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Theorem nmofval 22719
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 6822 . . . 4 ((𝑠 = 𝑆𝑡 = 𝑇) → (𝑠 GrpHom 𝑡) = (𝑆 GrpHom 𝑇))
3 simpl 474 . . . . . . . . 9 ((𝑠 = 𝑆𝑡 = 𝑇) → 𝑠 = 𝑆)
43fveq2d 6356 . . . . . . . 8 ((𝑠 = 𝑆𝑡 = 𝑇) → (Base‘𝑠) = (Base‘𝑆))
5 nmofval.2 . . . . . . . 8 𝑉 = (Base‘𝑆)
64, 5syl6eqr 2812 . . . . . . 7 ((𝑠 = 𝑆𝑡 = 𝑇) → (Base‘𝑠) = 𝑉)
7 simpr 479 . . . . . . . . . . 11 ((𝑠 = 𝑆𝑡 = 𝑇) → 𝑡 = 𝑇)
87fveq2d 6356 . . . . . . . . . 10 ((𝑠 = 𝑆𝑡 = 𝑇) → (norm‘𝑡) = (norm‘𝑇))
9 nmofval.4 . . . . . . . . . 10 𝑀 = (norm‘𝑇)
108, 9syl6eqr 2812 . . . . . . . . 9 ((𝑠 = 𝑆𝑡 = 𝑇) → (norm‘𝑡) = 𝑀)
1110fveq1d 6354 . . . . . . . 8 ((𝑠 = 𝑆𝑡 = 𝑇) → ((norm‘𝑡)‘(𝑓𝑥)) = (𝑀‘(𝑓𝑥)))
123fveq2d 6356 . . . . . . . . . . 11 ((𝑠 = 𝑆𝑡 = 𝑇) → (norm‘𝑠) = (norm‘𝑆))
13 nmofval.3 . . . . . . . . . . 11 𝐿 = (norm‘𝑆)
1412, 13syl6eqr 2812 . . . . . . . . . 10 ((𝑠 = 𝑆𝑡 = 𝑇) → (norm‘𝑠) = 𝐿)
1514fveq1d 6354 . . . . . . . . 9 ((𝑠 = 𝑆𝑡 = 𝑇) → ((norm‘𝑠)‘𝑥) = (𝐿𝑥))
1615oveq2d 6829 . . . . . . . 8 ((𝑠 = 𝑆𝑡 = 𝑇) → (𝑟 · ((norm‘𝑠)‘𝑥)) = (𝑟 · (𝐿𝑥)))
1711, 16breq12d 4817 . . . . . . 7 ((𝑠 = 𝑆𝑡 = 𝑇) → (((norm‘𝑡)‘(𝑓𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥)) ↔ (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))))
186, 17raleqbidv 3291 . . . . . 6 ((𝑠 = 𝑆𝑡 = 𝑇) → (∀𝑥 ∈ (Base‘𝑠)((norm‘𝑡)‘(𝑓𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥)) ↔ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))))
1918rabbidv 3329 . . . . 5 ((𝑠 = 𝑆𝑡 = 𝑇) → {𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ (Base‘𝑠)((norm‘𝑡)‘(𝑓𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥))} = {𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))})
2019infeq1d 8548 . . . 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 4885 . . 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 22713 . . 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 2760 . . . . 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 3828 . . . . . . 7 {𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))} ⊆ (0[,)+∞)
25 icossxr 12451 . . . . . . 7 (0[,)+∞) ⊆ ℝ*
2624, 25sstri 3753 . . . . . 6 {𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))} ⊆ ℝ*
27 infxrcl 12356 . . . . . 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 6546 . . . 4 (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))}, ℝ*, < )):(𝑆 GrpHom 𝑇)⟶ℝ*
30 ovex 6841 . . . 4 (𝑆 GrpHom 𝑇) ∈ V
31 xrex 12022 . . . 4 * ∈ V
32 fex2 7286 . . . 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 1573 . . 3 (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))}, ℝ*, < )) ∈ V
3421, 22, 33ovmpt2a 6956 . 2 ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 normOp 𝑇) = (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))}, ℝ*, < )))
351, 34syl5eq 2806 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 383   = wceq 1632  wcel 2139  wral 3050  {crab 3054  Vcvv 3340  wss 3715   class class class wbr 4804  cmpt 4881  wf 6045  cfv 6049  (class class class)co 6813  infcinf 8512  0cc0 10128   · cmul 10133  +∞cpnf 10263  *cxr 10265   < clt 10266  cle 10267  [,)cico 12370  Basecbs 16059   GrpHom cghm 17858  normcnm 22582  NrmGrpcngp 22583   normOp cnmo 22710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-pre-sup 10206
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-po 5187  df-so 5188  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-sup 8513  df-inf 8514  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-ico 12374  df-nmo 22713
This theorem is referenced by:  nmoval  22720  nmof  22724
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