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Theorem nmofval 22428
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 6613 . . . 4 ((𝑠 = 𝑆𝑡 = 𝑇) → (𝑠 GrpHom 𝑡) = (𝑆 GrpHom 𝑇))
3 simpl 473 . . . . . . . . 9 ((𝑠 = 𝑆𝑡 = 𝑇) → 𝑠 = 𝑆)
43fveq2d 6152 . . . . . . . 8 ((𝑠 = 𝑆𝑡 = 𝑇) → (Base‘𝑠) = (Base‘𝑆))
5 nmofval.2 . . . . . . . 8 𝑉 = (Base‘𝑆)
64, 5syl6eqr 2673 . . . . . . 7 ((𝑠 = 𝑆𝑡 = 𝑇) → (Base‘𝑠) = 𝑉)
7 simpr 477 . . . . . . . . . . 11 ((𝑠 = 𝑆𝑡 = 𝑇) → 𝑡 = 𝑇)
87fveq2d 6152 . . . . . . . . . 10 ((𝑠 = 𝑆𝑡 = 𝑇) → (norm‘𝑡) = (norm‘𝑇))
9 nmofval.4 . . . . . . . . . 10 𝑀 = (norm‘𝑇)
108, 9syl6eqr 2673 . . . . . . . . 9 ((𝑠 = 𝑆𝑡 = 𝑇) → (norm‘𝑡) = 𝑀)
1110fveq1d 6150 . . . . . . . 8 ((𝑠 = 𝑆𝑡 = 𝑇) → ((norm‘𝑡)‘(𝑓𝑥)) = (𝑀‘(𝑓𝑥)))
123fveq2d 6152 . . . . . . . . . . 11 ((𝑠 = 𝑆𝑡 = 𝑇) → (norm‘𝑠) = (norm‘𝑆))
13 nmofval.3 . . . . . . . . . . 11 𝐿 = (norm‘𝑆)
1412, 13syl6eqr 2673 . . . . . . . . . 10 ((𝑠 = 𝑆𝑡 = 𝑇) → (norm‘𝑠) = 𝐿)
1514fveq1d 6150 . . . . . . . . 9 ((𝑠 = 𝑆𝑡 = 𝑇) → ((norm‘𝑠)‘𝑥) = (𝐿𝑥))
1615oveq2d 6620 . . . . . . . 8 ((𝑠 = 𝑆𝑡 = 𝑇) → (𝑟 · ((norm‘𝑠)‘𝑥)) = (𝑟 · (𝐿𝑥)))
1711, 16breq12d 4626 . . . . . . 7 ((𝑠 = 𝑆𝑡 = 𝑇) → (((norm‘𝑡)‘(𝑓𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥)) ↔ (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))))
186, 17raleqbidv 3141 . . . . . 6 ((𝑠 = 𝑆𝑡 = 𝑇) → (∀𝑥 ∈ (Base‘𝑠)((norm‘𝑡)‘(𝑓𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥)) ↔ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))))
1918rabbidv 3177 . . . . 5 ((𝑠 = 𝑆𝑡 = 𝑇) → {𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ (Base‘𝑠)((norm‘𝑡)‘(𝑓𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥))} = {𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))})
2019infeq1d 8327 . . . 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 4693 . . 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 22422 . . 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 2621 . . . . 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 3666 . . . . . . 7 {𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))} ⊆ (0[,)+∞)
25 icossxr 12200 . . . . . . 7 (0[,)+∞) ⊆ ℝ*
2624, 25sstri 3592 . . . . . 6 {𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))} ⊆ ℝ*
27 infxrcl 12106 . . . . . 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 6339 . . . 4 (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))}, ℝ*, < )):(𝑆 GrpHom 𝑇)⟶ℝ*
30 ovex 6632 . . . 4 (𝑆 GrpHom 𝑇) ∈ V
31 xrex 11773 . . . 4 * ∈ V
32 fex2 7068 . . . 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 1421 . . 3 (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))}, ℝ*, < )) ∈ V
3421, 22, 33ovmpt2a 6744 . 2 ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 normOp 𝑇) = (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))}, ℝ*, < )))
351, 34syl5eq 2667 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 384   = wceq 1480  wcel 1987  wral 2907  {crab 2911  Vcvv 3186  wss 3555   class class class wbr 4613  cmpt 4673  wf 5843  cfv 5847  (class class class)co 6604  infcinf 8291  0cc0 9880   · cmul 9885  +∞cpnf 10015  *cxr 10017   < clt 10018  cle 10019  [,)cico 12119  Basecbs 15781   GrpHom cghm 17578  normcnm 22291  NrmGrpcngp 22292   normOp cnmo 22419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-po 4995  df-so 4996  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-sup 8292  df-inf 8293  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-ico 12123  df-nmo 22422
This theorem is referenced by:  nmoval  22429  nmof  22433
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