Theorem pserdvlem1 24181

Description: Lemma for pserdv 24183. (Contributed by Mario Carneiro, 7-May-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 ) )
psercn.m	|- M = if ( R e. RR , ( ( ( abs ` a ) + R ) / 2 ) , ( ( abs ` a ) + 1 ) )

Assertion

Ref	Expression
pserdvlem1	|- ( ( ph /\ a e. S ) -> ( ( ( ( abs ` a ) + M ) / 2 ) e. RR+ /\ ( abs ` a ) < ( ( ( abs ` a ) + M ) / 2 ) /\ ( ( ( abs ` a ) + M ) / 2 ) < R ) )

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 pserdvlem1

Step	Hyp	Ref	Expression
1		psercn.s	. . . . . . . . 9 |- S = ( `' abs " ( 0 [,) R ) )
2		cnvimass 5485	. . . . . . . . . 10 |- ( `' abs " ( 0 [,) R ) ) C_ dom abs
3		absf 14077	. . . . . . . . . . 11 |- abs : CC --> RR
4	3	fdmi 6052	. . . . . . . . . 10 |- dom abs = CC
5	2, 4	sseqtri 3637	. . . . . . . . 9 |- ( `' abs " ( 0 [,) R ) ) C_ CC
6	1, 5	eqsstri 3635	. . . . . . . 8 |- S C_ CC
7	6	a1i 11	. . . . . . 7 |- ( ph -> S C_ CC )
8	7	sselda 3603	. . . . . 6 |- ( ( ph /\ a e. S ) -> a e. CC )
9	8	abscld 14175	. . . . 5 |- ( ( ph /\ a e. S ) -> ( abs ` a ) e. RR )
10		pserf.g	. . . . . . . 8 |- G = ( x e. CC |-> ( n e. NN0 |-> ( ( A ` n ) x. ( x ^ n ) ) ) )
11		pserf.f	. . . . . . . 8 |- F = ( y e. S |-> sum_ j e. NN0 ( ( G ` y ) ` j ) )
12		pserf.a	. . . . . . . 8 |- ( ph -> A : NN0 --> CC )
13		pserf.r	. . . . . . . 8 |- R = sup ( { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } , RR* , < )
14		psercn.m	. . . . . . . 8 |- M = if ( R e. RR , ( ( ( abs ` a ) + R ) / 2 ) , ( ( abs ` a ) + 1 ) )
15	10, 11, 12, 13, 1, 14	psercnlem1 24179	. . . . . . 7 |- ( ( ph /\ a e. S ) -> ( M e. RR+ /\ ( abs ` a ) < M /\ M < R ) )
16	15	simp1d 1073	. . . . . 6 |- ( ( ph /\ a e. S ) -> M e. RR+ )
17	16	rpred 11872	. . . . 5 |- ( ( ph /\ a e. S ) -> M e. RR )
18	9, 17	readdcld 10069	. . . 4 |- ( ( ph /\ a e. S ) -> ( ( abs ` a ) + M ) e. RR )
19		0red 10041	. . . . 5 |- ( ( ph /\ a e. S ) -> 0 e. RR )
20	8	absge0d 14183	. . . . 5 |- ( ( ph /\ a e. S ) -> 0 <_ ( abs ` a ) )
21	9, 16	ltaddrpd 11905	. . . . 5 |- ( ( ph /\ a e. S ) -> ( abs ` a ) < ( ( abs ` a ) + M ) )
22	19, 9, 18, 20, 21	lelttrd 10195	. . . 4 |- ( ( ph /\ a e. S ) -> 0 < ( ( abs ` a ) + M ) )
23	18, 22	elrpd 11869	. . 3 |- ( ( ph /\ a e. S ) -> ( ( abs ` a ) + M ) e. RR+ )
24	23	rphalfcld 11884	. 2 |- ( ( ph /\ a e. S ) -> ( ( ( abs ` a ) + M ) / 2 ) e. RR+ )
25	15	simp2d 1074	. . 3 |- ( ( ph /\ a e. S ) -> ( abs ` a ) < M )
26		avglt1 11270	. . . 4 |- ( ( ( abs ` a ) e. RR /\ M e. RR ) -> ( ( abs ` a ) < M <-> ( abs ` a ) < ( ( ( abs ` a ) + M ) / 2 ) ) )
27	9, 17, 26	syl2anc 693	. . 3 |- ( ( ph /\ a e. S ) -> ( ( abs ` a ) < M <-> ( abs ` a ) < ( ( ( abs ` a ) + M ) / 2 ) ) )
28	25, 27	mpbid 222	. 2 |- ( ( ph /\ a e. S ) -> ( abs ` a ) < ( ( ( abs ` a ) + M ) / 2 ) )
29	18	rehalfcld 11279	. . . 4 |- ( ( ph /\ a e. S ) -> ( ( ( abs ` a ) + M ) / 2 ) e. RR )
30	29	rexrd 10089	. . 3 |- ( ( ph /\ a e. S ) -> ( ( ( abs ` a ) + M ) / 2 ) e. RR* )
31	17	rexrd 10089	. . 3 |- ( ( ph /\ a e. S ) -> M e. RR* )
32		iccssxr 12256	. . . . 5 |- ( 0 [,] +oo ) C_ RR*
33	10, 12, 13	radcnvcl 24171	. . . . 5 |- ( ph -> R e. ( 0 [,] +oo ) )
34	32, 33	sseldi 3601	. . . 4 |- ( ph -> R e. RR* )
35	34	adantr 481	. . 3 |- ( ( ph /\ a e. S ) -> R e. RR* )
36		avglt2 11271	. . . . 5 |- ( ( ( abs ` a ) e. RR /\ M e. RR ) -> ( ( abs ` a ) < M <-> ( ( ( abs ` a ) + M ) / 2 ) < M ) )
37	9, 17, 36	syl2anc 693	. . . 4 |- ( ( ph /\ a e. S ) -> ( ( abs ` a ) < M <-> ( ( ( abs ` a ) + M ) / 2 ) < M ) )
38	25, 37	mpbid 222	. . 3 |- ( ( ph /\ a e. S ) -> ( ( ( abs ` a ) + M ) / 2 ) < M )
39	15	simp3d 1075	. . 3 |- ( ( ph /\ a e. S ) -> M < R )
40	30, 31, 35, 38, 39	xrlttrd 11990	. 2 |- ( ( ph /\ a e. S ) -> ( ( ( abs ` a ) + M ) / 2 ) < R )
41	24, 28, 40	3jca 1242	1 |- ( ( ph /\ a e. S ) -> ( ( ( ( abs ` a ) + M ) / 2 ) e. RR+ /\ ( abs ` a ) < ( ( ( abs ` a ) + M ) / 2 ) /\ ( ( ( abs ` a ) + M ) / 2 ) < R ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   C_ wss 3574   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   `'ccnv 5113   dom cdm 5114   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650   supcsup 8346   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   +oocpnf 10071   RR*cxr 10073   < clt 10074   / cdiv 10684   2c2 11070   NN0cn0 11292   RR+crp 11832   [,)cico 12177   [,]cicc 12178   seqcseq 12801   ^cexp 12860   abscabs 13974   ~~> cli 14215   sum_csu 14416

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-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-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

This theorem is referenced by:  pserdvlem2  24182  pserdv  24183