Theorem dvlog2lem 24398

Description: Lemma for dvlog2 24399. (Contributed by Mario Carneiro, 1-Mar-2015.)

Hypothesis

Ref	Expression
dvlog2.s	|- S = ( 1 ( ball ` ( abs o. - ) ) 1 )

Assertion

Ref	Expression
dvlog2lem	|- S C_ ( CC \ ( -oo (,] 0 ) )

Proof of Theorem dvlog2lem

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvlog2.s	. . . . 5 |- S = ( 1 ( ball ` ( abs o. - ) ) 1 )
2		cnxmet 22576	. . . . . 6 |- ( abs o. - ) e. ( *Met ` CC )
3		ax-1cn 9994	. . . . . 6 |- 1 e. CC
4		1re 10039	. . . . . . 7 |- 1 e. RR
5	4	rexri 10097	. . . . . 6 |- 1 e. RR*
6		blssm 22223	. . . . . 6 |- ( ( ( abs o. - ) e. ( *Met ` CC ) /\ 1 e. CC /\ 1 e. RR* ) -> ( 1 ( ball ` ( abs o. - ) ) 1 ) C_ CC )
7	2, 3, 5, 6	mp3an 1424	. . . . 5 |- ( 1 ( ball ` ( abs o. - ) ) 1 ) C_ CC
8	1, 7	eqsstri 3635	. . . 4 |- S C_ CC
9	8	sseli 3599	. . 3 |- ( x e. S -> x e. CC )
10		1m0e1 11131	. . . . . . . . 9 |- ( 1 - 0 ) = 1
11		mnfxr 10096	. . . . . . . . . . . 12 |- -oo e. RR*
12		0re 10040	. . . . . . . . . . . 12 |- 0 e. RR
13		iocssre 12253	. . . . . . . . . . . 12 |- ( ( -oo e. RR* /\ 0 e. RR ) -> ( -oo (,] 0 ) C_ RR )
14	11, 12, 13	mp2an 708	. . . . . . . . . . 11 |- ( -oo (,] 0 ) C_ RR
15	14	sseli 3599	. . . . . . . . . 10 |- ( x e. ( -oo (,] 0 ) -> x e. RR )
16	12	a1i 11	. . . . . . . . . 10 |- ( x e. ( -oo (,] 0 ) -> 0 e. RR )
17	4	a1i 11	. . . . . . . . . 10 |- ( x e. ( -oo (,] 0 ) -> 1 e. RR )
18		elioc2 12236	. . . . . . . . . . . 12 |- ( ( -oo e. RR* /\ 0 e. RR ) -> ( x e. ( -oo (,] 0 ) <-> ( x e. RR /\ -oo < x /\ x <_ 0 ) ) )
19	11, 12, 18	mp2an 708	. . . . . . . . . . 11 |- ( x e. ( -oo (,] 0 ) <-> ( x e. RR /\ -oo < x /\ x <_ 0 ) )
20	19	simp3bi 1078	. . . . . . . . . 10 |- ( x e. ( -oo (,] 0 ) -> x <_ 0 )
21	15, 16, 17, 20	lesub2dd 10644	. . . . . . . . 9 |- ( x e. ( -oo (,] 0 ) -> ( 1 - 0 ) <_ ( 1 - x ) )
22	10, 21	syl5eqbrr 4689	. . . . . . . 8 |- ( x e. ( -oo (,] 0 ) -> 1 <_ ( 1 - x ) )
23		ax-resscn 9993	. . . . . . . . . . . 12 |- RR C_ CC
24	14, 23	sstri 3612	. . . . . . . . . . 11 |- ( -oo (,] 0 ) C_ CC
25	24	sseli 3599	. . . . . . . . . 10 |- ( x e. ( -oo (,] 0 ) -> x e. CC )
26		eqid 2622	. . . . . . . . . . 11 |- ( abs o. - ) = ( abs o. - )
27	26	cnmetdval 22574	. . . . . . . . . 10 |- ( ( 1 e. CC /\ x e. CC ) -> ( 1 ( abs o. - ) x ) = ( abs ` ( 1 - x ) ) )
28	3, 25, 27	sylancr 695	. . . . . . . . 9 |- ( x e. ( -oo (,] 0 ) -> ( 1 ( abs o. - ) x ) = ( abs ` ( 1 - x ) ) )
29		0le1 10551	. . . . . . . . . . . 12 |- 0 <_ 1
30	29	a1i 11	. . . . . . . . . . 11 |- ( x e. ( -oo (,] 0 ) -> 0 <_ 1 )
31	15, 16, 17, 20, 30	letrd 10194	. . . . . . . . . 10 |- ( x e. ( -oo (,] 0 ) -> x <_ 1 )
32	15, 17, 31	abssubge0d 14170	. . . . . . . . 9 |- ( x e. ( -oo (,] 0 ) -> ( abs ` ( 1 - x ) ) = ( 1 - x ) )
33	28, 32	eqtrd 2656	. . . . . . . 8 |- ( x e. ( -oo (,] 0 ) -> ( 1 ( abs o. - ) x ) = ( 1 - x ) )
34	22, 33	breqtrrd 4681	. . . . . . 7 |- ( x e. ( -oo (,] 0 ) -> 1 <_ ( 1 ( abs o. - ) x ) )
35		cnmet 22575	. . . . . . . . . 10 |- ( abs o. - ) e. ( Met ` CC )
36	35	a1i 11	. . . . . . . . 9 |- ( x e. ( -oo (,] 0 ) -> ( abs o. - ) e. ( Met ` CC ) )
37	3	a1i 11	. . . . . . . . 9 |- ( x e. ( -oo (,] 0 ) -> 1 e. CC )
38		metcl 22137	. . . . . . . . 9 |- ( ( ( abs o. - ) e. ( Met ` CC ) /\ 1 e. CC /\ x e. CC ) -> ( 1 ( abs o. - ) x ) e. RR )
39	36, 37, 25, 38	syl3anc 1326	. . . . . . . 8 |- ( x e. ( -oo (,] 0 ) -> ( 1 ( abs o. - ) x ) e. RR )
40		lenlt 10116	. . . . . . . 8 |- ( ( 1 e. RR /\ ( 1 ( abs o. - ) x ) e. RR ) -> ( 1 <_ ( 1 ( abs o. - ) x ) <-> -. ( 1 ( abs o. - ) x ) < 1 ) )
41	4, 39, 40	sylancr 695	. . . . . . 7 |- ( x e. ( -oo (,] 0 ) -> ( 1 <_ ( 1 ( abs o. - ) x ) <-> -. ( 1 ( abs o. - ) x ) < 1 ) )
42	34, 41	mpbid 222	. . . . . 6 |- ( x e. ( -oo (,] 0 ) -> -. ( 1 ( abs o. - ) x ) < 1 )
43	2	a1i 11	. . . . . . 7 |- ( x e. ( -oo (,] 0 ) -> ( abs o. - ) e. ( *Met ` CC ) )
44	5	a1i 11	. . . . . . 7 |- ( x e. ( -oo (,] 0 ) -> 1 e. RR* )
45		elbl2 22195	. . . . . . 7 |- ( ( ( ( abs o. - ) e. ( *Met ` CC ) /\ 1 e. RR* ) /\ ( 1 e. CC /\ x e. CC ) ) -> ( x e. ( 1 ( ball ` ( abs o. - ) ) 1 ) <-> ( 1 ( abs o. - ) x ) < 1 ) )
46	43, 44, 37, 25, 45	syl22anc 1327	. . . . . 6 |- ( x e. ( -oo (,] 0 ) -> ( x e. ( 1 ( ball ` ( abs o. - ) ) 1 ) <-> ( 1 ( abs o. - ) x ) < 1 ) )
47	42, 46	mtbird 315	. . . . 5 |- ( x e. ( -oo (,] 0 ) -> -. x e. ( 1 ( ball ` ( abs o. - ) ) 1 ) )
48	47	con2i 134	. . . 4 |- ( x e. ( 1 ( ball ` ( abs o. - ) ) 1 ) -> -. x e. ( -oo (,] 0 ) )
49	48, 1	eleq2s 2719	. . 3 |- ( x e. S -> -. x e. ( -oo (,] 0 ) )
50	9, 49	eldifd 3585	. 2 |- ( x e. S -> x e. ( CC \ ( -oo (,] 0 ) ) )
51	50	ssriv 3607	1 |- S C_ ( CC \ ( -oo (,] 0 ) )

Syntax hints:   -. wn 3   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   \ cdif 3571   C_ wss 3574   class class class wbr 4653   o. ccom 5118   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   -oocmnf 10072   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   (,]cioc 12176   abscabs 13974   *Metcxmt 19731   Metcme 19732   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-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  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-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  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-ioc 12180  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741

This theorem is referenced by:  dvlog2  24399  logtayl  24406  logtayl2  24408  efrlim  24696  lgamcvg2  24781