Theorem climresmpt 39891

Description: A function restricted to upper integers converges iff the original function converges. (Contributed by Glauco Siliprandi, 8-Apr-2021.)

Hypotheses

Ref	Expression
climresmpt.z	|- Z = ( ZZ>= ` M )
climresmpt.f	|- F = ( x e. Z |-> A )
climresmpt.n	|- ( ph -> N e. Z )
climresmpt.g	|- G = ( x e. ( ZZ>= ` N ) |-> A )

Assertion

Ref	Expression
climresmpt	|- ( ph -> ( G ~~> B <-> F ~~> B ) )

Distinct variable groups:   x, N   x, Z

Allowed substitution hints:   ph( x)   A( x)   B( x)   F( x)   G( x)   M( x)

Proof of Theorem climresmpt

Step	Hyp	Ref	Expression
1		climresmpt.f	. . . . . 6 |- F = ( x e. Z |-> A )
2	1	reseq1i 5392	. . . . 5 |- ( F |` ( ZZ>= ` N ) ) = ( ( x e. Z |-> A ) |` ( ZZ>= ` N ) )
3	2	a1i 11	. . . 4 |- ( ph -> ( F |` ( ZZ>= ` N ) ) = ( ( x e. Z |-> A ) |` ( ZZ>= ` N ) ) )
4		climresmpt.n	. . . . . . . 8 |- ( ph -> N e. Z )
5		climresmpt.z	. . . . . . . 8 |- Z = ( ZZ>= ` M )
6	4, 5	syl6eleq 2711	. . . . . . 7 |- ( ph -> N e. ( ZZ>= ` M ) )
7		uzss 11708	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> ( ZZ>= ` N ) C_ ( ZZ>= ` M ) )
8	6, 7	syl 17	. . . . . 6 |- ( ph -> ( ZZ>= ` N ) C_ ( ZZ>= ` M ) )
9	8, 5	syl6sseqr 3652	. . . . 5 |- ( ph -> ( ZZ>= ` N ) C_ Z )
10		resmpt 5449	. . . . 5 |- ( ( ZZ>= ` N ) C_ Z -> ( ( x e. Z |-> A ) |` ( ZZ>= ` N ) ) = ( x e. ( ZZ>= ` N ) |-> A ) )
11	9, 10	syl 17	. . . 4 |- ( ph -> ( ( x e. Z |-> A ) |` ( ZZ>= ` N ) ) = ( x e. ( ZZ>= ` N ) |-> A ) )
12		climresmpt.g	. . . . . 6 |- G = ( x e. ( ZZ>= ` N ) |-> A )
13	12	eqcomi 2631	. . . . 5 |- ( x e. ( ZZ>= ` N ) |-> A ) = G
14	13	a1i 11	. . . 4 |- ( ph -> ( x e. ( ZZ>= ` N ) |-> A ) = G )
15	3, 11, 14	3eqtrrd 2661	. . 3 |- ( ph -> G = ( F |` ( ZZ>= ` N ) ) )
16	15	breq1d 4663	. 2 |- ( ph -> ( G ~~> B <-> ( F |` ( ZZ>= ` N ) ) ~~> B ) )
17		eluzelz 11697	. . . 4 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
18	6, 17	syl 17	. . 3 |- ( ph -> N e. ZZ )
19		fvex 6201	. . . . . . 7 |- ( ZZ>= ` M ) e. _V
20	5, 19	eqeltri 2697	. . . . . 6 |- Z e. _V
21	20	mptex 6486	. . . . 5 |- ( x e. Z |-> A ) e. _V
22	21	a1i 11	. . . 4 |- ( ph -> ( x e. Z |-> A ) e. _V )
23	1, 22	syl5eqel 2705	. . 3 |- ( ph -> F e. _V )
24		climres 14306	. . 3 |- ( ( N e. ZZ /\ F e. _V ) -> ( ( F |` ( ZZ>= ` N ) ) ~~> B <-> F ~~> B ) )
25	18, 23, 24	syl2anc 693	. 2 |- ( ph -> ( ( F |` ( ZZ>= ` N ) ) ~~> B <-> F ~~> B ) )
26	16, 25	bitrd 268	1 |- ( ph -> ( G ~~> B <-> F ~~> B ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   |` cres 5116   ` cfv 5888   ZZcz 11377   ZZ>=cuz 11687   ~~> cli 14215

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-pre-lttri 10010  ax-pre-lttrn 10011

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-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-neg 10269  df-z 11378  df-uz 11688  df-clim 14219

This theorem is referenced by:  meaiininclem  40700