Theorem nmofval 24756

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 7457	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑠 GrpHom 𝑡) = (𝑆 GrpHom 𝑇))
3		simpl 482	. . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → 𝑠 = 𝑆)
4	3	fveq2d 6924	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (Base‘𝑠) = (Base‘𝑆))
5		nmofval.2	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑆)
6	4, 5	eqtr4di 2798	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (Base‘𝑠) = 𝑉)
7		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → 𝑡 = 𝑇)
8	7	fveq2d 6924	. . . . . . . . . 10 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (norm‘𝑡) = (norm‘𝑇))
9		nmofval.4	. . . . . . . . . 10 ⊢ 𝑀 = (norm‘𝑇)
10	8, 9	eqtr4di 2798	. . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (norm‘𝑡) = 𝑀)
11	10	fveq1d 6922	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → ((norm‘𝑡)‘(𝑓‘𝑥)) = (𝑀‘(𝑓‘𝑥)))
12	3	fveq2d 6924	. . . . . . . . . . 11 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (norm‘𝑠) = (norm‘𝑆))
13		nmofval.3	. . . . . . . . . . 11 ⊢ 𝐿 = (norm‘𝑆)
14	12, 13	eqtr4di 2798	. . . . . . . . . 10 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (norm‘𝑠) = 𝐿)
15	14	fveq1d 6922	. . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → ((norm‘𝑠)‘𝑥) = (𝐿‘𝑥))
16	15	oveq2d 7464	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑟 · ((norm‘𝑠)‘𝑥)) = (𝑟 · (𝐿‘𝑥)))
17	11, 16	breq12d 5179	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (((norm‘𝑡)‘(𝑓‘𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥)) ↔ (𝑀‘(𝑓‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))))
18	6, 17	raleqbidv 3354	. . . . . 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 3451	. . . . 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 9546	. . . 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 5257	. . 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 24750	. . 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 2740	. . . . 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 4103	. . . . . . 7 ⊢ {𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝑓‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))} ⊆ (0[,)+∞)
25		icossxr 13492	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℝ*
26	24, 25	sstri 4018	. . . . . 6 ⊢ {𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝑓‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))} ⊆ ℝ*
27		infxrcl 13395	. . . . . 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 7146	. . . 4 ⊢ (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝑓‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))}, ℝ*, < )):(𝑆 GrpHom 𝑇)⟶ℝ*
30		ovex 7481	. . . 4 ⊢ (𝑆 GrpHom 𝑇) ∈ V
31		xrex 13052	. . . 4 ⊢ ℝ* ∈ V
32		fex2 7974	. . . 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 1461	. . 3 ⊢ (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝑓‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))}, ℝ*, < )) ∈ V
34	21, 22, 33	ovmpoa 7605	. 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	eqtrid 2792	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 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  {crab 3443  Vcvv 3488   ⊆ wss 3976   class class class wbr 5166   ↦ cmpt 5249  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  infcinf 9510  0cc0 11184   · cmul 11189  +∞cpnf 11321  ℝ^*cxr 11323   < clt 11324   ≤ cle 11325  [,)cico 13409  Basecbs 17258   GrpHom cghm 19252  normcnm 24610  NrmGrpcngp 24611   normOp cnmo 24747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-po 5607  df-so 5608  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-sup 9511  df-inf 9512  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-ico 13413  df-nmo 24750

This theorem is referenced by:  nmoval  24757  nmof  24761