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Theorem nmofval 24094
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 = (Baseβ€˜π‘†)
nmofval.3 𝐿 = (normβ€˜π‘†)
nmofval.4 𝑀 = (normβ€˜π‘‡)
Assertion
Ref Expression
nmofval ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) β†’ 𝑁 = (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({π‘Ÿ ∈ (0[,)+∞) ∣ βˆ€π‘₯ ∈ 𝑉 (π‘€β€˜(π‘“β€˜π‘₯)) ≀ (π‘Ÿ Β· (πΏβ€˜π‘₯))}, ℝ*, < )))
Distinct variable groups:   𝑓,π‘Ÿ,π‘₯,𝐿   𝑓,𝑀,π‘Ÿ,π‘₯   𝑆,𝑓,π‘Ÿ,π‘₯   𝑇,𝑓,π‘Ÿ,π‘₯   𝑓,𝑉,π‘Ÿ,π‘₯   𝑁,π‘Ÿ,π‘₯
Allowed substitution hint:   𝑁(𝑓)

Proof of Theorem nmofval
Dummy variables 𝑠 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nmofval.1 . 2 𝑁 = (𝑆 normOp 𝑇)
2 oveq12 7371 . . . 4 ((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) β†’ (𝑠 GrpHom 𝑑) = (𝑆 GrpHom 𝑇))
3 simpl 484 . . . . . . . . 9 ((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) β†’ 𝑠 = 𝑆)
43fveq2d 6851 . . . . . . . 8 ((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) β†’ (Baseβ€˜π‘ ) = (Baseβ€˜π‘†))
5 nmofval.2 . . . . . . . 8 𝑉 = (Baseβ€˜π‘†)
64, 5eqtr4di 2795 . . . . . . 7 ((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) β†’ (Baseβ€˜π‘ ) = 𝑉)
7 simpr 486 . . . . . . . . . . 11 ((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) β†’ 𝑑 = 𝑇)
87fveq2d 6851 . . . . . . . . . 10 ((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) β†’ (normβ€˜π‘‘) = (normβ€˜π‘‡))
9 nmofval.4 . . . . . . . . . 10 𝑀 = (normβ€˜π‘‡)
108, 9eqtr4di 2795 . . . . . . . . 9 ((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) β†’ (normβ€˜π‘‘) = 𝑀)
1110fveq1d 6849 . . . . . . . 8 ((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) β†’ ((normβ€˜π‘‘)β€˜(π‘“β€˜π‘₯)) = (π‘€β€˜(π‘“β€˜π‘₯)))
123fveq2d 6851 . . . . . . . . . . 11 ((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) β†’ (normβ€˜π‘ ) = (normβ€˜π‘†))
13 nmofval.3 . . . . . . . . . . 11 𝐿 = (normβ€˜π‘†)
1412, 13eqtr4di 2795 . . . . . . . . . 10 ((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) β†’ (normβ€˜π‘ ) = 𝐿)
1514fveq1d 6849 . . . . . . . . 9 ((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) β†’ ((normβ€˜π‘ )β€˜π‘₯) = (πΏβ€˜π‘₯))
1615oveq2d 7378 . . . . . . . 8 ((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) β†’ (π‘Ÿ Β· ((normβ€˜π‘ )β€˜π‘₯)) = (π‘Ÿ Β· (πΏβ€˜π‘₯)))
1711, 16breq12d 5123 . . . . . . 7 ((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) β†’ (((normβ€˜π‘‘)β€˜(π‘“β€˜π‘₯)) ≀ (π‘Ÿ Β· ((normβ€˜π‘ )β€˜π‘₯)) ↔ (π‘€β€˜(π‘“β€˜π‘₯)) ≀ (π‘Ÿ Β· (πΏβ€˜π‘₯))))
186, 17raleqbidv 3322 . . . . . 6 ((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) β†’ (βˆ€π‘₯ ∈ (Baseβ€˜π‘ )((normβ€˜π‘‘)β€˜(π‘“β€˜π‘₯)) ≀ (π‘Ÿ Β· ((normβ€˜π‘ )β€˜π‘₯)) ↔ βˆ€π‘₯ ∈ 𝑉 (π‘€β€˜(π‘“β€˜π‘₯)) ≀ (π‘Ÿ Β· (πΏβ€˜π‘₯))))
1918rabbidv 3418 . . . . 5 ((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) β†’ {π‘Ÿ ∈ (0[,)+∞) ∣ βˆ€π‘₯ ∈ (Baseβ€˜π‘ )((normβ€˜π‘‘)β€˜(π‘“β€˜π‘₯)) ≀ (π‘Ÿ Β· ((normβ€˜π‘ )β€˜π‘₯))} = {π‘Ÿ ∈ (0[,)+∞) ∣ βˆ€π‘₯ ∈ 𝑉 (π‘€β€˜(π‘“β€˜π‘₯)) ≀ (π‘Ÿ Β· (πΏβ€˜π‘₯))})
2019infeq1d 9420 . . . 4 ((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) β†’ inf({π‘Ÿ ∈ (0[,)+∞) ∣ βˆ€π‘₯ ∈ (Baseβ€˜π‘ )((normβ€˜π‘‘)β€˜(π‘“β€˜π‘₯)) ≀ (π‘Ÿ Β· ((normβ€˜π‘ )β€˜π‘₯))}, ℝ*, < ) = inf({π‘Ÿ ∈ (0[,)+∞) ∣ βˆ€π‘₯ ∈ 𝑉 (π‘€β€˜(π‘“β€˜π‘₯)) ≀ (π‘Ÿ Β· (πΏβ€˜π‘₯))}, ℝ*, < ))
212, 20mpteq12dv 5201 . . 3 ((𝑠 = 𝑆 ∧ 𝑑 = 𝑇) β†’ (𝑓 ∈ (𝑠 GrpHom 𝑑) ↦ inf({π‘Ÿ ∈ (0[,)+∞) ∣ βˆ€π‘₯ ∈ (Baseβ€˜π‘ )((normβ€˜π‘‘)β€˜(π‘“β€˜π‘₯)) ≀ (π‘Ÿ Β· ((normβ€˜π‘ )β€˜π‘₯))}, ℝ*, < )) = (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({π‘Ÿ ∈ (0[,)+∞) ∣ βˆ€π‘₯ ∈ 𝑉 (π‘€β€˜(π‘“β€˜π‘₯)) ≀ (π‘Ÿ Β· (πΏβ€˜π‘₯))}, ℝ*, < )))
22 df-nmo 24088 . . 3 normOp = (𝑠 ∈ NrmGrp, 𝑑 ∈ NrmGrp ↦ (𝑓 ∈ (𝑠 GrpHom 𝑑) ↦ inf({π‘Ÿ ∈ (0[,)+∞) ∣ βˆ€π‘₯ ∈ (Baseβ€˜π‘ )((normβ€˜π‘‘)β€˜(π‘“β€˜π‘₯)) ≀ (π‘Ÿ Β· ((normβ€˜π‘ )β€˜π‘₯))}, ℝ*, < )))
23 eqid 2737 . . . . 5 (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({π‘Ÿ ∈ (0[,)+∞) ∣ βˆ€π‘₯ ∈ 𝑉 (π‘€β€˜(π‘“β€˜π‘₯)) ≀ (π‘Ÿ Β· (πΏβ€˜π‘₯))}, ℝ*, < )) = (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({π‘Ÿ ∈ (0[,)+∞) ∣ βˆ€π‘₯ ∈ 𝑉 (π‘€β€˜(π‘“β€˜π‘₯)) ≀ (π‘Ÿ Β· (πΏβ€˜π‘₯))}, ℝ*, < ))
24 ssrab2 4042 . . . . . . 7 {π‘Ÿ ∈ (0[,)+∞) ∣ βˆ€π‘₯ ∈ 𝑉 (π‘€β€˜(π‘“β€˜π‘₯)) ≀ (π‘Ÿ Β· (πΏβ€˜π‘₯))} βŠ† (0[,)+∞)
25 icossxr 13356 . . . . . . 7 (0[,)+∞) βŠ† ℝ*
2624, 25sstri 3958 . . . . . 6 {π‘Ÿ ∈ (0[,)+∞) ∣ βˆ€π‘₯ ∈ 𝑉 (π‘€β€˜(π‘“β€˜π‘₯)) ≀ (π‘Ÿ Β· (πΏβ€˜π‘₯))} βŠ† ℝ*
27 infxrcl 13259 . . . . . 6 ({π‘Ÿ ∈ (0[,)+∞) ∣ βˆ€π‘₯ ∈ 𝑉 (π‘€β€˜(π‘“β€˜π‘₯)) ≀ (π‘Ÿ Β· (πΏβ€˜π‘₯))} βŠ† ℝ* β†’ inf({π‘Ÿ ∈ (0[,)+∞) ∣ βˆ€π‘₯ ∈ 𝑉 (π‘€β€˜(π‘“β€˜π‘₯)) ≀ (π‘Ÿ Β· (πΏβ€˜π‘₯))}, ℝ*, < ) ∈ ℝ*)
2826, 27mp1i 13 . . . . 5 (𝑓 ∈ (𝑆 GrpHom 𝑇) β†’ inf({π‘Ÿ ∈ (0[,)+∞) ∣ βˆ€π‘₯ ∈ 𝑉 (π‘€β€˜(π‘“β€˜π‘₯)) ≀ (π‘Ÿ Β· (πΏβ€˜π‘₯))}, ℝ*, < ) ∈ ℝ*)
2923, 28fmpti 7065 . . . 4 (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({π‘Ÿ ∈ (0[,)+∞) ∣ βˆ€π‘₯ ∈ 𝑉 (π‘€β€˜(π‘“β€˜π‘₯)) ≀ (π‘Ÿ Β· (πΏβ€˜π‘₯))}, ℝ*, < )):(𝑆 GrpHom 𝑇)βŸΆβ„*
30 ovex 7395 . . . 4 (𝑆 GrpHom 𝑇) ∈ V
31 xrex 12919 . . . 4 ℝ* ∈ V
32 fex2 7875 . . . 4 (((𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({π‘Ÿ ∈ (0[,)+∞) ∣ βˆ€π‘₯ ∈ 𝑉 (π‘€β€˜(π‘“β€˜π‘₯)) ≀ (π‘Ÿ Β· (πΏβ€˜π‘₯))}, ℝ*, < )):(𝑆 GrpHom 𝑇)βŸΆβ„* ∧ (𝑆 GrpHom 𝑇) ∈ V ∧ ℝ* ∈ V) β†’ (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({π‘Ÿ ∈ (0[,)+∞) ∣ βˆ€π‘₯ ∈ 𝑉 (π‘€β€˜(π‘“β€˜π‘₯)) ≀ (π‘Ÿ Β· (πΏβ€˜π‘₯))}, ℝ*, < )) ∈ V)
3329, 30, 31, 32mp3an 1462 . . 3 (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({π‘Ÿ ∈ (0[,)+∞) ∣ βˆ€π‘₯ ∈ 𝑉 (π‘€β€˜(π‘“β€˜π‘₯)) ≀ (π‘Ÿ Β· (πΏβ€˜π‘₯))}, ℝ*, < )) ∈ V
3421, 22, 33ovmpoa 7515 . 2 ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) β†’ (𝑆 normOp 𝑇) = (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({π‘Ÿ ∈ (0[,)+∞) ∣ βˆ€π‘₯ ∈ 𝑉 (π‘€β€˜(π‘“β€˜π‘₯)) ≀ (π‘Ÿ Β· (πΏβ€˜π‘₯))}, ℝ*, < )))
351, 34eqtrid 2789 1 ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) β†’ 𝑁 = (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({π‘Ÿ ∈ (0[,)+∞) ∣ βˆ€π‘₯ ∈ 𝑉 (π‘€β€˜(π‘“β€˜π‘₯)) ≀ (π‘Ÿ Β· (πΏβ€˜π‘₯))}, ℝ*, < )))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  {crab 3410  Vcvv 3448   βŠ† wss 3915   class class class wbr 5110   ↦ cmpt 5193  βŸΆwf 6497  β€˜cfv 6501  (class class class)co 7362  infcinf 9384  0cc0 11058   Β· cmul 11063  +∞cpnf 11193  β„*cxr 11195   < clt 11196   ≀ cle 11197  [,)cico 13273  Basecbs 17090   GrpHom cghm 19012  normcnm 23948  NrmGrpcngp 23949   normOp cnmo 24085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135  ax-pre-sup 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-po 5550  df-so 5551  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7926  df-2nd 7927  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-sup 9385  df-inf 9386  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-ico 13277  df-nmo 24088
This theorem is referenced by:  nmoval  24095  nmof  24099
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