Theorem psercnlem2 24178

Description: Lemma for psercn 24180. (Contributed by Mario Carneiro, 18-Mar-2015.)

Hypotheses

Ref	Expression
pserf.g	|- G = ( x e. CC |-> ( n e. NN0 |-> ( ( A ` n ) x. ( x ^ n ) ) ) )
pserf.f	|- F = ( y e. S |-> sum_ j e. NN0 ( ( G ` y ) ` j ) )
pserf.a	|- ( ph -> A : NN0 --> CC )
pserf.r	|- R = sup ( { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } , RR* , < )
psercn.s	|- S = ( `' abs " ( 0 [,) R ) )
psercnlem2.i	|- ( ( ph /\ a e. S ) -> ( M e. RR+ /\ ( abs ` a ) < M /\ M < R ) )

Assertion

Ref	Expression
psercnlem2	|- ( ( ph /\ a e. S ) -> ( a e. ( 0 ( ball ` ( abs o. - ) ) M ) /\ ( 0 ( ball ` ( abs o. - ) ) M ) C_ ( `' abs " ( 0 [,] M ) ) /\ ( `' abs " ( 0 [,] M ) ) C_ S ) )

Distinct variable groups:   j, a, n, r, x, y, A   j, M, y   j, G, r, y   S, a, j, y   F, a   ph, a, j, y

Allowed substitution hints:   ph( x, n, r)   R( x, y, j, n, r, a)   S( x, n, r)   F( x, y, j, n, r)   G( x, n, a)   M( x, n, r, a)

Proof of Theorem psercnlem2

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

Step	Hyp	Ref	Expression
1		psercn.s	. . . . . . 7 |- S = ( `' abs " ( 0 [,) R ) )
2		cnvimass 5485	. . . . . . . 8 |- ( `' abs " ( 0 [,) R ) ) C_ dom abs
3		absf 14077	. . . . . . . . 9 |- abs : CC --> RR
4	3	fdmi 6052	. . . . . . . 8 |- dom abs = CC
5	2, 4	sseqtri 3637	. . . . . . 7 |- ( `' abs " ( 0 [,) R ) ) C_ CC
6	1, 5	eqsstri 3635	. . . . . 6 |- S C_ CC
7	6	a1i 11	. . . . 5 |- ( ph -> S C_ CC )
8	7	sselda 3603	. . . 4 |- ( ( ph /\ a e. S ) -> a e. CC )
9	8	abscld 14175	. . . . 5 |- ( ( ph /\ a e. S ) -> ( abs ` a ) e. RR )
10	8	absge0d 14183	. . . . 5 |- ( ( ph /\ a e. S ) -> 0 <_ ( abs ` a ) )
11		psercnlem2.i	. . . . . 6 |- ( ( ph /\ a e. S ) -> ( M e. RR+ /\ ( abs ` a ) < M /\ M < R ) )
12	11	simp2d 1074	. . . . 5 |- ( ( ph /\ a e. S ) -> ( abs ` a ) < M )
13		0re 10040	. . . . . 6 |- 0 e. RR
14	11	simp1d 1073	. . . . . . 7 |- ( ( ph /\ a e. S ) -> M e. RR+ )
15	14	rpxrd 11873	. . . . . 6 |- ( ( ph /\ a e. S ) -> M e. RR* )
16		elico2 12237	. . . . . 6 |- ( ( 0 e. RR /\ M e. RR* ) -> ( ( abs ` a ) e. ( 0 [,) M ) <-> ( ( abs ` a ) e. RR /\ 0 <_ ( abs ` a ) /\ ( abs ` a ) < M ) ) )
17	13, 15, 16	sylancr 695	. . . . 5 |- ( ( ph /\ a e. S ) -> ( ( abs ` a ) e. ( 0 [,) M ) <-> ( ( abs ` a ) e. RR /\ 0 <_ ( abs ` a ) /\ ( abs ` a ) < M ) ) )
18	9, 10, 12, 17	mpbir3and 1245	. . . 4 |- ( ( ph /\ a e. S ) -> ( abs ` a ) e. ( 0 [,) M ) )
19		ffn 6045	. . . . 5 |- ( abs : CC --> RR -> abs Fn CC )
20		elpreima 6337	. . . . 5 |- ( abs Fn CC -> ( a e. ( `' abs " ( 0 [,) M ) ) <-> ( a e. CC /\ ( abs ` a ) e. ( 0 [,) M ) ) ) )
21	3, 19, 20	mp2b 10	. . . 4 |- ( a e. ( `' abs " ( 0 [,) M ) ) <-> ( a e. CC /\ ( abs ` a ) e. ( 0 [,) M ) ) )
22	8, 18, 21	sylanbrc 698	. . 3 |- ( ( ph /\ a e. S ) -> a e. ( `' abs " ( 0 [,) M ) ) )
23		eqid 2622	. . . . 5 |- ( abs o. - ) = ( abs o. - )
24	23	cnbl0 22577	. . . 4 |- ( M e. RR* -> ( `' abs " ( 0 [,) M ) ) = ( 0 ( ball ` ( abs o. - ) ) M ) )
25	15, 24	syl 17	. . 3 |- ( ( ph /\ a e. S ) -> ( `' abs " ( 0 [,) M ) ) = ( 0 ( ball ` ( abs o. - ) ) M ) )
26	22, 25	eleqtrd 2703	. 2 |- ( ( ph /\ a e. S ) -> a e. ( 0 ( ball ` ( abs o. - ) ) M ) )
27		icossicc 12260	. . . 4 |- ( 0 [,) M ) C_ ( 0 [,] M )
28		imass2 5501	. . . 4 |- ( ( 0 [,) M ) C_ ( 0 [,] M ) -> ( `' abs " ( 0 [,) M ) ) C_ ( `' abs " ( 0 [,] M ) ) )
29	27, 28	mp1i 13	. . 3 |- ( ( ph /\ a e. S ) -> ( `' abs " ( 0 [,) M ) ) C_ ( `' abs " ( 0 [,] M ) ) )
30	25, 29	eqsstr3d 3640	. 2 |- ( ( ph /\ a e. S ) -> ( 0 ( ball ` ( abs o. - ) ) M ) C_ ( `' abs " ( 0 [,] M ) ) )
31		iccssxr 12256	. . . . . 6 |- ( 0 [,] +oo ) C_ RR*
32		pserf.g	. . . . . . . 8 |- G = ( x e. CC |-> ( n e. NN0 |-> ( ( A ` n ) x. ( x ^ n ) ) ) )
33		pserf.a	. . . . . . . 8 |- ( ph -> A : NN0 --> CC )
34		pserf.r	. . . . . . . 8 |- R = sup ( { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } , RR* , < )
35	32, 33, 34	radcnvcl 24171	. . . . . . 7 |- ( ph -> R e. ( 0 [,] +oo ) )
36	35	adantr 481	. . . . . 6 |- ( ( ph /\ a e. S ) -> R e. ( 0 [,] +oo ) )
37	31, 36	sseldi 3601	. . . . 5 |- ( ( ph /\ a e. S ) -> R e. RR* )
38	11	simp3d 1075	. . . . 5 |- ( ( ph /\ a e. S ) -> M < R )
39		df-ico 12181	. . . . . 6 |- [,) = ( u e. RR* , v e. RR* |-> { w e. RR* | ( u <_ w /\ w < v ) } )
40		df-icc 12182	. . . . . 6 |- [,] = ( u e. RR* , v e. RR* |-> { w e. RR* | ( u <_ w /\ w <_ v ) } )
41		xrlelttr 11987	. . . . . 6 |- ( ( z e. RR* /\ M e. RR* /\ R e. RR* ) -> ( ( z <_ M /\ M < R ) -> z < R ) )
42	39, 40, 41	ixxss2 12194	. . . . 5 |- ( ( R e. RR* /\ M < R ) -> ( 0 [,] M ) C_ ( 0 [,) R ) )
43	37, 38, 42	syl2anc 693	. . . 4 |- ( ( ph /\ a e. S ) -> ( 0 [,] M ) C_ ( 0 [,) R ) )
44		imass2 5501	. . . 4 |- ( ( 0 [,] M ) C_ ( 0 [,) R ) -> ( `' abs " ( 0 [,] M ) ) C_ ( `' abs " ( 0 [,) R ) ) )
45	43, 44	syl 17	. . 3 |- ( ( ph /\ a e. S ) -> ( `' abs " ( 0 [,] M ) ) C_ ( `' abs " ( 0 [,) R ) ) )
46	45, 1	syl6sseqr 3652	. 2 |- ( ( ph /\ a e. S ) -> ( `' abs " ( 0 [,] M ) ) C_ S )
47	26, 30, 46	3jca 1242	1 |- ( ( ph /\ a e. S ) -> ( a e. ( 0 ( ball ` ( abs o. - ) ) M ) /\ ( 0 ( ball ` ( abs o. - ) ) M ) C_ ( `' abs " ( 0 [,] M ) ) /\ ( `' abs " ( 0 [,] M ) ) C_ S ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   `'ccnv 5113   dom cdm 5114   "cima 5117   o. ccom 5118   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   supcsup 8346   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   x. cmul 9941   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   NN0cn0 11292   RR+crp 11832   [,)cico 12177   [,]cicc 12178   seqcseq 12801   ^cexp 12860   abscabs 13974   ~~> cli 14215   sum_csu 14416   ballcbl 19733

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-inf2 8538  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-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-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  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-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-xadd 11947  df-ico 12181  df-icc 12182  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-clim 14219  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741

This theorem is referenced by:  psercn  24180  pserdvlem2  24182  pserdv  24183