ( 1 / B ) - ( 1 / C ) ) )
49	48	mpteq2dva 4744	. . . 4 |- ( ph -> ( x e. A |-> ( ( C - B ) / ( B x. C ) ) ) = ( x e. A |-> ( ( 1 / B ) - ( 1 / C ) ) ) )
50	49	oveq1d 6665	. . 3 |- ( ph -> ( ( x e. A |-> ( ( C - B ) / ( B x. C ) ) ) lim CC D ) = ( ( x e. A |-> ( ( 1 / B ) - ( 1 / C ) ) ) lim CC D ) )
51	41, 50	eleqtrd 2703	. 2 |- ( ph -> 0 e. ( ( x e. A |-> ( ( 1 / B ) - ( 1 / C ) ) ) lim CC D ) )
52		reclimc.g	. . 3 |- G = ( x e. A |-> ( 1 / B ) )
53		eqid 2622	. . 3 |- ( x e. A |-> ( ( 1 / B ) - ( 1 / C ) ) ) = ( x e. A |-> ( ( 1 / B ) - ( 1 / C ) ) )
54	9, 13	reccld 10794	. . 3 |- ( ( ph /\ x e. A ) -> ( 1 / B ) e. CC )
55	6, 14	reccld 10794	. . 3 |- ( ph -> ( 1 / C ) e. CC )
56	52, 53, 27, 54, 30, 55	ellimcabssub0 39849	. 2 |- ( ph -> ( ( 1 / C ) e. ( G lim CC D ) <-> 0 e. ( ( x e. A |-> ( ( 1 / B ) - ( 1 / C ) ) ) lim CC D ) ) )
57	51, 56	mpbird 247	1 |- ( ph -> ( 1 / C ) e. ( G lim CC D ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   C_ wss 3574   {csn 4177   |-> cmpt 4729   dom cdm 5114   -->wf 5884  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   / cdiv 10684   lim CC climc 23626

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-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-fz 12327  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-rest 16083  df-topn 16084  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cnp 21032  df-xms 22125  df-ms 22126  df-limc 23630

This theorem is referenced by:  divlimc  39888  fourierdlem62  40385