Theorem nmofval 24940

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 7422	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑠 GrpHom 𝑡) = (𝑆 GrpHom 𝑇))
3		simpl 488	. . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → 𝑠 = 𝑆)
4	3	fveq2d 6882	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (Base‘𝑠) = (Base‘𝑆))
5		nmofval.2	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑆)
6	4, 5	eqtr4di 2813	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (Base‘𝑠) = 𝑉)
7		simpr 490	. . . . . . . . . . 11 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → 𝑡 = 𝑇)
8	7	fveq2d 6882	. . . . . . . . . 10 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (norm‘𝑡) = (norm‘𝑇))
9		nmofval.4	. . . . . . . . . 10 ⊢ 𝑀 = (norm‘𝑇)
10	8, 9	eqtr4di 2813	. . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (norm‘𝑡) = 𝑀)
11	10	fveq1d 6880	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → ((norm‘𝑡)‘(𝑓‘𝑥)) = (𝑀‘(𝑓‘𝑥)))
12	3	fveq2d 6882	. . . . . . . . . . 11 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (norm‘𝑠) = (norm‘𝑆))
13		nmofval.3	. . . . . . . . . . 11 ⊢ 𝐿 = (norm‘𝑆)
14	12, 13	eqtr4di 2813	. . . . . . . . . 10 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (norm‘𝑠) = 𝐿)
15	14	fveq1d 6880	. . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → ((norm‘𝑠)‘𝑥) = (𝐿‘𝑥))
16	15	oveq2d 7429	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑟 · ((norm‘𝑠)‘𝑥)) = (𝑟 · (𝐿‘𝑥)))
17	11, 16	breq12d 5116	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (((norm‘𝑡)‘(𝑓‘𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥)) ↔ (𝑀‘(𝑓‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))))
18	6, 17	raleqbidv 3334	. . . . . 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 3419	. . . . 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 9448	. . . 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 5192	. . 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 24934	. . 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 2760	. . . . 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 4028	. . . . . . 7 ⊢ {𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝑓‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))} ⊆ (0[,)+∞)
25		icossxr 13485	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℝ*
26	24, 25	sstri 3940	. . . . . 6 ⊢ {𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝑓‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))} ⊆ ℝ*
27		infxrcl 13386	. . . . . 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 14	. . . . 5 ⊢ (𝑓 ∈ (𝑆 GrpHom 𝑇) → inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝑓‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))}, ℝ*, < ) ∈ ℝ*)
29	23, 28	fmpti 7105	. . . 4 ⊢ (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝑓‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))}, ℝ*, < )):(𝑆 GrpHom 𝑇)⟶ℝ*
30		ovex 7446	. . . 4 ⊢ (𝑆 GrpHom 𝑇) ∈ V
31		xrex 13037	. . . 4 ⊢ ℝ* ∈ V
32		fex2 7933	. . . 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 1490	. . 3 ⊢ (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝑓‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))}, ℝ*, < )) ∈ V
34	21, 22, 33	ovmpoa 7568	. 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 2807	1 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 𝑁 = (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝑓‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))}, ℝ*, < )))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  {crab 3412  Vcvv 3450   ⊆ wss 3899   class class class wbr 5103   ↦ cmpt 5186  ⟶wf 6529  ‘cfv 6533  (class class class)co 7413  infcinf 9411  0cc0 11124   · cmul 11129  +∞cpnf 11264  ℝ^*cxr 11266   < clt 11267   ≤ cle 11268  [,)cico 13400  Basecbs 17301   GrpHom cghm 19340  normcnm 24802  NrmGrpcngp 24803   normOp cnmo 24931

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201  ax-pre-sup 11202

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-po 5563  df-so 5564  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7986  df-2nd 7987  df-er 8696  df-en 8953  df-dom 8954  df-sdom 8955  df-sup 9412  df-inf 9413  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-ico 13404  df-nmo 24934

This theorem is used by:  nmoval  24941  nmof  24945