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Theorem nmoofval 22255
 Description: The operator norm function. (Contributed by NM, 6-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
nmoofval.1
nmoofval.2
nmoofval.3 CV
nmoofval.4 CV
nmoofval.6
Assertion
Ref Expression
nmoofval
Distinct variable groups:   ,,,   ,,,   ,,   ,,   ,   ,
Allowed substitution hints:   (,)   (,)   (,,)   ()   ()

Proof of Theorem nmoofval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nmoofval.6 . 2
2 fveq2 5720 . . . . . 6
3 nmoofval.1 . . . . . 6
42, 3syl6eqr 2485 . . . . 5
54oveq2d 6089 . . . 4
6 fveq2 5720 . . . . . . . . . . 11 CV CV
7 nmoofval.3 . . . . . . . . . . 11 CV
86, 7syl6eqr 2485 . . . . . . . . . 10 CV
98fveq1d 5722 . . . . . . . . 9 CV
109breq1d 4214 . . . . . . . 8 CV
1110anbi1d 686 . . . . . . 7 CV CV CV
124, 11rexeqbidv 2909 . . . . . 6 CV CV CV
1312abbidv 2549 . . . . 5 CV CV CV
1413supeq1d 7443 . . . 4 CV CV CV
155, 14mpteq12dv 4279 . . 3 CV CV CV
16 fveq2 5720 . . . . . 6
17 nmoofval.2 . . . . . 6
1816, 17syl6eqr 2485 . . . . 5
1918oveq1d 6088 . . . 4
20 fveq2 5720 . . . . . . . . . . 11 CV CV
21 nmoofval.4 . . . . . . . . . . 11 CV
2220, 21syl6eqr 2485 . . . . . . . . . 10 CV
2322fveq1d 5722 . . . . . . . . 9 CV
2423eqeq2d 2446 . . . . . . . 8 CV
2524anbi2d 685 . . . . . . 7 CV
2625rexbidv 2718 . . . . . 6 CV
2726abbidv 2549 . . . . 5 CV
2827supeq1d 7443 . . . 4 CV
2919, 28mpteq12dv 4279 . . 3 CV
30 df-nmoo 22238 . . 3 CV CV
31 ovex 6098 . . . 4
3231mptex 5958 . . 3
3315, 29, 30, 32ovmpt2 6201 . 2
341, 33syl5eq 2479 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725  cab 2421  wrex 2698   class class class wbr 4204   cmpt 4258  cfv 5446  (class class class)co 6073   cmap 7010  csup 7437  c1 8983  cxr 9111   clt 9112   cle 9113  cnv 22055  cba 22057  CVcnmcv 22061  cnmoo 22234 This theorem is referenced by:  nmooval  22256  hhnmoi  23396 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-sup 7438  df-nmoo 22238
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