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Theorem nmofval 23318
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 7155 . . . 4 ((𝑠 = 𝑆𝑡 = 𝑇) → (𝑠 GrpHom 𝑡) = (𝑆 GrpHom 𝑇))
3 simpl 486 . . . . . . . . 9 ((𝑠 = 𝑆𝑡 = 𝑇) → 𝑠 = 𝑆)
43fveq2d 6663 . . . . . . . 8 ((𝑠 = 𝑆𝑡 = 𝑇) → (Base‘𝑠) = (Base‘𝑆))
5 nmofval.2 . . . . . . . 8 𝑉 = (Base‘𝑆)
64, 5syl6eqr 2877 . . . . . . 7 ((𝑠 = 𝑆𝑡 = 𝑇) → (Base‘𝑠) = 𝑉)
7 simpr 488 . . . . . . . . . . 11 ((𝑠 = 𝑆𝑡 = 𝑇) → 𝑡 = 𝑇)
87fveq2d 6663 . . . . . . . . . 10 ((𝑠 = 𝑆𝑡 = 𝑇) → (norm‘𝑡) = (norm‘𝑇))
9 nmofval.4 . . . . . . . . . 10 𝑀 = (norm‘𝑇)
108, 9syl6eqr 2877 . . . . . . . . 9 ((𝑠 = 𝑆𝑡 = 𝑇) → (norm‘𝑡) = 𝑀)
1110fveq1d 6661 . . . . . . . 8 ((𝑠 = 𝑆𝑡 = 𝑇) → ((norm‘𝑡)‘(𝑓𝑥)) = (𝑀‘(𝑓𝑥)))
123fveq2d 6663 . . . . . . . . . . 11 ((𝑠 = 𝑆𝑡 = 𝑇) → (norm‘𝑠) = (norm‘𝑆))
13 nmofval.3 . . . . . . . . . . 11 𝐿 = (norm‘𝑆)
1412, 13syl6eqr 2877 . . . . . . . . . 10 ((𝑠 = 𝑆𝑡 = 𝑇) → (norm‘𝑠) = 𝐿)
1514fveq1d 6661 . . . . . . . . 9 ((𝑠 = 𝑆𝑡 = 𝑇) → ((norm‘𝑠)‘𝑥) = (𝐿𝑥))
1615oveq2d 7162 . . . . . . . 8 ((𝑠 = 𝑆𝑡 = 𝑇) → (𝑟 · ((norm‘𝑠)‘𝑥)) = (𝑟 · (𝐿𝑥)))
1711, 16breq12d 5066 . . . . . . 7 ((𝑠 = 𝑆𝑡 = 𝑇) → (((norm‘𝑡)‘(𝑓𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥)) ↔ (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))))
186, 17raleqbidv 3393 . . . . . 6 ((𝑠 = 𝑆𝑡 = 𝑇) → (∀𝑥 ∈ (Base‘𝑠)((norm‘𝑡)‘(𝑓𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥)) ↔ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))))
1918rabbidv 3466 . . . . 5 ((𝑠 = 𝑆𝑡 = 𝑇) → {𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ (Base‘𝑠)((norm‘𝑡)‘(𝑓𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥))} = {𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))})
2019infeq1d 8934 . . . 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 5138 . . 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 23312 . . 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 4042 . . . . . . 7 {𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))} ⊆ (0[,)+∞)
25 icossxr 12817 . . . . . . 7 (0[,)+∞) ⊆ ℝ*
2624, 25sstri 3962 . . . . . 6 {𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))} ⊆ ℝ*
27 infxrcl 12721 . . . . . 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 6865 . . . 4 (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))}, ℝ*, < )):(𝑆 GrpHom 𝑇)⟶ℝ*
30 ovex 7179 . . . 4 (𝑆 GrpHom 𝑇) ∈ V
31 xrex 12381 . . . 4 * ∈ V
32 fex2 7630 . . . 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 1458 . . 3 (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))}, ℝ*, < )) ∈ V
3421, 22, 33ovmpoa 7295 . 2 ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 normOp 𝑇) = (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))}, ℝ*, < )))
351, 34syl5eq 2871 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 399   = wceq 1538  wcel 2115  wral 3133  {crab 3137  Vcvv 3480  wss 3919   class class class wbr 5053  cmpt 5133  wf 6340  cfv 6344  (class class class)co 7146  infcinf 8898  0cc0 10531   · cmul 10536  +∞cpnf 10666  *cxr 10668   < clt 10669  cle 10670  [,)cico 12735  Basecbs 16481   GrpHom cghm 18353  normcnm 23181  NrmGrpcngp 23182   normOp cnmo 23309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-po 5462  df-so 5463  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-riota 7104  df-ov 7149  df-oprab 7150  df-mpo 7151  df-1st 7681  df-2nd 7682  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-sup 8899  df-inf 8900  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-ico 12739  df-nmo 23312
This theorem is referenced by:  nmoval  23319  nmof  23323
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