Theorem nmoi2 22534

Description: The operator norm is a bound on the growth of a vector under the action of the operator. (Contributed by Mario Carneiro, 19-Oct-2015.)

Hypotheses

Ref	Expression
nmofval.1	|- N = ( S normOp T )
nmoi.2	|- V = ( Base ` S )
nmoi.3	|- L = ( norm ` S )
nmoi.4	|- M = ( norm ` T )
nmoi2.z	|- .0. = ( 0g ` S )

Assertion

Ref	Expression
nmoi2	|- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( X e. V /\ X =/= .0. ) ) -> ( ( M ` ( F ` X ) ) / ( L ` X ) ) <_ ( N ` F ) )

Proof of Theorem nmoi2

Step	Hyp	Ref	Expression
1		simpl2 1065	. . . . 5 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( X e. V /\ X =/= .0. ) ) -> T e. NrmGrp )
2		simpl3 1066	. . . . . . 7 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( X e. V /\ X =/= .0. ) ) -> F e. ( S GrpHom T ) )
3		nmoi.2	. . . . . . . 8 |- V = ( Base ` S )
4		eqid 2622	. . . . . . . 8 |- ( Base ` T ) = ( Base ` T )
5	3, 4	ghmf 17664	. . . . . . 7 |- ( F e. ( S GrpHom T ) -> F : V --> ( Base ` T ) )
6	2, 5	syl 17	. . . . . 6 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( X e. V /\ X =/= .0. ) ) -> F : V --> ( Base ` T ) )
7		simprl 794	. . . . . 6 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( X e. V /\ X =/= .0. ) ) -> X e. V )
8	6, 7	ffvelrnd 6360	. . . . 5 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( X e. V /\ X =/= .0. ) ) -> ( F ` X ) e. ( Base ` T ) )
9		nmoi.4	. . . . . 6 |- M = ( norm ` T )
10	4, 9	nmcl 22420	. . . . 5 |- ( ( T e. NrmGrp /\ ( F ` X ) e. ( Base ` T ) ) -> ( M ` ( F ` X ) ) e. RR )
11	1, 8, 10	syl2anc 693	. . . 4 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( X e. V /\ X =/= .0. ) ) -> ( M ` ( F ` X ) ) e. RR )
12	11	rexrd 10089	. . 3 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( X e. V /\ X =/= .0. ) ) -> ( M ` ( F ` X ) ) e. RR* )
13		nmofval.1	. . . . . 6 |- N = ( S normOp T )
14	13	nmocl 22524	. . . . 5 |- ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) -> ( N ` F ) e. RR* )
15	14	adantr 481	. . . 4 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( X e. V /\ X =/= .0. ) ) -> ( N ` F ) e. RR* )
16		nmoi.3	. . . . . . . 8 |- L = ( norm ` S )
17		nmoi2.z	. . . . . . . 8 |- .0. = ( 0g ` S )
18	3, 16, 17	nmrpcl 22424	. . . . . . 7 |- ( ( S e. NrmGrp /\ X e. V /\ X =/= .0. ) -> ( L ` X ) e. RR+ )
19	18	3expb 1266	. . . . . 6 |- ( ( S e. NrmGrp /\ ( X e. V /\ X =/= .0. ) ) -> ( L ` X ) e. RR+ )
20	19	3ad2antl1 1223	. . . . 5 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( X e. V /\ X =/= .0. ) ) -> ( L ` X ) e. RR+ )
21	20	rpxrd 11873	. . . 4 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( X e. V /\ X =/= .0. ) ) -> ( L ` X ) e. RR* )
22	15, 21	xmulcld 12132	. . 3 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( X e. V /\ X =/= .0. ) ) -> ( ( N ` F ) xe ( L ` X ) ) e. RR* )
23	20	rpreccld 11882	. . . . 5 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( X e. V /\ X =/= .0. ) ) -> ( 1 / ( L ` X ) ) e. RR+ )
24	23	rpxrd 11873	. . . 4 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( X e. V /\ X =/= .0. ) ) -> ( 1 / ( L ` X ) ) e. RR* )
25	23	rpge0d 11876	. . . 4 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( X e. V /\ X =/= .0. ) ) -> 0 <_ ( 1 / ( L ` X ) ) )
26	24, 25	jca 554	. . 3 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( X e. V /\ X =/= .0. ) ) -> ( ( 1 / ( L ` X ) ) e. RR* /\ 0 <_ ( 1 / ( L ` X ) ) ) )
27	13, 3, 16, 9	nmoix 22533	. . . 4 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ X e. V ) -> ( M ` ( F ` X ) ) <_ ( ( N ` F ) xe ( L ` X ) ) )
28	27	adantrr 753	. . 3 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( X e. V /\ X =/= .0. ) ) -> ( M ` ( F ` X ) ) <_ ( ( N ` F ) xe ( L ` X ) ) )
29		xlemul1a 12118	. . 3 |- ( ( ( ( M ` ( F ` X ) ) e. RR* /\ ( ( N ` F ) xe ( L ` X ) ) e. RR* /\ ( ( 1 / ( L ` X ) ) e. RR* /\ 0 <_ ( 1 / ( L ` X ) ) ) ) /\ ( M ` ( F ` X ) ) <_ ( ( N ` F ) xe ( L ` X ) ) ) -> ( ( M ` ( F ` X ) ) xe ( 1 / ( L ` X ) ) ) <_ ( ( ( N ` F ) xe ( L ` X ) ) xe ( 1 / ( L ` X ) ) ) )
30	12, 22, 26, 28, 29	syl31anc 1329	. 2 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( X e. V /\ X =/= .0. ) ) -> ( ( M ` ( F ` X ) ) xe ( 1 / ( L ` X ) ) ) <_ ( ( ( N ` F ) xe ( L ` X ) ) xe ( 1 / ( L ` X ) ) ) )
31	23	rpred 11872	. . . 4 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( X e. V /\ X =/= .0. ) ) -> ( 1 / ( L ` X ) ) e. RR )
32		rexmul 12101	. . . 4 |- ( ( ( M ` ( F ` X ) ) e. RR /\ ( 1 / ( L ` X ) ) e. RR ) -> ( ( M ` ( F ` X ) ) xe ( 1 / ( L ` X ) ) ) = ( ( M ` ( F ` X ) ) x. ( 1 / ( L ` X ) ) ) )
33	11, 31, 32	syl2anc 693	. . 3 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( X e. V /\ X =/= .0. ) ) -> ( ( M ` ( F ` X ) ) xe ( 1 / ( L ` X ) ) ) = ( ( M ` ( F ` X ) ) x. ( 1 / ( L ` X ) ) ) )
34	11	recnd 10068	. . . 4 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( X e. V /\ X =/= .0. ) ) -> ( M ` ( F ` X ) ) e. CC )
35	20	rpcnd 11874	. . . 4 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( X e. V /\ X =/= .0. ) ) -> ( L ` X ) e. CC )
36	20	rpne0d 11877	. . . 4 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( X e. V /\ X =/= .0. ) ) -> ( L ` X ) =/= 0 )
37	34, 35, 36	divrecd 10804	. . 3 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( X e. V /\ X =/= .0. ) ) -> ( ( M ` ( F ` X ) ) / ( L ` X ) ) = ( ( M ` ( F ` X ) ) x. ( 1 / ( L ` X ) ) ) )
38	33, 37	eqtr4d 2659	. 2 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( X e. V /\ X =/= .0. ) ) -> ( ( M ` ( F ` X ) ) xe ( 1 / ( L ` X ) ) ) = ( ( M ` ( F ` X ) ) / ( L ` X ) ) )
39		xmulass 12117	. . . 4 |- ( ( ( N ` F ) e. RR* /\ ( L ` X ) e. RR* /\ ( 1 / ( L ` X ) ) e. RR* ) -> ( ( ( N ` F ) xe ( L ` X ) ) xe ( 1 / ( L ` X ) ) ) = ( ( N ` F ) xe ( ( L ` X ) xe ( 1 / ( L ` X ) ) ) ) )
40	15, 21, 24, 39	syl3anc 1326	. . 3 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( X e. V /\ X =/= .0. ) ) -> ( ( ( N ` F ) xe ( L ` X ) ) xe ( 1 / ( L ` X ) ) ) = ( ( N ` F ) xe ( ( L ` X ) xe ( 1 / ( L ` X ) ) ) ) )
41	20	rpred 11872	. . . . . 6 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( X e. V /\ X =/= .0. ) ) -> ( L ` X ) e. RR )
42		rexmul 12101	. . . . . 6 |- ( ( ( L ` X ) e. RR /\ ( 1 / ( L ` X ) ) e. RR ) -> ( ( L ` X ) xe ( 1 / ( L ` X ) ) ) = ( ( L ` X ) x. ( 1 / ( L ` X ) ) ) )
43	41, 31, 42	syl2anc 693	. . . . 5 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( X e. V /\ X =/= .0. ) ) -> ( ( L ` X ) xe ( 1 / ( L ` X ) ) ) = ( ( L ` X ) x. ( 1 / ( L ` X ) ) ) )
44	35, 36	recidd 10796	. . . . 5 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( X e. V /\ X =/= .0. ) ) -> ( ( L ` X ) x. ( 1 / ( L ` X ) ) ) = 1 )
45	43, 44	eqtrd 2656	. . . 4 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( X e. V /\ X =/= .0. ) ) -> ( ( L ` X ) xe ( 1 / ( L ` X ) ) ) = 1 )
46	45	oveq2d 6666	. . 3 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( X e. V /\ X =/= .0. ) ) -> ( ( N ` F ) xe ( ( L ` X ) xe ( 1 / ( L ` X ) ) ) ) = ( ( N ` F ) xe 1 ) )
47		xmulid1 12109	. . . 4 |- ( ( N ` F ) e. RR* -> ( ( N ` F ) xe 1 ) = ( N ` F ) )
48	15, 47	syl 17	. . 3 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( X e. V /\ X =/= .0. ) ) -> ( ( N ` F ) xe 1 ) = ( N ` F ) )
49	40, 46, 48	3eqtrd 2660	. 2 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( X e. V /\ X =/= .0. ) ) -> ( ( ( N ` F ) xe ( L ` X ) ) xe ( 1 / ( L ` X ) ) ) = ( N ` F ) )
50	30, 38, 49	3brtr3d 4684	1 |- ( ( ( S e. NrmGrp /\ T e. NrmGrp /\ F e. ( S GrpHom T ) ) /\ ( X e. V /\ X =/= .0. ) ) -> ( ( M ` ( F ` X ) ) / ( L ` X ) ) <_ ( N ` F ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   -->wf 5884   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   RR*cxr 10073   <_ cle 10075   / cdiv 10684   RR+crp 11832   xecxmu 11945   Basecbs 15857   0gc0g 16100   GrpHom cghm 17657   normcnm 22381  NrmGrpcngp 22382   normOpcnmo 22509

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-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-inf 8349  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-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ico 12181  df-0g 16102  df-topgen 16104  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-ghm 17658  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-xms 22125  df-ms 22126  df-nm 22387  df-ngp 22388  df-nmo 22512  df-nghm 22513

This theorem is referenced by:  nmoleub  22535