Theorem constlimc 39856

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

Hypotheses

Ref	Expression
constlimc.f	|- F = ( x e. A |-> B )
constlimc.a	|- ( ph -> A C_ CC )
constlimc.b	|- ( ph -> B e. CC )
constlimc.c	|- ( ph -> C e. CC )

Assertion

Ref	Expression
constlimc	|- ( ph -> B e. ( F lim CC C ) )

Distinct variable groups:   x, A   x, B   ph, x

Allowed substitution hints:   C( x)   F( x)

Proof of Theorem constlimc

Dummy variables v w y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		constlimc.b	. 2 |- ( ph -> B e. CC )
2		1rp 11836	. . . . 5 |- 1 e. RR+
3	2	a1i 11	. . . 4 |- ( ( ph /\ y e. RR+ ) -> 1 e. RR+ )
4		simpr 477	. . . . . . . . . . . . 13 |- ( ( ph /\ v e. A ) -> v e. A )
5		vex 3203	. . . . . . . . . . . . . . . 16 |- v e. _V
6		nfcv 2764	. . . . . . . . . . . . . . . 16 |- F/_ x B
7		csbtt 3544	. . . . . . . . . . . . . . . 16 |- ( ( v e. _V /\ F/_ x B ) -> [_ v / x ]_ B = B )
8	5, 6, 7	mp2an 708	. . . . . . . . . . . . . . 15 |- [_ v / x ]_ B = B
9	8, 1	syl5eqel 2705	. . . . . . . . . . . . . 14 |- ( ph -> [_ v / x ]_ B e. CC )
10	9	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ v e. A ) -> [_ v / x ]_ B e. CC )
11		constlimc.f	. . . . . . . . . . . . . 14 |- F = ( x e. A |-> B )
12	11	fvmpts 6285	. . . . . . . . . . . . 13 |- ( ( v e. A /\ [_ v / x ]_ B e. CC ) -> ( F ` v ) = [_ v / x ]_ B )
13	4, 10, 12	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ph /\ v e. A ) -> ( F ` v ) = [_ v / x ]_ B )
14	13	oveq1d 6665	. . . . . . . . . . 11 |- ( ( ph /\ v e. A ) -> ( ( F ` v ) - B ) = ( [_ v / x ]_ B - B ) )
15	8	oveq1i 6660	. . . . . . . . . . 11 |- ( [_ v / x ]_ B - B ) = ( B - B )
16	14, 15	syl6eq 2672	. . . . . . . . . 10 |- ( ( ph /\ v e. A ) -> ( ( F ` v ) - B ) = ( B - B ) )
17	16	fveq2d 6195	. . . . . . . . 9 |- ( ( ph /\ v e. A ) -> ( abs ` ( ( F ` v ) - B ) ) = ( abs ` ( B - B ) ) )
18	1	subidd 10380	. . . . . . . . . . 11 |- ( ph -> ( B - B ) = 0 )
19	18	fveq2d 6195	. . . . . . . . . 10 |- ( ph -> ( abs ` ( B - B ) ) = ( abs ` 0 ) )
20	19	adantr 481	. . . . . . . . 9 |- ( ( ph /\ v e. A ) -> ( abs ` ( B - B ) ) = ( abs ` 0 ) )
21		abs0 14025	. . . . . . . . . 10 |- ( abs ` 0 ) = 0
22	21	a1i 11	. . . . . . . . 9 |- ( ( ph /\ v e. A ) -> ( abs ` 0 ) = 0 )
23	17, 20, 22	3eqtrd 2660	. . . . . . . 8 |- ( ( ph /\ v e. A ) -> ( abs ` ( ( F ` v ) - B ) ) = 0 )
24	23	adantlr 751	. . . . . . 7 |- ( ( ( ph /\ y e. RR+ ) /\ v e. A ) -> ( abs ` ( ( F ` v ) - B ) ) = 0 )
25		rpgt0 11844	. . . . . . . 8 |- ( y e. RR+ -> 0 < y )
26	25	ad2antlr 763	. . . . . . 7 |- ( ( ( ph /\ y e. RR+ ) /\ v e. A ) -> 0 < y )
27	24, 26	eqbrtrd 4675	. . . . . 6 |- ( ( ( ph /\ y e. RR+ ) /\ v e. A ) -> ( abs ` ( ( F ` v ) - B ) ) < y )
28	27	a1d 25	. . . . 5 |- ( ( ( ph /\ y e. RR+ ) /\ v e. A ) -> ( ( v =/= C /\ ( abs ` ( v - C ) ) < 1 ) -> ( abs ` ( ( F ` v ) - B ) ) < y ) )
29	28	ralrimiva 2966	. . . 4 |- ( ( ph /\ y e. RR+ ) -> A. v e. A ( ( v =/= C /\ ( abs ` ( v - C ) ) < 1 ) -> ( abs ` ( ( F ` v ) - B ) ) < y ) )
30		breq2 4657	. . . . . . . 8 |- ( w = 1 -> ( ( abs ` ( v - C ) ) < w <-> ( abs ` ( v - C ) ) < 1 ) )
31	30	anbi2d 740	. . . . . . 7 |- ( w = 1 -> ( ( v =/= C /\ ( abs ` ( v - C ) ) < w ) <-> ( v =/= C /\ ( abs ` ( v - C ) ) < 1 ) ) )
32	31	imbi1d 331	. . . . . 6 |- ( w = 1 -> ( ( ( v =/= C /\ ( abs ` ( v - C ) ) < w ) -> ( abs ` ( ( F ` v ) - B ) ) < y ) <-> ( ( v =/= C /\ ( abs ` ( v - C ) ) < 1 ) -> ( abs ` ( ( F ` v ) - B ) ) < y ) ) )
33	32	ralbidv 2986	. . . . 5 |- ( w = 1 -> ( A. v e. A ( ( v =/= C /\ ( abs ` ( v - C ) ) < w ) -> ( abs ` ( ( F ` v ) - B ) ) < y ) <-> A. v e. A ( ( v =/= C /\ ( abs ` ( v - C ) ) < 1 ) -> ( abs ` ( ( F ` v ) - B ) ) < y ) ) )
34	33	rspcev 3309	. . . 4 |- ( ( 1 e. RR+ /\ A. v e. A ( ( v =/= C /\ ( abs ` ( v - C ) ) < 1 ) -> ( abs ` ( ( F ` v ) - B ) ) < y ) ) -> E. w e. RR+ A. v e. A ( ( v =/= C /\ ( abs ` ( v - C ) ) < w ) -> ( abs ` ( ( F ` v ) - B ) ) < y ) )
35	3, 29, 34	syl2anc 693	. . 3 |- ( ( ph /\ y e. RR+ ) -> E. w e. RR+ A. v e. A ( ( v =/= C /\ ( abs ` ( v - C ) ) < w ) -> ( abs ` ( ( F ` v ) - B ) ) < y ) )
36	35	ralrimiva 2966	. 2 |- ( ph -> A. y e. RR+ E. w e. RR+ A. v e. A ( ( v =/= C /\ ( abs ` ( v - C ) ) < w ) -> ( abs ` ( ( F ` v ) - B ) ) < y ) )
37	1	adantr 481	. . . 4 |- ( ( ph /\ x e. A ) -> B e. CC )
38	37, 11	fmptd 6385	. . 3 |- ( ph -> F : A --> CC )
39		constlimc.a	. . 3 |- ( ph -> A C_ CC )
40		constlimc.c	. . 3 |- ( ph -> C e. CC )
41	38, 39, 40	ellimc3 23643	. 2 |- ( ph -> ( B e. ( F lim CC C ) <-> ( B e. CC /\ A. y e. RR+ E. w e. RR+ A. v e. A ( ( v =/= C /\ ( abs ` ( v - C ) ) < w ) -> ( abs ` ( ( F ` v ) - B ) ) < y ) ) ) )
42	1, 36, 41	mpbir2and 957	1 |- ( ph -> B e. ( F lim CC C ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   F/_wnfc 2751   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   [_csb 3533   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   < clt 10074   - cmin 10266   RR+crp 11832   abscabs 13974   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-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:  reclimc  39885  fourierdlem53  40376  fourierdlem60  40383  fourierdlem61  40384  fourierdlem73  40396  fourierdlem74  40397  fourierdlem75  40398  fourierdlem76  40399  fouriersw  40448