Theorem nmooval 27618

Description: The operator norm function. (Contributed by NM, 27-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nmoofval.1	|- X = ( BaseSet ` U )
nmoofval.2	|- Y = ( BaseSet ` W )
nmoofval.3	|- L = ( normCV ` U )
nmoofval.4	|- M = ( normCV ` W )
nmoofval.6	|- N = ( U normOpOLD W )

Assertion

Ref	Expression
nmooval	|- ( ( U e. NrmCVec /\ W e. NrmCVec /\ T : X --> Y ) -> ( N ` T ) = sup ( { x | E. z e. X ( ( L ` z ) <_ 1 /\ x = ( M ` ( T ` z ) ) ) } , RR* , < ) )

Distinct variable groups:   x, z, U   x, W, z   z, X   x, Y   x, T, z

Allowed substitution hints:   L( x, z)   M( x, z)   N( x, z)   X( x)   Y( z)

Proof of Theorem nmooval

Dummy variable t is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nmoofval.2	. . . . 5 |- Y = ( BaseSet ` W )
2		fvex 6201	. . . . 5 |- ( BaseSet ` W ) e. _V
3	1, 2	eqeltri 2697	. . . 4 |- Y e. _V
4		nmoofval.1	. . . . 5 |- X = ( BaseSet ` U )
5		fvex 6201	. . . . 5 |- ( BaseSet ` U ) e. _V
6	4, 5	eqeltri 2697	. . . 4 |- X e. _V
7	3, 6	elmap 7886	. . 3 |- ( T e. ( Y ^m X ) <-> T : X --> Y )
8		nmoofval.3	. . . . . 6 |- L = ( normCV ` U )
9		nmoofval.4	. . . . . 6 |- M = ( normCV ` W )
10		nmoofval.6	. . . . . 6 |- N = ( U normOpOLD W )
11	4, 1, 8, 9, 10	nmoofval 27617	. . . . 5 |- ( ( U e. NrmCVec /\ W e. NrmCVec ) -> N = ( t e. ( Y ^m X ) |-> sup ( { x | E. z e. X ( ( L ` z ) <_ 1 /\ x = ( M ` ( t ` z ) ) ) } , RR* , < ) ) )
12	11	fveq1d 6193	. . . 4 |- ( ( U e. NrmCVec /\ W e. NrmCVec ) -> ( N ` T ) = ( ( t e. ( Y ^m X ) |-> sup ( { x | E. z e. X ( ( L ` z ) <_ 1 /\ x = ( M ` ( t ` z ) ) ) } , RR* , < ) ) ` T ) )
13		fveq1 6190	. . . . . . . . . . 11 |- ( t = T -> ( t ` z ) = ( T ` z ) )
14	13	fveq2d 6195	. . . . . . . . . 10 |- ( t = T -> ( M ` ( t ` z ) ) = ( M ` ( T ` z ) ) )
15	14	eqeq2d 2632	. . . . . . . . 9 |- ( t = T -> ( x = ( M ` ( t ` z ) ) <-> x = ( M ` ( T ` z ) ) ) )
16	15	anbi2d 740	. . . . . . . 8 |- ( t = T -> ( ( ( L ` z ) <_ 1 /\ x = ( M ` ( t ` z ) ) ) <-> ( ( L ` z ) <_ 1 /\ x = ( M ` ( T ` z ) ) ) ) )
17	16	rexbidv 3052	. . . . . . 7 |- ( t = T -> ( E. z e. X ( ( L ` z ) <_ 1 /\ x = ( M ` ( t ` z ) ) ) <-> E. z e. X ( ( L ` z ) <_ 1 /\ x = ( M ` ( T ` z ) ) ) ) )
18	17	abbidv 2741	. . . . . 6 |- ( t = T -> { x | E. z e. X ( ( L ` z ) <_ 1 /\ x = ( M ` ( t ` z ) ) ) } = { x | E. z e. X ( ( L ` z ) <_ 1 /\ x = ( M ` ( T ` z ) ) ) } )
19	18	supeq1d 8352	. . . . 5 |- ( t = T -> sup ( { x | E. z e. X ( ( L ` z ) <_ 1 /\ x = ( M ` ( t ` z ) ) ) } , RR* , < ) = sup ( { x | E. z e. X ( ( L ` z ) <_ 1 /\ x = ( M ` ( T ` z ) ) ) } , RR* , < ) )
20		eqid 2622	. . . . 5 |- ( t e. ( Y ^m X ) |-> sup ( { x | E. z e. X ( ( L ` z ) <_ 1 /\ x = ( M ` ( t ` z ) ) ) } , RR* , < ) ) = ( t e. ( Y ^m X ) |-> sup ( { x | E. z e. X ( ( L ` z ) <_ 1 /\ x = ( M ` ( t ` z ) ) ) } , RR* , < ) )
21		xrltso 11974	. . . . . 6 |- < Or RR*
22	21	supex 8369	. . . . 5 |- sup ( { x | E. z e. X ( ( L ` z ) <_ 1 /\ x = ( M ` ( T ` z ) ) ) } , RR* , < ) e. _V
23	19, 20, 22	fvmpt 6282	. . . 4 |- ( T e. ( Y ^m X ) -> ( ( t e. ( Y ^m X ) |-> sup ( { x | E. z e. X ( ( L ` z ) <_ 1 /\ x = ( M ` ( t ` z ) ) ) } , RR* , < ) ) ` T ) = sup ( { x | E. z e. X ( ( L ` z ) <_ 1 /\ x = ( M ` ( T ` z ) ) ) } , RR* , < ) )
24	12, 23	sylan9eq 2676	. . 3 |- ( ( ( U e. NrmCVec /\ W e. NrmCVec ) /\ T e. ( Y ^m X ) ) -> ( N ` T ) = sup ( { x | E. z e. X ( ( L ` z ) <_ 1 /\ x = ( M ` ( T ` z ) ) ) } , RR* , < ) )
25	7, 24	sylan2br 493	. 2 |- ( ( ( U e. NrmCVec /\ W e. NrmCVec ) /\ T : X --> Y ) -> ( N ` T ) = sup ( { x | E. z e. X ( ( L ` z ) <_ 1 /\ x = ( M ` ( T ` z ) ) ) } , RR* , < ) )
26	25	3impa 1259	1 |- ( ( U e. NrmCVec /\ W e. NrmCVec /\ T : X --> Y ) -> ( N ` T ) = sup ( { x | E. z e. X ( ( L ` z ) <_ 1 /\ x = ( M ` ( T ` z ) ) ) } , RR* , < ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   _Vcvv 3200   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   supcsup 8346   1c1 9937   RR*cxr 10073   < clt 10074   <_ cle 10075   NrmCVeccnv 27439   BaseSetcba 27441   norm_CVcnmcv 27445   normOpOLDcnmoo 27596

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-pre-lttri 10010  ax-pre-lttrn 10011

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-ov 6653  df-oprab 6654  df-mpt2 6655  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-nmoo 27600

This theorem is referenced by:  nmoxr  27621  nmooge0  27622  nmorepnf  27623  nmoolb  27626  nmoubi  27627  nmoo0  27646  nmlno0lem  27648