Theorem psercnlem1 24179

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 ) )
psercn.m	|- M = if ( R e. RR , ( ( ( abs ` a ) + R ) / 2 ) , ( ( abs ` a ) + 1 ) )

Assertion

Ref	Expression
psercnlem1	|- ( ( ph /\ a e. S ) -> ( M e. RR+ /\ ( abs ` a ) < M /\ M < 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 psercnlem1

Step	Hyp	Ref	Expression
1		psercn.m	. . . 4 |- M = if ( R e. RR , ( ( ( abs ` a ) + R ) / 2 ) , ( ( abs ` a ) + 1 ) )
2		psercn.s	. . . . . . . . . . 11 |- S = ( `' abs " ( 0 [,) R ) )
3		cnvimass 5485	. . . . . . . . . . . 12 |- ( `' abs " ( 0 [,) R ) ) C_ dom abs
4		absf 14077	. . . . . . . . . . . . 13 |- abs : CC --> RR
5	4	fdmi 6052	. . . . . . . . . . . 12 |- dom abs = CC
6	3, 5	sseqtri 3637	. . . . . . . . . . 11 |- ( `' abs " ( 0 [,) R ) ) C_ CC
7	2, 6	eqsstri 3635	. . . . . . . . . 10 |- S C_ CC
8	7	a1i 11	. . . . . . . . 9 |- ( ph -> S C_ CC )
9	8	sselda 3603	. . . . . . . 8 |- ( ( ph /\ a e. S ) -> a e. CC )
10	9	abscld 14175	. . . . . . 7 |- ( ( ph /\ a e. S ) -> ( abs ` a ) e. RR )
11		readdcl 10019	. . . . . . 7 |- ( ( ( abs ` a ) e. RR /\ R e. RR ) -> ( ( abs ` a ) + R ) e. RR )
12	10, 11	sylan 488	. . . . . 6 |- ( ( ( ph /\ a e. S ) /\ R e. RR ) -> ( ( abs ` a ) + R ) e. RR )
13	12	rehalfcld 11279	. . . . 5 |- ( ( ( ph /\ a e. S ) /\ R e. RR ) -> ( ( ( abs ` a ) + R ) / 2 ) e. RR )
14		peano2re 10209	. . . . . . 7 |- ( ( abs ` a ) e. RR -> ( ( abs ` a ) + 1 ) e. RR )
15	10, 14	syl 17	. . . . . 6 |- ( ( ph /\ a e. S ) -> ( ( abs ` a ) + 1 ) e. RR )
16	15	adantr 481	. . . . 5 |- ( ( ( ph /\ a e. S ) /\ -. R e. RR ) -> ( ( abs ` a ) + 1 ) e. RR )
17	13, 16	ifclda 4120	. . . 4 |- ( ( ph /\ a e. S ) -> if ( R e. RR , ( ( ( abs ` a ) + R ) / 2 ) , ( ( abs ` a ) + 1 ) ) e. RR )
18	1, 17	syl5eqel 2705	. . 3 |- ( ( ph /\ a e. S ) -> M e. RR )
19		0re 10040	. . . . 5 |- 0 e. RR
20	19	a1i 11	. . . 4 |- ( ( ph /\ a e. S ) -> 0 e. RR )
21	9	absge0d 14183	. . . 4 |- ( ( ph /\ a e. S ) -> 0 <_ ( abs ` a ) )
22		breq2 4657	. . . . . 6 |- ( ( ( ( abs ` a ) + R ) / 2 ) = if ( R e. RR , ( ( ( abs ` a ) + R ) / 2 ) , ( ( abs ` a ) + 1 ) ) -> ( ( abs ` a ) < ( ( ( abs ` a ) + R ) / 2 ) <-> ( abs ` a ) < if ( R e. RR , ( ( ( abs ` a ) + R ) / 2 ) , ( ( abs ` a ) + 1 ) ) ) )
23		breq2 4657	. . . . . 6 |- ( ( ( abs ` a ) + 1 ) = if ( R e. RR , ( ( ( abs ` a ) + R ) / 2 ) , ( ( abs ` a ) + 1 ) ) -> ( ( abs ` a ) < ( ( abs ` a ) + 1 ) <-> ( abs ` a ) < if ( R e. RR , ( ( ( abs ` a ) + R ) / 2 ) , ( ( abs ` a ) + 1 ) ) ) )
24		simpr 477	. . . . . . . . . . . . 13 |- ( ( ph /\ a e. S ) -> a e. S )
25	24, 2	syl6eleq 2711	. . . . . . . . . . . 12 |- ( ( ph /\ a e. S ) -> a e. ( `' abs " ( 0 [,) R ) ) )
26		ffn 6045	. . . . . . . . . . . . 13 |- ( abs : CC --> RR -> abs Fn CC )
27		elpreima 6337	. . . . . . . . . . . . 13 |- ( abs Fn CC -> ( a e. ( `' abs " ( 0 [,) R ) ) <-> ( a e. CC /\ ( abs ` a ) e. ( 0 [,) R ) ) ) )
28	4, 26, 27	mp2b 10	. . . . . . . . . . . 12 |- ( a e. ( `' abs " ( 0 [,) R ) ) <-> ( a e. CC /\ ( abs ` a ) e. ( 0 [,) R ) ) )
29	25, 28	sylib 208	. . . . . . . . . . 11 |- ( ( ph /\ a e. S ) -> ( a e. CC /\ ( abs ` a ) e. ( 0 [,) R ) ) )
30	29	simprd 479	. . . . . . . . . 10 |- ( ( ph /\ a e. S ) -> ( abs ` a ) e. ( 0 [,) R ) )
31		iccssxr 12256	. . . . . . . . . . . 12 |- ( 0 [,] +oo ) C_ RR*
32		pserf.g	. . . . . . . . . . . . . 14 |- G = ( x e. CC |-> ( n e. NN0 |-> ( ( A ` n ) x. ( x ^ n ) ) ) )
33		pserf.a	. . . . . . . . . . . . . 14 |- ( ph -> A : NN0 --> CC )
34		pserf.r	. . . . . . . . . . . . . 14 |- R = sup ( { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } , RR* , < )
35	32, 33, 34	radcnvcl 24171	. . . . . . . . . . . . 13 |- ( ph -> R e. ( 0 [,] +oo ) )
36	35	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ a e. S ) -> R e. ( 0 [,] +oo ) )
37	31, 36	sseldi 3601	. . . . . . . . . . 11 |- ( ( ph /\ a e. S ) -> R e. RR* )
38		elico2 12237	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ R e. RR* ) -> ( ( abs ` a ) e. ( 0 [,) R ) <-> ( ( abs ` a ) e. RR /\ 0 <_ ( abs ` a ) /\ ( abs ` a ) < R ) ) )
39	19, 37, 38	sylancr 695	. . . . . . . . . 10 |- ( ( ph /\ a e. S ) -> ( ( abs ` a ) e. ( 0 [,) R ) <-> ( ( abs ` a ) e. RR /\ 0 <_ ( abs ` a ) /\ ( abs ` a ) < R ) ) )
40	30, 39	mpbid 222	. . . . . . . . 9 |- ( ( ph /\ a e. S ) -> ( ( abs ` a ) e. RR /\ 0 <_ ( abs ` a ) /\ ( abs ` a ) < R ) )
41	40	simp3d 1075	. . . . . . . 8 |- ( ( ph /\ a e. S ) -> ( abs ` a ) < R )
42	41	adantr 481	. . . . . . 7 |- ( ( ( ph /\ a e. S ) /\ R e. RR ) -> ( abs ` a ) < R )
43		avglt1 11270	. . . . . . . 8 |- ( ( ( abs ` a ) e. RR /\ R e. RR ) -> ( ( abs ` a ) < R <-> ( abs ` a ) < ( ( ( abs ` a ) + R ) / 2 ) ) )
44	10, 43	sylan 488	. . . . . . 7 |- ( ( ( ph /\ a e. S ) /\ R e. RR ) -> ( ( abs ` a ) < R <-> ( abs ` a ) < ( ( ( abs ` a ) + R ) / 2 ) ) )
45	42, 44	mpbid 222	. . . . . 6 |- ( ( ( ph /\ a e. S ) /\ R e. RR ) -> ( abs ` a ) < ( ( ( abs ` a ) + R ) / 2 ) )
46	10	ltp1d 10954	. . . . . . 7 |- ( ( ph /\ a e. S ) -> ( abs ` a ) < ( ( abs ` a ) + 1 ) )
47	46	adantr 481	. . . . . 6 |- ( ( ( ph /\ a e. S ) /\ -. R e. RR ) -> ( abs ` a ) < ( ( abs ` a ) + 1 ) )
48	22, 23, 45, 47	ifbothda 4123	. . . . 5 |- ( ( ph /\ a e. S ) -> ( abs ` a ) < if ( R e. RR , ( ( ( abs ` a ) + R ) / 2 ) , ( ( abs ` a ) + 1 ) ) )
49	48, 1	syl6breqr 4695	. . . 4 |- ( ( ph /\ a e. S ) -> ( abs ` a ) < M )
50	20, 10, 18, 21, 49	lelttrd 10195	. . 3 |- ( ( ph /\ a e. S ) -> 0 < M )
51	18, 50	elrpd 11869	. 2 |- ( ( ph /\ a e. S ) -> M e. RR+ )
52		breq1 4656	. . . 4 |- ( ( ( ( abs ` a ) + R ) / 2 ) = if ( R e. RR , ( ( ( abs ` a ) + R ) / 2 ) , ( ( abs ` a ) + 1 ) ) -> ( ( ( ( abs ` a ) + R ) / 2 ) < R <-> if ( R e. RR , ( ( ( abs ` a ) + R ) / 2 ) , ( ( abs ` a ) + 1 ) ) < R ) )
53		breq1 4656	. . . 4 |- ( ( ( abs ` a ) + 1 ) = if ( R e. RR , ( ( ( abs ` a ) + R ) / 2 ) , ( ( abs ` a ) + 1 ) ) -> ( ( ( abs ` a ) + 1 ) < R <-> if ( R e. RR , ( ( ( abs ` a ) + R ) / 2 ) , ( ( abs ` a ) + 1 ) ) < R ) )
54		avglt2 11271	. . . . . 6 |- ( ( ( abs ` a ) e. RR /\ R e. RR ) -> ( ( abs ` a ) < R <-> ( ( ( abs ` a ) + R ) / 2 ) < R ) )
55	10, 54	sylan 488	. . . . 5 |- ( ( ( ph /\ a e. S ) /\ R e. RR ) -> ( ( abs ` a ) < R <-> ( ( ( abs ` a ) + R ) / 2 ) < R ) )
56	42, 55	mpbid 222	. . . 4 |- ( ( ( ph /\ a e. S ) /\ R e. RR ) -> ( ( ( abs ` a ) + R ) / 2 ) < R )
57	15	rexrd 10089	. . . . . . . 8 |- ( ( ph /\ a e. S ) -> ( ( abs ` a ) + 1 ) e. RR* )
58		xrlenlt 10103	. . . . . . . 8 |- ( ( R e. RR* /\ ( ( abs ` a ) + 1 ) e. RR* ) -> ( R <_ ( ( abs ` a ) + 1 ) <-> -. ( ( abs ` a ) + 1 ) < R ) )
59	37, 57, 58	syl2anc 693	. . . . . . 7 |- ( ( ph /\ a e. S ) -> ( R <_ ( ( abs ` a ) + 1 ) <-> -. ( ( abs ` a ) + 1 ) < R ) )
60		0xr 10086	. . . . . . . . . . . . 13 |- 0 e. RR*
61		pnfxr 10092	. . . . . . . . . . . . 13 |- +oo e. RR*
62		elicc1 12219	. . . . . . . . . . . . 13 |- ( ( 0 e. RR* /\ +oo e. RR* ) -> ( R e. ( 0 [,] +oo ) <-> ( R e. RR* /\ 0 <_ R /\ R <_ +oo ) ) )
63	60, 61, 62	mp2an 708	. . . . . . . . . . . 12 |- ( R e. ( 0 [,] +oo ) <-> ( R e. RR* /\ 0 <_ R /\ R <_ +oo ) )
64	35, 63	sylib 208	. . . . . . . . . . 11 |- ( ph -> ( R e. RR* /\ 0 <_ R /\ R <_ +oo ) )
65	64	simp2d 1074	. . . . . . . . . 10 |- ( ph -> 0 <_ R )
66	65	adantr 481	. . . . . . . . 9 |- ( ( ph /\ a e. S ) -> 0 <_ R )
67		ge0gtmnf 12003	. . . . . . . . 9 |- ( ( R e. RR* /\ 0 <_ R ) -> -oo < R )
68	37, 66, 67	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ a e. S ) -> -oo < R )
69		xrre 12000	. . . . . . . . 9 |- ( ( ( R e. RR* /\ ( ( abs ` a ) + 1 ) e. RR ) /\ ( -oo < R /\ R <_ ( ( abs ` a ) + 1 ) ) ) -> R e. RR )
70	69	expr 643	. . . . . . . 8 |- ( ( ( R e. RR* /\ ( ( abs ` a ) + 1 ) e. RR ) /\ -oo < R ) -> ( R <_ ( ( abs ` a ) + 1 ) -> R e. RR ) )
71	37, 15, 68, 70	syl21anc 1325	. . . . . . 7 |- ( ( ph /\ a e. S ) -> ( R <_ ( ( abs ` a ) + 1 ) -> R e. RR ) )
72	59, 71	sylbird 250	. . . . . 6 |- ( ( ph /\ a e. S ) -> ( -. ( ( abs ` a ) + 1 ) < R -> R e. RR ) )
73	72	con1d 139	. . . . 5 |- ( ( ph /\ a e. S ) -> ( -. R e. RR -> ( ( abs ` a ) + 1 ) < R ) )
74	73	imp 445	. . . 4 |- ( ( ( ph /\ a e. S ) /\ -. R e. RR ) -> ( ( abs ` a ) + 1 ) < R )
75	52, 53, 56, 74	ifbothda 4123	. . 3 |- ( ( ph /\ a e. S ) -> if ( R e. RR , ( ( ( abs ` a ) + R ) / 2 ) , ( ( abs ` a ) + 1 ) ) < R )
76	1, 75	syl5eqbr 4688	. 2 |- ( ( ph /\ a e. S ) -> M < R )
77	51, 49, 76	3jca 1242	1 |- ( ( ph /\ a e. S ) -> ( M e. RR+ /\ ( abs ` a ) < M /\ M < R ) )

Syntax hints:   -. wn 3   -> 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   Fn wfn 5883   -->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   -oocmnf 10072   RR*cxr 10073   < clt 10074   <_ cle 10075   / 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:  psercn  24180  pserdvlem1  24181  pserdvlem2  24182  pserdv  24183