MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nmofval Structured version   Visualization version   GIF version

Theorem nmofval 22887
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 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 𝑁 = (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))}, ℝ*, < )))
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.
StepHypRef Expression
1 nmofval.1 . 2 𝑁 = (𝑆 normOp 𝑇)
2 oveq12 6913 . . . 4 ((𝑠 = 𝑆𝑡 = 𝑇) → (𝑠 GrpHom 𝑡) = (𝑆 GrpHom 𝑇))
3 simpl 476 . . . . . . . . 9 ((𝑠 = 𝑆𝑡 = 𝑇) → 𝑠 = 𝑆)
43fveq2d 6436 . . . . . . . 8 ((𝑠 = 𝑆𝑡 = 𝑇) → (Base‘𝑠) = (Base‘𝑆))
5 nmofval.2 . . . . . . . 8 𝑉 = (Base‘𝑆)
64, 5syl6eqr 2878 . . . . . . 7 ((𝑠 = 𝑆𝑡 = 𝑇) → (Base‘𝑠) = 𝑉)
7 simpr 479 . . . . . . . . . . 11 ((𝑠 = 𝑆𝑡 = 𝑇) → 𝑡 = 𝑇)
87fveq2d 6436 . . . . . . . . . 10 ((𝑠 = 𝑆𝑡 = 𝑇) → (norm‘𝑡) = (norm‘𝑇))
9 nmofval.4 . . . . . . . . . 10 𝑀 = (norm‘𝑇)
108, 9syl6eqr 2878 . . . . . . . . 9 ((𝑠 = 𝑆𝑡 = 𝑇) → (norm‘𝑡) = 𝑀)
1110fveq1d 6434 . . . . . . . 8 ((𝑠 = 𝑆𝑡 = 𝑇) → ((norm‘𝑡)‘(𝑓𝑥)) = (𝑀‘(𝑓𝑥)))
123fveq2d 6436 . . . . . . . . . . 11 ((𝑠 = 𝑆𝑡 = 𝑇) → (norm‘𝑠) = (norm‘𝑆))
13 nmofval.3 . . . . . . . . . . 11 𝐿 = (norm‘𝑆)
1412, 13syl6eqr 2878 . . . . . . . . . 10 ((𝑠 = 𝑆𝑡 = 𝑇) → (norm‘𝑠) = 𝐿)
1514fveq1d 6434 . . . . . . . . 9 ((𝑠 = 𝑆𝑡 = 𝑇) → ((norm‘𝑠)‘𝑥) = (𝐿𝑥))
1615oveq2d 6920 . . . . . . . 8 ((𝑠 = 𝑆𝑡 = 𝑇) → (𝑟 · ((norm‘𝑠)‘𝑥)) = (𝑟 · (𝐿𝑥)))
1711, 16breq12d 4885 . . . . . . 7 ((𝑠 = 𝑆𝑡 = 𝑇) → (((norm‘𝑡)‘(𝑓𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥)) ↔ (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))))
186, 17raleqbidv 3363 . . . . . 6 ((𝑠 = 𝑆𝑡 = 𝑇) → (∀𝑥 ∈ (Base‘𝑠)((norm‘𝑡)‘(𝑓𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥)) ↔ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))))
1918rabbidv 3401 . . . . 5 ((𝑠 = 𝑆𝑡 = 𝑇) → {𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ (Base‘𝑠)((norm‘𝑡)‘(𝑓𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥))} = {𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))})
2019infeq1d 8651 . . . 4 ((𝑠 = 𝑆𝑡 = 𝑇) → inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ (Base‘𝑠)((norm‘𝑡)‘(𝑓𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥))}, ℝ*, < ) = inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))}, ℝ*, < ))
212, 20mpteq12dv 4955 . . 3 ((𝑠 = 𝑆𝑡 = 𝑇) → (𝑓 ∈ (𝑠 GrpHom 𝑡) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ (Base‘𝑠)((norm‘𝑡)‘(𝑓𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥))}, ℝ*, < )) = (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))}, ℝ*, < )))
22 df-nmo 22881 . . 3 normOp = (𝑠 ∈ NrmGrp, 𝑡 ∈ NrmGrp ↦ (𝑓 ∈ (𝑠 GrpHom 𝑡) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ (Base‘𝑠)((norm‘𝑡)‘(𝑓𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥))}, ℝ*, < )))
23 eqid 2824 . . . . 5 (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))}, ℝ*, < )) = (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))}, ℝ*, < ))
24 ssrab2 3911 . . . . . . 7 {𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))} ⊆ (0[,)+∞)
25 icossxr 12545 . . . . . . 7 (0[,)+∞) ⊆ ℝ*
2624, 25sstri 3835 . . . . . 6 {𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))} ⊆ ℝ*
27 infxrcl 12450 . . . . . 6 ({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))} ⊆ ℝ* → inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))}, ℝ*, < ) ∈ ℝ*)
2826, 27mp1i 13 . . . . 5 (𝑓 ∈ (𝑆 GrpHom 𝑇) → inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))}, ℝ*, < ) ∈ ℝ*)
2923, 28fmpti 6630 . . . 4 (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))}, ℝ*, < )):(𝑆 GrpHom 𝑇)⟶ℝ*
30 ovex 6936 . . . 4 (𝑆 GrpHom 𝑇) ∈ V
31 xrex 12108 . . . 4 * ∈ V
32 fex2 7382 . . . 4 (((𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))}, ℝ*, < )):(𝑆 GrpHom 𝑇)⟶ℝ* ∧ (𝑆 GrpHom 𝑇) ∈ V ∧ ℝ* ∈ V) → (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))}, ℝ*, < )) ∈ V)
3329, 30, 31, 32mp3an 1591 . . 3 (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))}, ℝ*, < )) ∈ V
3421, 22, 33ovmpt2a 7050 . 2 ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 normOp 𝑇) = (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))}, ℝ*, < )))
351, 34syl5eq 2872 1 ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 𝑁 = (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))}, ℝ*, < )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1658  wcel 2166  wral 3116  {crab 3120  Vcvv 3413  wss 3797   class class class wbr 4872  cmpt 4951  wf 6118  cfv 6122  (class class class)co 6904  infcinf 8615  0cc0 10251   · cmul 10256  +∞cpnf 10387  *cxr 10389   < clt 10390  cle 10391  [,)cico 12464  Basecbs 16221   GrpHom cghm 18007  normcnm 22750  NrmGrpcngp 22751   normOp cnmo 22878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126  ax-un 7208  ax-cnex 10307  ax-resscn 10308  ax-1cn 10309  ax-icn 10310  ax-addcl 10311  ax-addrcl 10312  ax-mulcl 10313  ax-mulrcl 10314  ax-mulcom 10315  ax-addass 10316  ax-mulass 10317  ax-distr 10318  ax-i2m1 10319  ax-1ne0 10320  ax-1rid 10321  ax-rnegex 10322  ax-rrecex 10323  ax-cnre 10324  ax-pre-lttri 10325  ax-pre-lttrn 10326  ax-pre-ltadd 10327  ax-pre-mulgt0 10328  ax-pre-sup 10329
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-nel 3102  df-ral 3121  df-rex 3122  df-reu 3123  df-rmo 3124  df-rab 3125  df-v 3415  df-sbc 3662  df-csb 3757  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-pw 4379  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-iun 4741  df-br 4873  df-opab 4935  df-mpt 4952  df-id 5249  df-po 5262  df-so 5263  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-f1 6127  df-fo 6128  df-f1o 6129  df-fv 6130  df-riota 6865  df-ov 6907  df-oprab 6908  df-mpt2 6909  df-1st 7427  df-2nd 7428  df-er 8008  df-en 8222  df-dom 8223  df-sdom 8224  df-sup 8616  df-inf 8617  df-pnf 10392  df-mnf 10393  df-xr 10394  df-ltxr 10395  df-le 10396  df-sub 10586  df-neg 10587  df-ico 12468  df-nmo 22881
This theorem is referenced by:  nmoval  22888  nmof  22892
  Copyright terms: Public domain W3C validator