Theorem nmofval 22518

Description: Value of the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.) (Revised by AV, 26-Sep-2020.)

Hypotheses

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

Assertion

Ref	Expression
nmofval	|- ( ( S e. NrmGrp /\ T e. NrmGrp ) -> N = ( f e. ( S GrpHom T ) |-> inf ( { r e. ( 0 [,) +oo ) | A. x e. V ( M ` ( f ` x ) ) <_ ( r x. ( L ` x ) ) } , RR* , < ) ) )

Distinct variable groups:   f, r, x, L   f, M, r, x   S, f, r, x   T, f, r, x   f, V, r, x   N, r, x

Allowed substitution hint:   N( f)

Proof of Theorem nmofval

Dummy variables s t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nmofval.1	. 2 |- N = ( S normOp T )
2		oveq12 6659	. . . 4 |- ( ( s = S /\ t = T ) -> ( s GrpHom t ) = ( S GrpHom T ) )
3		simpl 473	. . . . . . . . 9 |- ( ( s = S /\ t = T ) -> s = S )
4	3	fveq2d 6195	. . . . . . . 8 |- ( ( s = S /\ t = T ) -> ( Base ` s ) = ( Base ` S ) )
5		nmofval.2	. . . . . . . 8 |- V = ( Base ` S )
6	4, 5	syl6eqr 2674	. . . . . . 7 |- ( ( s = S /\ t = T ) -> ( Base ` s ) = V )
7		simpr 477	. . . . . . . . . . 11 |- ( ( s = S /\ t = T ) -> t = T )
8	7	fveq2d 6195	. . . . . . . . . 10 |- ( ( s = S /\ t = T ) -> ( norm ` t ) = ( norm ` T ) )
9		nmofval.4	. . . . . . . . . 10 |- M = ( norm ` T )
10	8, 9	syl6eqr 2674	. . . . . . . . 9 |- ( ( s = S /\ t = T ) -> ( norm ` t ) = M )
11	10	fveq1d 6193	. . . . . . . 8 |- ( ( s = S /\ t = T ) -> ( ( norm ` t ) ` ( f ` x ) ) = ( M ` ( f ` x ) ) )
12	3	fveq2d 6195	. . . . . . . . . . 11 |- ( ( s = S /\ t = T ) -> ( norm ` s ) = ( norm ` S ) )
13		nmofval.3	. . . . . . . . . . 11 |- L = ( norm ` S )
14	12, 13	syl6eqr 2674	. . . . . . . . . 10 |- ( ( s = S /\ t = T ) -> ( norm ` s ) = L )
15	14	fveq1d 6193	. . . . . . . . 9 |- ( ( s = S /\ t = T ) -> ( ( norm ` s ) ` x ) = ( L ` x ) )
16	15	oveq2d 6666	. . . . . . . 8 |- ( ( s = S /\ t = T ) -> ( r x. ( ( norm ` s ) ` x ) ) = ( r x. ( L ` x ) ) )
17	11, 16	breq12d 4666	. . . . . . 7 |- ( ( s = S /\ t = T ) -> ( ( ( norm ` t ) ` ( f ` x ) ) <_ ( r x. ( ( norm ` s ) ` x ) ) <-> ( M ` ( f ` x ) ) <_ ( r x. ( L ` x ) ) ) )
18	6, 17	raleqbidv 3152	. . . . . 6 |- ( ( s = S /\ t = T ) -> ( A. x e. ( Base ` s ) ( ( norm ` t ) ` ( f ` x ) ) <_ ( r x. ( ( norm ` s ) ` x ) ) <-> A. x e. V ( M ` ( f ` x ) ) <_ ( r x. ( L ` x ) ) ) )
19	18	rabbidv 3189	. . . . 5 |- ( ( s = S /\ t = T ) -> { r e. ( 0 [,) +oo ) | A. x e. ( Base ` s ) ( ( norm ` t ) ` ( f ` x ) ) <_ ( r x. ( ( norm ` s ) ` x ) ) } = { r e. ( 0 [,) +oo ) | A. x e. V ( M ` ( f ` x ) ) <_ ( r x. ( L ` x ) ) } )
20	19	infeq1d 8383	. . . 4 |- ( ( s = S /\ t = T ) -> inf ( { r e. ( 0 [,) +oo ) | A. x e. ( Base ` s ) ( ( norm ` t ) ` ( f ` x ) ) <_ ( r x. ( ( norm ` s ) ` x ) ) } , RR* , < ) = inf ( { r e. ( 0 [,) +oo ) | A. x e. V ( M ` ( f ` x ) ) <_ ( r x. ( L ` x ) ) } , RR* , < ) )
21	2, 20	mpteq12dv 4733	. . 3 |- ( ( s = S /\ t = T ) -> ( f e. ( s GrpHom t ) |-> inf ( { r e. ( 0 [,) +oo ) | A. x e. ( Base ` s ) ( ( norm ` t ) ` ( f ` x ) ) <_ ( r x. ( ( norm ` s ) ` x ) ) } , RR* , < ) ) = ( f e. ( S GrpHom T ) |-> inf ( { r e. ( 0 [,) +oo ) | A. x e. V ( M ` ( f ` x ) ) <_ ( r x. ( L ` x ) ) } , RR* , < ) ) )
22		df-nmo 22512	. . 3 |- normOp = ( s e. NrmGrp , t e. NrmGrp |-> ( f e. ( s GrpHom t ) |-> inf ( { r e. ( 0 [,) +oo ) | A. x e. ( Base ` s ) ( ( norm ` t ) ` ( f ` x ) ) <_ ( r x. ( ( norm ` s ) ` x ) ) } , RR* , < ) ) )
23		eqid 2622	. . . . 5 |- ( f e. ( S GrpHom T ) |-> inf ( { r e. ( 0 [,) +oo ) | A. x e. V ( M ` ( f ` x ) ) <_ ( r x. ( L ` x ) ) } , RR* , < ) ) = ( f e. ( S GrpHom T ) |-> inf ( { r e. ( 0 [,) +oo ) | A. x e. V ( M ` ( f ` x ) ) <_ ( r x. ( L ` x ) ) } , RR* , < ) )
24		ssrab2 3687	. . . . . . 7 |- { r e. ( 0 [,) +oo ) | A. x e. V ( M ` ( f ` x ) ) <_ ( r x. ( L ` x ) ) } C_ ( 0 [,) +oo )
25		icossxr 12258	. . . . . . 7 |- ( 0 [,) +oo ) C_ RR*
26	24, 25	sstri 3612	. . . . . 6 |- { r e. ( 0 [,) +oo ) | A. x e. V ( M ` ( f ` x ) ) <_ ( r x. ( L ` x ) ) } C_ RR*
27		infxrcl 12163	. . . . . 6 |- ( { r e. ( 0 [,) +oo ) | A. x e. V ( M ` ( f ` x ) ) <_ ( r x. ( L ` x ) ) } C_ RR* -> inf ( { r e. ( 0 [,) +oo ) | A. x e. V ( M ` ( f ` x ) ) <_ ( r x. ( L ` x ) ) } , RR* , < ) e. RR* )
28	26, 27	mp1i 13	. . . . 5 |- ( f e. ( S GrpHom T ) -> inf ( { r e. ( 0 [,) +oo ) | A. x e. V ( M ` ( f ` x ) ) <_ ( r x. ( L ` x ) ) } , RR* , < ) e. RR* )
29	23, 28	fmpti 6383	. . . 4 |- ( f e. ( S GrpHom T ) |-> inf ( { r e. ( 0 [,) +oo ) | A. x e. V ( M ` ( f ` x ) ) <_ ( r x. ( L ` x ) ) } , RR* , < ) ) : ( S GrpHom T ) --> RR*
30		ovex 6678	. . . 4 |- ( S GrpHom T ) e. _V
31		xrex 11829	. . . 4 |- RR* e. _V
32		fex2 7121	. . . 4 |- ( ( ( f e. ( S GrpHom T ) |-> inf ( { r e. ( 0 [,) +oo ) | A. x e. V ( M ` ( f ` x ) ) <_ ( r x. ( L ` x ) ) } , RR* , < ) ) : ( S GrpHom T ) --> RR* /\ ( S GrpHom T ) e. _V /\ RR* e. _V ) -> ( f e. ( S GrpHom T ) |-> inf ( { r e. ( 0 [,) +oo ) | A. x e. V ( M ` ( f ` x ) ) <_ ( r x. ( L ` x ) ) } , RR* , < ) ) e. _V )
33	29, 30, 31, 32	mp3an 1424	. . 3 |- ( f e. ( S GrpHom T ) |-> inf ( { r e. ( 0 [,) +oo ) | A. x e. V ( M ` ( f ` x ) ) <_ ( r x. ( L ` x ) ) } , RR* , < ) ) e. _V
34	21, 22, 33	ovmpt2a 6791	. 2 |- ( ( S e. NrmGrp /\ T e. NrmGrp ) -> ( S normOp T ) = ( f e. ( S GrpHom T ) |-> inf ( { r e. ( 0 [,) +oo ) | A. x e. V ( M ` ( f ` x ) ) <_ ( r x. ( L ` x ) ) } , RR* , < ) ) )
35	1, 34	syl5eq 2668	1 |- ( ( S e. NrmGrp /\ T e. NrmGrp ) -> N = ( f e. ( S GrpHom T ) |-> inf ( { r e. ( 0 [,) +oo ) | A. x e. V ( M ` ( f ` x ) ) <_ ( r x. ( L ` x ) ) } , RR* , < ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650  infcinf 8347   0cc0 9936   x. cmul 9941   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   [,)cico 12177   Basecbs 15857   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-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-iun 4522  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-1st 7168  df-2nd 7169  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-ico 12181  df-nmo 22512

This theorem is referenced by:  nmoval  22519  nmof  22523