Theorem nmoid 22546

Description: The operator norm of the identity function on a nontrivial group. (Contributed by Mario Carneiro, 20-Oct-2015.)

Hypotheses

Ref	Expression
nmoid.1	|- N = ( S normOp S )
nmoid.2	|- V = ( Base ` S )
nmoid.3	|- .0. = ( 0g ` S )

Assertion

Ref	Expression
nmoid	|- ( ( S e. NrmGrp /\ { .0. } C. V ) -> ( N ` ( _I |` V ) ) = 1 )

Proof of Theorem nmoid

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nmoid.1	. . 3 |- N = ( S normOp S )
2		nmoid.2	. . 3 |- V = ( Base ` S )
3		eqid 2622	. . 3 |- ( norm ` S ) = ( norm ` S )
4		nmoid.3	. . 3 |- .0. = ( 0g ` S )
5		simpl 473	. . 3 |- ( ( S e. NrmGrp /\ { .0. } C. V ) -> S e. NrmGrp )
6		ngpgrp 22403	. . . . 5 |- ( S e. NrmGrp -> S e. Grp )
7	6	adantr 481	. . . 4 |- ( ( S e. NrmGrp /\ { .0. } C. V ) -> S e. Grp )
8	2	idghm 17675	. . . 4 |- ( S e. Grp -> ( _I |` V ) e. ( S GrpHom S ) )
9	7, 8	syl 17	. . 3 |- ( ( S e. NrmGrp /\ { .0. } C. V ) -> ( _I |` V ) e. ( S GrpHom S ) )
10		1red 10055	. . 3 |- ( ( S e. NrmGrp /\ { .0. } C. V ) -> 1 e. RR )
11		0le1 10551	. . . 4 |- 0 <_ 1
12	11	a1i 11	. . 3 |- ( ( S e. NrmGrp /\ { .0. } C. V ) -> 0 <_ 1 )
13	2, 3	nmcl 22420	. . . . . 6 |- ( ( S e. NrmGrp /\ x e. V ) -> ( ( norm ` S ) ` x ) e. RR )
14	13	ad2ant2r 783	. . . . 5 |- ( ( ( S e. NrmGrp /\ { .0. } C. V ) /\ ( x e. V /\ x =/= .0. ) ) -> ( ( norm ` S ) ` x ) e. RR )
15	14	leidd 10594	. . . 4 |- ( ( ( S e. NrmGrp /\ { .0. } C. V ) /\ ( x e. V /\ x =/= .0. ) ) -> ( ( norm ` S ) ` x ) <_ ( ( norm ` S ) ` x ) )
16		fvresi 6439	. . . . . 6 |- ( x e. V -> ( ( _I |` V ) ` x ) = x )
17	16	ad2antrl 764	. . . . 5 |- ( ( ( S e. NrmGrp /\ { .0. } C. V ) /\ ( x e. V /\ x =/= .0. ) ) -> ( ( _I |` V ) ` x ) = x )
18	17	fveq2d 6195	. . . 4 |- ( ( ( S e. NrmGrp /\ { .0. } C. V ) /\ ( x e. V /\ x =/= .0. ) ) -> ( ( norm ` S ) ` ( ( _I |` V ) ` x ) ) = ( ( norm ` S ) ` x ) )
19	14	recnd 10068	. . . . 5 |- ( ( ( S e. NrmGrp /\ { .0. } C. V ) /\ ( x e. V /\ x =/= .0. ) ) -> ( ( norm ` S ) ` x ) e. CC )
20	19	mulid2d 10058	. . . 4 |- ( ( ( S e. NrmGrp /\ { .0. } C. V ) /\ ( x e. V /\ x =/= .0. ) ) -> ( 1 x. ( ( norm ` S ) ` x ) ) = ( ( norm ` S ) ` x ) )
21	15, 18, 20	3brtr4d 4685	. . 3 |- ( ( ( S e. NrmGrp /\ { .0. } C. V ) /\ ( x e. V /\ x =/= .0. ) ) -> ( ( norm ` S ) ` ( ( _I |` V ) ` x ) ) <_ ( 1 x. ( ( norm ` S ) ` x ) ) )
22	1, 2, 3, 3, 4, 5, 5, 9, 10, 12, 21	nmolb2d 22522	. 2 |- ( ( S e. NrmGrp /\ { .0. } C. V ) -> ( N ` ( _I |` V ) ) <_ 1 )
23		pssnel 4039	. . . 4 |- ( { .0. } C. V -> E. x ( x e. V /\ -. x e. { .0. } ) )
24	23	adantl 482	. . 3 |- ( ( S e. NrmGrp /\ { .0. } C. V ) -> E. x ( x e. V /\ -. x e. { .0. } ) )
25		velsn 4193	. . . . . 6 |- ( x e. { .0. } <-> x = .0. )
26	25	biimpri 218	. . . . 5 |- ( x = .0. -> x e. { .0. } )
27	26	necon3bi 2820	. . . 4 |- ( -. x e. { .0. } -> x =/= .0. )
28	20, 18	eqtr4d 2659	. . . . . 6 |- ( ( ( S e. NrmGrp /\ { .0. } C. V ) /\ ( x e. V /\ x =/= .0. ) ) -> ( 1 x. ( ( norm ` S ) ` x ) ) = ( ( norm ` S ) ` ( ( _I |` V ) ` x ) ) )
29	1	nmocl 22524	. . . . . . . . . . 11 |- ( ( S e. NrmGrp /\ S e. NrmGrp /\ ( _I |` V ) e. ( S GrpHom S ) ) -> ( N ` ( _I |` V ) ) e. RR* )
30	5, 5, 9, 29	syl3anc 1326	. . . . . . . . . 10 |- ( ( S e. NrmGrp /\ { .0. } C. V ) -> ( N ` ( _I |` V ) ) e. RR* )
31	1	nmoge0 22525	. . . . . . . . . . 11 |- ( ( S e. NrmGrp /\ S e. NrmGrp /\ ( _I |` V ) e. ( S GrpHom S ) ) -> 0 <_ ( N ` ( _I |` V ) ) )
32	5, 5, 9, 31	syl3anc 1326	. . . . . . . . . 10 |- ( ( S e. NrmGrp /\ { .0. } C. V ) -> 0 <_ ( N ` ( _I |` V ) ) )
33		xrrege0 12005	. . . . . . . . . 10 |- ( ( ( ( N ` ( _I |` V ) ) e. RR* /\ 1 e. RR ) /\ ( 0 <_ ( N ` ( _I |` V ) ) /\ ( N ` ( _I |` V ) ) <_ 1 ) ) -> ( N ` ( _I |` V ) ) e. RR )
34	30, 10, 32, 22, 33	syl22anc 1327	. . . . . . . . 9 |- ( ( S e. NrmGrp /\ { .0. } C. V ) -> ( N ` ( _I |` V ) ) e. RR )
35	1	isnghm2 22528	. . . . . . . . . 10 |- ( ( S e. NrmGrp /\ S e. NrmGrp /\ ( _I |` V ) e. ( S GrpHom S ) ) -> ( ( _I |` V ) e. ( S NGHom S ) <-> ( N ` ( _I |` V ) ) e. RR ) )
36	5, 5, 9, 35	syl3anc 1326	. . . . . . . . 9 |- ( ( S e. NrmGrp /\ { .0. } C. V ) -> ( ( _I |` V ) e. ( S NGHom S ) <-> ( N ` ( _I |` V ) ) e. RR ) )
37	34, 36	mpbird 247	. . . . . . . 8 |- ( ( S e. NrmGrp /\ { .0. } C. V ) -> ( _I |` V ) e. ( S NGHom S ) )
38	37	adantr 481	. . . . . . 7 |- ( ( ( S e. NrmGrp /\ { .0. } C. V ) /\ ( x e. V /\ x =/= .0. ) ) -> ( _I |` V ) e. ( S NGHom S ) )
39		simprl 794	. . . . . . 7 |- ( ( ( S e. NrmGrp /\ { .0. } C. V ) /\ ( x e. V /\ x =/= .0. ) ) -> x e. V )
40	1, 2, 3, 3	nmoi 22532	. . . . . . 7 |- ( ( ( _I |` V ) e. ( S NGHom S ) /\ x e. V ) -> ( ( norm ` S ) ` ( ( _I |` V ) ` x ) ) <_ ( ( N ` ( _I |` V ) ) x. ( ( norm ` S ) ` x ) ) )
41	38, 39, 40	syl2anc 693	. . . . . 6 |- ( ( ( S e. NrmGrp /\ { .0. } C. V ) /\ ( x e. V /\ x =/= .0. ) ) -> ( ( norm ` S ) ` ( ( _I |` V ) ` x ) ) <_ ( ( N ` ( _I |` V ) ) x. ( ( norm ` S ) ` x ) ) )
42	28, 41	eqbrtrd 4675	. . . . 5 |- ( ( ( S e. NrmGrp /\ { .0. } C. V ) /\ ( x e. V /\ x =/= .0. ) ) -> ( 1 x. ( ( norm ` S ) ` x ) ) <_ ( ( N ` ( _I |` V ) ) x. ( ( norm ` S ) ` x ) ) )
43		1red 10055	. . . . . 6 |- ( ( ( S e. NrmGrp /\ { .0. } C. V ) /\ ( x e. V /\ x =/= .0. ) ) -> 1 e. RR )
44	34	adantr 481	. . . . . 6 |- ( ( ( S e. NrmGrp /\ { .0. } C. V ) /\ ( x e. V /\ x =/= .0. ) ) -> ( N ` ( _I |` V ) ) e. RR )
45	2, 3, 4	nmrpcl 22424	. . . . . . . 8 |- ( ( S e. NrmGrp /\ x e. V /\ x =/= .0. ) -> ( ( norm ` S ) ` x ) e. RR+ )
46	45	3expb 1266	. . . . . . 7 |- ( ( S e. NrmGrp /\ ( x e. V /\ x =/= .0. ) ) -> ( ( norm ` S ) ` x ) e. RR+ )
47	46	adantlr 751	. . . . . 6 |- ( ( ( S e. NrmGrp /\ { .0. } C. V ) /\ ( x e. V /\ x =/= .0. ) ) -> ( ( norm ` S ) ` x ) e. RR+ )
48	43, 44, 47	lemul1d 11915	. . . . 5 |- ( ( ( S e. NrmGrp /\ { .0. } C. V ) /\ ( x e. V /\ x =/= .0. ) ) -> ( 1 <_ ( N ` ( _I |` V ) ) <-> ( 1 x. ( ( norm ` S ) ` x ) ) <_ ( ( N ` ( _I |` V ) ) x. ( ( norm ` S ) ` x ) ) ) )
49	42, 48	mpbird 247	. . . 4 |- ( ( ( S e. NrmGrp /\ { .0. } C. V ) /\ ( x e. V /\ x =/= .0. ) ) -> 1 <_ ( N ` ( _I |` V ) ) )
50	27, 49	sylanr2 685	. . 3 |- ( ( ( S e. NrmGrp /\ { .0. } C. V ) /\ ( x e. V /\ -. x e. { .0. } ) ) -> 1 <_ ( N ` ( _I |` V ) ) )
51	24, 50	exlimddv 1863	. 2 |- ( ( S e. NrmGrp /\ { .0. } C. V ) -> 1 <_ ( N ` ( _I |` V ) ) )
52		1re 10039	. . . 4 |- 1 e. RR
53	52	rexri 10097	. . 3 |- 1 e. RR*
54		xrletri3 11985	. . 3 |- ( ( ( N ` ( _I |` V ) ) e. RR* /\ 1 e. RR* ) -> ( ( N ` ( _I |` V ) ) = 1 <-> ( ( N ` ( _I |` V ) ) <_ 1 /\ 1 <_ ( N ` ( _I |` V ) ) ) ) )
55	30, 53, 54	sylancl 694	. 2 |- ( ( S e. NrmGrp /\ { .0. } C. V ) -> ( ( N ` ( _I |` V ) ) = 1 <-> ( ( N ` ( _I |` V ) ) <_ 1 /\ 1 <_ ( N ` ( _I |` V ) ) ) ) )
56	22, 51, 55	mpbir2and 957	1 |- ( ( S e. NrmGrp /\ { .0. } C. V ) -> ( N ` ( _I |` V ) ) = 1 )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   C. wpss 3575   {csn 4177   class class class wbr 4653   _I cid 5023   |` cres 5116   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   RR*cxr 10073   <_ cle 10075   RR+crp 11832   Basecbs 15857   0gc0g 16100   Grpcgrp 17422   GrpHom cghm 17657   normcnm 22381  NrmGrpcngp 22382   normOpcnmo 22509   NGHom cnghm 22510

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:  idnghm  22547