Theorem ovolsslem 23252

Description: Lemma for ovolss 23253. (Contributed by Mario Carneiro, 16-Mar-2014.) (Proof shortened by AV, 17-Sep-2020.)

Hypotheses

Ref	Expression
ovolss.1	|- M = { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) }
ovolss.2	|- N = { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( B C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) }

Assertion

Ref	Expression
ovolsslem	|- ( ( A C_ B /\ B C_ RR ) -> ( vol* ` A ) <_ ( vol* ` B ) )

Distinct variable groups:   y, f, A   B, f, y

Allowed substitution hints:   M( y, f)   N( y, f)

Proof of Theorem ovolsslem

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sstr2 3610	. . . . . . . . 9 |- ( A C_ B -> ( B C_ U. ran ( (,) o. f ) -> A C_ U. ran ( (,) o. f ) ) )
2	1	ad2antrr 762	. . . . . . . 8 |- ( ( ( A C_ B /\ B C_ RR ) /\ y e. RR* ) -> ( B C_ U. ran ( (,) o. f ) -> A C_ U. ran ( (,) o. f ) ) )
3	2	anim1d 588	. . . . . . 7 |- ( ( ( A C_ B /\ B C_ RR ) /\ y e. RR* ) -> ( ( B C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) -> ( A C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) ) )
4	3	reximdv 3016	. . . . . 6 |- ( ( ( A C_ B /\ B C_ RR ) /\ y e. RR* ) -> ( E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( B C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) -> E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) ) )
5	4	ss2rabdv 3683	. . . . 5 |- ( ( A C_ B /\ B C_ RR ) -> { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( B C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) } C_ { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) } )
6		ovolss.2	. . . . 5 |- N = { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( B C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) }
7		ovolss.1	. . . . 5 |- M = { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) }
8	5, 6, 7	3sstr4g 3646	. . . 4 |- ( ( A C_ B /\ B C_ RR ) -> N C_ M )
9		sstr 3611	. . . . 5 |- ( ( A C_ B /\ B C_ RR ) -> A C_ RR )
10	7	ovolval 23242	. . . . . . . 8 |- ( A C_ RR -> ( vol* ` A ) = inf ( M , RR* , < ) )
11	10	adantr 481	. . . . . . 7 |- ( ( A C_ RR /\ x e. M ) -> ( vol* ` A ) = inf ( M , RR* , < ) )
12		ssrab2 3687	. . . . . . . . . 10 |- { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) } C_ RR*
13	7, 12	eqsstri 3635	. . . . . . . . 9 |- M C_ RR*
14		infxrlb 12164	. . . . . . . . 9 |- ( ( M C_ RR* /\ x e. M ) -> inf ( M , RR* , < ) <_ x )
15	13, 14	mpan 706	. . . . . . . 8 |- ( x e. M -> inf ( M , RR* , < ) <_ x )
16	15	adantl 482	. . . . . . 7 |- ( ( A C_ RR /\ x e. M ) -> inf ( M , RR* , < ) <_ x )
17	11, 16	eqbrtrd 4675	. . . . . 6 |- ( ( A C_ RR /\ x e. M ) -> ( vol* ` A ) <_ x )
18	17	ralrimiva 2966	. . . . 5 |- ( A C_ RR -> A. x e. M ( vol* ` A ) <_ x )
19	9, 18	syl 17	. . . 4 |- ( ( A C_ B /\ B C_ RR ) -> A. x e. M ( vol* ` A ) <_ x )
20		ssralv 3666	. . . 4 |- ( N C_ M -> ( A. x e. M ( vol* ` A ) <_ x -> A. x e. N ( vol* ` A ) <_ x ) )
21	8, 19, 20	sylc 65	. . 3 |- ( ( A C_ B /\ B C_ RR ) -> A. x e. N ( vol* ` A ) <_ x )
22		ssrab2 3687	. . . . 5 |- { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( B C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) } C_ RR*
23	6, 22	eqsstri 3635	. . . 4 |- N C_ RR*
24		ovolcl 23246	. . . . 5 |- ( A C_ RR -> ( vol* ` A ) e. RR* )
25	9, 24	syl 17	. . . 4 |- ( ( A C_ B /\ B C_ RR ) -> ( vol* ` A ) e. RR* )
26		infxrgelb 12165	. . . 4 |- ( ( N C_ RR* /\ ( vol* ` A ) e. RR* ) -> ( ( vol* ` A ) <_ inf ( N , RR* , < ) <-> A. x e. N ( vol* ` A ) <_ x ) )
27	23, 25, 26	sylancr 695	. . 3 |- ( ( A C_ B /\ B C_ RR ) -> ( ( vol* ` A ) <_ inf ( N , RR* , < ) <-> A. x e. N ( vol* ` A ) <_ x ) )
28	21, 27	mpbird 247	. 2 |- ( ( A C_ B /\ B C_ RR ) -> ( vol* ` A ) <_ inf ( N , RR* , < ) )
29	6	ovolval 23242	. . 3 |- ( B C_ RR -> ( vol* ` B ) = inf ( N , RR* , < ) )
30	29	adantl 482	. 2 |- ( ( A C_ B /\ B C_ RR ) -> ( vol* ` B ) = inf ( N , RR* , < ) )
31	28, 30	breqtrrd 4681	1 |- ( ( A C_ B /\ B C_ RR ) -> ( vol* ` A ) <_ ( vol* ` B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   i^i cin 3573   C_ wss 3574   U.cuni 4436   class class class wbr 4653   X. cxp 5112   ran crn 5115   o. ccom 5118   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   supcsup 8346  infcinf 8347   RRcr 9935   1c1 9937   + caddc 9939   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   NNcn 11020   (,)cioo 12175   seqcseq 12801   abscabs 13974   vol*covol 23231

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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-ovol 23233

This theorem is referenced by:  ovolss  23253