Theorem climsuselem1 39839

Description: The subsequence index I has the expected properties: it belongs to the same upper integers as the original index, and it is always larger or equal than the original index. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Hypotheses

Ref	Expression
climsuselem1.1	|- Z = ( ZZ>= ` M )
climsuselem1.2	|- ( ph -> M e. ZZ )
climsuselem1.3	|- ( ph -> ( I ` M ) e. Z )
climsuselem1.4	|- ( ( ph /\ k e. Z ) -> ( I ` ( k + 1 ) ) e. ( ZZ>= ` ( ( I ` k ) + 1 ) ) )

Assertion

Ref	Expression
climsuselem1	|- ( ( ph /\ K e. Z ) -> ( I ` K ) e. ( ZZ>= ` K ) )

Distinct variable groups:   ph, k   k, I   k, M   k, Z

Allowed substitution hint:   K( k)

Proof of Theorem climsuselem1

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		climsuselem1.1	. . . . 5 |- Z = ( ZZ>= ` M )
2	1	eleq2i 2693	. . . 4 |- ( K e. Z <-> K e. ( ZZ>= ` M ) )
3	2	biimpi 206	. . 3 |- ( K e. Z -> K e. ( ZZ>= ` M ) )
4	3	adantl 482	. 2 |- ( ( ph /\ K e. Z ) -> K e. ( ZZ>= ` M ) )
5		simpl 473	. 2 |- ( ( ph /\ K e. Z ) -> ph )
6		fveq2 6191	. . . . 5 |- ( j = M -> ( I ` j ) = ( I ` M ) )
7		fveq2 6191	. . . . 5 |- ( j = M -> ( ZZ>= ` j ) = ( ZZ>= ` M ) )
8	6, 7	eleq12d 2695	. . . 4 |- ( j = M -> ( ( I ` j ) e. ( ZZ>= ` j ) <-> ( I ` M ) e. ( ZZ>= ` M ) ) )
9	8	imbi2d 330	. . 3 |- ( j = M -> ( ( ph -> ( I ` j ) e. ( ZZ>= ` j ) ) <-> ( ph -> ( I ` M ) e. ( ZZ>= ` M ) ) ) )
10		fveq2 6191	. . . . 5 |- ( j = k -> ( I ` j ) = ( I ` k ) )
11		fveq2 6191	. . . . 5 |- ( j = k -> ( ZZ>= ` j ) = ( ZZ>= ` k ) )
12	10, 11	eleq12d 2695	. . . 4 |- ( j = k -> ( ( I ` j ) e. ( ZZ>= ` j ) <-> ( I ` k ) e. ( ZZ>= ` k ) ) )
13	12	imbi2d 330	. . 3 |- ( j = k -> ( ( ph -> ( I ` j ) e. ( ZZ>= ` j ) ) <-> ( ph -> ( I ` k ) e. ( ZZ>= ` k ) ) ) )
14		fveq2 6191	. . . . 5 |- ( j = ( k + 1 ) -> ( I ` j ) = ( I ` ( k + 1 ) ) )
15		fveq2 6191	. . . . 5 |- ( j = ( k + 1 ) -> ( ZZ>= ` j ) = ( ZZ>= ` ( k + 1 ) ) )
16	14, 15	eleq12d 2695	. . . 4 |- ( j = ( k + 1 ) -> ( ( I ` j ) e. ( ZZ>= ` j ) <-> ( I ` ( k + 1 ) ) e. ( ZZ>= ` ( k + 1 ) ) ) )
17	16	imbi2d 330	. . 3 |- ( j = ( k + 1 ) -> ( ( ph -> ( I ` j ) e. ( ZZ>= ` j ) ) <-> ( ph -> ( I ` ( k + 1 ) ) e. ( ZZ>= ` ( k + 1 ) ) ) ) )
18		fveq2 6191	. . . . 5 |- ( j = K -> ( I ` j ) = ( I ` K ) )
19		fveq2 6191	. . . . 5 |- ( j = K -> ( ZZ>= ` j ) = ( ZZ>= ` K ) )
20	18, 19	eleq12d 2695	. . . 4 |- ( j = K -> ( ( I ` j ) e. ( ZZ>= ` j ) <-> ( I ` K ) e. ( ZZ>= ` K ) ) )
21	20	imbi2d 330	. . 3 |- ( j = K -> ( ( ph -> ( I ` j ) e. ( ZZ>= ` j ) ) <-> ( ph -> ( I ` K ) e. ( ZZ>= ` K ) ) ) )
22		climsuselem1.3	. . . . 5 |- ( ph -> ( I ` M ) e. Z )
23	22, 1	syl6eleq 2711	. . . 4 |- ( ph -> ( I ` M ) e. ( ZZ>= ` M ) )
24	23	a1i 11	. . 3 |- ( M e. ZZ -> ( ph -> ( I ` M ) e. ( ZZ>= ` M ) ) )
25		simpr 477	. . . . 5 |- ( ( ( k e. ( ZZ>= ` M ) /\ ( ph -> ( I ` k ) e. ( ZZ>= ` k ) ) ) /\ ph ) -> ph )
26		simpll 790	. . . . 5 |- ( ( ( k e. ( ZZ>= ` M ) /\ ( ph -> ( I ` k ) e. ( ZZ>= ` k ) ) ) /\ ph ) -> k e. ( ZZ>= ` M ) )
27		simplr 792	. . . . . 6 |- ( ( ( k e. ( ZZ>= ` M ) /\ ( ph -> ( I ` k ) e. ( ZZ>= ` k ) ) ) /\ ph ) -> ( ph -> ( I ` k ) e. ( ZZ>= ` k ) ) )
28	25, 27	mpd 15	. . . . 5 |- ( ( ( k e. ( ZZ>= ` M ) /\ ( ph -> ( I ` k ) e. ( ZZ>= ` k ) ) ) /\ ph ) -> ( I ` k ) e. ( ZZ>= ` k ) )
29		eluzelz 11697	. . . . . . . . . 10 |- ( k e. ( ZZ>= ` M ) -> k e. ZZ )
30	29	3ad2ant2 1083	. . . . . . . . 9 |- ( ( ph /\ k e. ( ZZ>= ` M ) /\ ( I ` k ) e. ( ZZ>= ` k ) ) -> k e. ZZ )
31	30	peano2zd 11485	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` M ) /\ ( I ` k ) e. ( ZZ>= ` k ) ) -> ( k + 1 ) e. ZZ )
32	31	zred 11482	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) /\ ( I ` k ) e. ( ZZ>= ` k ) ) -> ( k + 1 ) e. RR )
33		eluzelre 11698	. . . . . . . . 9 |- ( ( I ` k ) e. ( ZZ>= ` k ) -> ( I ` k ) e. RR )
34	33	3ad2ant3 1084	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` M ) /\ ( I ` k ) e. ( ZZ>= ` k ) ) -> ( I ` k ) e. RR )
35		1red 10055	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` M ) /\ ( I ` k ) e. ( ZZ>= ` k ) ) -> 1 e. RR )
36	34, 35	readdcld 10069	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) /\ ( I ` k ) e. ( ZZ>= ` k ) ) -> ( ( I ` k ) + 1 ) e. RR )
37	1	eqimss2i 3660	. . . . . . . . . . . . . 14 |- ( ZZ>= ` M ) C_ Z
38	37	a1i 11	. . . . . . . . . . . . 13 |- ( ph -> ( ZZ>= ` M ) C_ Z )
39	38	sseld 3602	. . . . . . . . . . . 12 |- ( ph -> ( k e. ( ZZ>= ` M ) -> k e. Z ) )
40	39	imdistani 726	. . . . . . . . . . 11 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ph /\ k e. Z ) )
41		climsuselem1.4	. . . . . . . . . . 11 |- ( ( ph /\ k e. Z ) -> ( I ` ( k + 1 ) ) e. ( ZZ>= ` ( ( I ` k ) + 1 ) ) )
42	40, 41	syl 17	. . . . . . . . . 10 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( I ` ( k + 1 ) ) e. ( ZZ>= ` ( ( I ` k ) + 1 ) ) )
43	42	3adant3 1081	. . . . . . . . 9 |- ( ( ph /\ k e. ( ZZ>= ` M ) /\ ( I ` k ) e. ( ZZ>= ` k ) ) -> ( I ` ( k + 1 ) ) e. ( ZZ>= ` ( ( I ` k ) + 1 ) ) )
44		eluzelz 11697	. . . . . . . . 9 |- ( ( I ` ( k + 1 ) ) e. ( ZZ>= ` ( ( I ` k ) + 1 ) ) -> ( I ` ( k + 1 ) ) e. ZZ )
45	43, 44	syl 17	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` M ) /\ ( I ` k ) e. ( ZZ>= ` k ) ) -> ( I ` ( k + 1 ) ) e. ZZ )
46	45	zred 11482	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) /\ ( I ` k ) e. ( ZZ>= ` k ) ) -> ( I ` ( k + 1 ) ) e. RR )
47	30	zred 11482	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` M ) /\ ( I ` k ) e. ( ZZ>= ` k ) ) -> k e. RR )
48		eluzle 11700	. . . . . . . . 9 |- ( ( I ` k ) e. ( ZZ>= ` k ) -> k <_ ( I ` k ) )
49	48	3ad2ant3 1084	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` M ) /\ ( I ` k ) e. ( ZZ>= ` k ) ) -> k <_ ( I ` k ) )
50	47, 34, 35, 49	leadd1dd 10641	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) /\ ( I ` k ) e. ( ZZ>= ` k ) ) -> ( k + 1 ) <_ ( ( I ` k ) + 1 ) )
51		eluzle 11700	. . . . . . . 8 |- ( ( I ` ( k + 1 ) ) e. ( ZZ>= ` ( ( I ` k ) + 1 ) ) -> ( ( I ` k ) + 1 ) <_ ( I ` ( k + 1 ) ) )
52	43, 51	syl 17	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) /\ ( I ` k ) e. ( ZZ>= ` k ) ) -> ( ( I ` k ) + 1 ) <_ ( I ` ( k + 1 ) ) )
53	32, 36, 46, 50, 52	letrd 10194	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) /\ ( I ` k ) e. ( ZZ>= ` k ) ) -> ( k + 1 ) <_ ( I ` ( k + 1 ) ) )
54		eluz 11701	. . . . . . 7 |- ( ( ( k + 1 ) e. ZZ /\ ( I ` ( k + 1 ) ) e. ZZ ) -> ( ( I ` ( k + 1 ) ) e. ( ZZ>= ` ( k + 1 ) ) <-> ( k + 1 ) <_ ( I ` ( k + 1 ) ) ) )
55	31, 45, 54	syl2anc 693	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) /\ ( I ` k ) e. ( ZZ>= ` k ) ) -> ( ( I ` ( k + 1 ) ) e. ( ZZ>= ` ( k + 1 ) ) <-> ( k + 1 ) <_ ( I ` ( k + 1 ) ) ) )
56	53, 55	mpbird 247	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` M ) /\ ( I ` k ) e. ( ZZ>= ` k ) ) -> ( I ` ( k + 1 ) ) e. ( ZZ>= ` ( k + 1 ) ) )
57	25, 26, 28, 56	syl3anc 1326	. . . 4 |- ( ( ( k e. ( ZZ>= ` M ) /\ ( ph -> ( I ` k ) e. ( ZZ>= ` k ) ) ) /\ ph ) -> ( I ` ( k + 1 ) ) e. ( ZZ>= ` ( k + 1 ) ) )
58	57	exp31 630	. . 3 |- ( k e. ( ZZ>= ` M ) -> ( ( ph -> ( I ` k ) e. ( ZZ>= ` k ) ) -> ( ph -> ( I ` ( k + 1 ) ) e. ( ZZ>= ` ( k + 1 ) ) ) ) )
59	9, 13, 17, 21, 24, 58	uzind4 11746	. 2 |- ( K e. ( ZZ>= ` M ) -> ( ph -> ( I ` K ) e. ( ZZ>= ` K ) ) )
60	4, 5, 59	sylc 65	1 |- ( ( ph /\ K e. Z ) -> ( I ` K ) e. ( ZZ>= ` K ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   RRcr 9935   1c1 9937   + caddc 9939   <_ cle 10075   ZZcz 11377   ZZ>=cuz 11687

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-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

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-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-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-wrecs 7407  df-recs 7468  df-rdg 7506  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-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688

This theorem is referenced by:  climsuse  39840