Theorem nmofval 23322

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[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝑓‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))}, 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.

Step	Hyp	Ref	Expression
1		nmofval.1	. 2 ⊢ 𝑁 = (𝑆 normOp 𝑇)
2		oveq12 7164	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑠 GrpHom 𝑡) = (𝑆 GrpHom 𝑇))
3		simpl 485	. . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → 𝑠 = 𝑆)
4	3	fveq2d 6673	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (Base‘𝑠) = (Base‘𝑆))
5		nmofval.2	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑆)
6	4, 5	syl6eqr 2874	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (Base‘𝑠) = 𝑉)
7		simpr 487	. . . . . . . . . . 11 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → 𝑡 = 𝑇)
8	7	fveq2d 6673	. . . . . . . . . 10 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (norm‘𝑡) = (norm‘𝑇))
9		nmofval.4	. . . . . . . . . 10 ⊢ 𝑀 = (norm‘𝑇)
10	8, 9	syl6eqr 2874	. . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (norm‘𝑡) = 𝑀)
11	10	fveq1d 6671	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → ((norm‘𝑡)‘(𝑓‘𝑥)) = (𝑀‘(𝑓‘𝑥)))
12	3	fveq2d 6673	. . . . . . . . . . 11 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (norm‘𝑠) = (norm‘𝑆))
13		nmofval.3	. . . . . . . . . . 11 ⊢ 𝐿 = (norm‘𝑆)
14	12, 13	syl6eqr 2874	. . . . . . . . . 10 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (norm‘𝑠) = 𝐿)
15	14	fveq1d 6671	. . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → ((norm‘𝑠)‘𝑥) = (𝐿‘𝑥))
16	15	oveq2d 7171	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑟 · ((norm‘𝑠)‘𝑥)) = (𝑟 · (𝐿‘𝑥)))
17	11, 16	breq12d 5078	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (((norm‘𝑡)‘(𝑓‘𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥)) ↔ (𝑀‘(𝑓‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))))
18	6, 17	raleqbidv 3401	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (∀𝑥 ∈ (Base‘𝑠)((norm‘𝑡)‘(𝑓‘𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥)) ↔ ∀𝑥 ∈ 𝑉 (𝑀‘(𝑓‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))))
19	18	rabbidv 3480	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → {𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ (Base‘𝑠)((norm‘𝑡)‘(𝑓‘𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥))} = {𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝑓‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))})
20	19	infeq1d 8940	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ (Base‘𝑠)((norm‘𝑡)‘(𝑓‘𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥))}, ℝ*, < ) = inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝑓‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))}, ℝ*, < ))
21	2, 20	mpteq12dv 5150	. . 3 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑓 ∈ (𝑠 GrpHom 𝑡) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ (Base‘𝑠)((norm‘𝑡)‘(𝑓‘𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥))}, ℝ*, < )) = (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝑓‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))}, ℝ*, < )))
22		df-nmo 23316	. . 3 ⊢ normOp = (𝑠 ∈ NrmGrp, 𝑡 ∈ NrmGrp ↦ (𝑓 ∈ (𝑠 GrpHom 𝑡) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ (Base‘𝑠)((norm‘𝑡)‘(𝑓‘𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥))}, ℝ*, < )))
23		eqid 2821	. . . . 5 ⊢ (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝑓‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))}, ℝ*, < )) = (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝑓‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))}, ℝ*, < ))
24		ssrab2 4055	. . . . . . 7 ⊢ {𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝑓‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))} ⊆ (0[,)+∞)
25		icossxr 12820	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℝ*
26	24, 25	sstri 3975	. . . . . 6 ⊢ {𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝑓‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))} ⊆ ℝ*
27		infxrcl 12725	. . . . . 6 ⊢ ({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝑓‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))} ⊆ ℝ* → inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝑓‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))}, ℝ*, < ) ∈ ℝ*)
28	26, 27	mp1i 13	. . . . 5 ⊢ (𝑓 ∈ (𝑆 GrpHom 𝑇) → inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝑓‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))}, ℝ*, < ) ∈ ℝ*)
29	23, 28	fmpti 6875	. . . 4 ⊢ (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝑓‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))}, ℝ*, < )):(𝑆 GrpHom 𝑇)⟶ℝ*
30		ovex 7188	. . . 4 ⊢ (𝑆 GrpHom 𝑇) ∈ V
31		xrex 12385	. . . 4 ⊢ ℝ* ∈ V
32		fex2 7637	. . . 4 ⊢ (((𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝑓‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))}, ℝ*, < )):(𝑆 GrpHom 𝑇)⟶ℝ* ∧ (𝑆 GrpHom 𝑇) ∈ V ∧ ℝ* ∈ V) → (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝑓‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))}, ℝ*, < )) ∈ V)
33	29, 30, 31, 32	mp3an 1457	. . 3 ⊢ (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝑓‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))}, ℝ*, < )) ∈ V
34	21, 22, 33	ovmpoa 7304	. 2 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 normOp 𝑇) = (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝑓‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))}, ℝ*, < )))
35	1, 34	syl5eq 2868	1 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 𝑁 = (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝑓‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))}, ℝ*, < )))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  {crab 3142  Vcvv 3494   ⊆ wss 3935   class class class wbr 5065   ↦ cmpt 5145  ⟶wf 6350  ‘cfv 6354  (class class class)co 7155  infcinf 8904  0cc0 10536   · cmul 10541  +∞cpnf 10671  ℝ^*cxr 10673   < clt 10674   ≤ cle 10675  [,)cico 12739  Basecbs 16482   GrpHom cghm 18354  normcnm 23185  NrmGrpcngp 23186   normOp cnmo 23313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613  ax-pre-sup 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-po 5473  df-so 5474  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-1st 7688  df-2nd 7689  df-er 8288  df-en 8509  df-dom 8510  df-sdom 8511  df-sup 8905  df-inf 8906  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-ico 12743  df-nmo 23316

This theorem is referenced by:  nmoval  23323  nmof  23327