Theorem nmoleub3 25247

Description: The operator norm is the supremum of the value of a linear operator on the unit sphere. (Contributed by Mario Carneiro, 19-Oct-2015.) (Proof shortened by AV, 29-Sep-2021.)

Hypotheses

Ref	Expression
nmoleub2.n	⊢ 𝑁 = (𝑆 normOp 𝑇)
nmoleub2.v	⊢ 𝑉 = (Base‘𝑆)
nmoleub2.l	⊢ 𝐿 = (norm‘𝑆)
nmoleub2.m	⊢ 𝑀 = (norm‘𝑇)
nmoleub2.g	⊢ 𝐺 = (Scalar‘𝑆)
nmoleub2.w	⊢ 𝐾 = (Base‘𝐺)
nmoleub2.s	⊢ (𝜑 → 𝑆 ∈ (NrmMod ∩ ℂMod))
nmoleub2.t	⊢ (𝜑 → 𝑇 ∈ (NrmMod ∩ ℂMod))
nmoleub2.f	⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇))
nmoleub2.a	⊢ (𝜑 → 𝐴 ∈ ℝ*)
nmoleub2.r	⊢ (𝜑 → 𝑅 ∈ ℝ+)
nmoleub3.5	⊢ (𝜑 → 0 ≤ 𝐴)
nmoleub3.6	⊢ (𝜑 → ℝ ⊆ 𝐾)

Assertion

Ref	Expression
nmoleub3	⊢ (𝜑 → ((𝑁‘𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐿   𝑥,𝑁   𝑥,𝑀   𝜑,𝑥   𝑥,𝑆   𝑥,𝑉   𝑥,𝑅

Allowed substitution hints:   𝑇(𝑥)   𝐺(𝑥)   𝐾(𝑥)

Proof of Theorem nmoleub3

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nmoleub2.n	. 2 ⊢ 𝑁 = (𝑆 normOp 𝑇)
2		nmoleub2.v	. 2 ⊢ 𝑉 = (Base‘𝑆)
3		nmoleub2.l	. 2 ⊢ 𝐿 = (norm‘𝑆)
4		nmoleub2.m	. 2 ⊢ 𝑀 = (norm‘𝑇)
5		nmoleub2.g	. 2 ⊢ 𝐺 = (Scalar‘𝑆)
6		nmoleub2.w	. 2 ⊢ 𝐾 = (Base‘𝐺)
7		nmoleub2.s	. 2 ⊢ (𝜑 → 𝑆 ∈ (NrmMod ∩ ℂMod))
8		nmoleub2.t	. 2 ⊢ (𝜑 → 𝑇 ∈ (NrmMod ∩ ℂMod))
9		nmoleub2.f	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇))
10		nmoleub2.a	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
11		nmoleub2.r	. 2 ⊢ (𝜑 → 𝑅 ∈ ℝ+)
12		nmoleub3.5	. . 3 ⊢ (𝜑 → 0 ≤ 𝐴)
13	12	adantr 485	. 2 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → 0 ≤ 𝐴)
14	9	ad3antrrr 742	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → 𝐹 ∈ (𝑆 LMHom 𝑇))
15		nmoleub3.6	. . . . . . . . . . 11 ⊢ (𝜑 → ℝ ⊆ 𝐾)
16	15	ad3antrrr 742	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → ℝ ⊆ 𝐾)
17	11	ad3antrrr 742	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → 𝑅 ∈ ℝ+)
18	7	elin1d 4165	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ∈ NrmMod)
19	18	ad3antrrr 742	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → 𝑆 ∈ NrmMod)
20		nlmngp 24803	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ NrmMod → 𝑆 ∈ NrmGrp)
21	19, 20	syl 18	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → 𝑆 ∈ NrmGrp)
22		simprl 782	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → 𝑦 ∈ 𝑉)
23		simprr 784	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → 𝑦 ≠ (0g‘𝑆))
24		eqid 2769	. . . . . . . . . . . . . 14 ⊢ (0g‘𝑆) = (0g‘𝑆)
25	2, 3, 24	nmrpcl 24746	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆)) → (𝐿‘𝑦) ∈ ℝ+)
26	21, 22, 23, 25	syl3anc 1396	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → (𝐿‘𝑦) ∈ ℝ+)
27	17, 26	rpdivcld 13077	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → (𝑅 / (𝐿‘𝑦)) ∈ ℝ+)
28	27	rpred 13060	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → (𝑅 / (𝐿‘𝑦)) ∈ ℝ)
29	16, 28	sseldd 3946	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → (𝑅 / (𝐿‘𝑦)) ∈ 𝐾)
30		eqid 2769	. . . . . . . . . 10 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
31		eqid 2769	. . . . . . . . . 10 ⊢ ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑇)
32	5, 6, 2, 30, 31	lmhmlin 21134	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (𝑅 / (𝐿‘𝑦)) ∈ 𝐾 ∧ 𝑦 ∈ 𝑉) → (𝐹‘((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦)) = ((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑇)(𝐹‘𝑦)))
33	14, 29, 22, 32	syl3anc 1396	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → (𝐹‘((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦)) = ((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑇)(𝐹‘𝑦)))
34	33	fveq2d 6886	. . . . . . 7 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → (𝑀‘(𝐹‘((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦))) = (𝑀‘((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑇)(𝐹‘𝑦))))
35	8	elin1d 4165	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ NrmMod)
36	35	ad3antrrr 742	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → 𝑇 ∈ NrmMod)
37		eqid 2769	. . . . . . . . . . . . 13 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
38	5, 37	lmhmsca 21129	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = 𝐺)
39	14, 38	syl 18	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → (Scalar‘𝑇) = 𝐺)
40	39	fveq2d 6886	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → (Base‘(Scalar‘𝑇)) = (Base‘𝐺))
41	40, 6	eqtr4di 2822	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → (Base‘(Scalar‘𝑇)) = 𝐾)
42	29, 41	eleqtrrd 2872	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → (𝑅 / (𝐿‘𝑦)) ∈ (Base‘(Scalar‘𝑇)))
43		eqid 2769	. . . . . . . . . . 11 ⊢ (Base‘𝑇) = (Base‘𝑇)
44	2, 43	lmhmf 21133	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:𝑉⟶(Base‘𝑇))
45	14, 44	syl 18	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → 𝐹:𝑉⟶(Base‘𝑇))
46	45, 22	ffvelcdmd 7081	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → (𝐹‘𝑦) ∈ (Base‘𝑇))
47		eqid 2769	. . . . . . . . 9 ⊢ (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
48		eqid 2769	. . . . . . . . 9 ⊢ (norm‘(Scalar‘𝑇)) = (norm‘(Scalar‘𝑇))
49	43, 4, 31, 37, 47, 48	nmvs 24802	. . . . . . . 8 ⊢ ((𝑇 ∈ NrmMod ∧ (𝑅 / (𝐿‘𝑦)) ∈ (Base‘(Scalar‘𝑇)) ∧ (𝐹‘𝑦) ∈ (Base‘𝑇)) → (𝑀‘((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑇)(𝐹‘𝑦))) = (((norm‘(Scalar‘𝑇))‘(𝑅 / (𝐿‘𝑦))) · (𝑀‘(𝐹‘𝑦))))
50	36, 42, 46, 49	syl3anc 1396	. . . . . . 7 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → (𝑀‘((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑇)(𝐹‘𝑦))) = (((norm‘(Scalar‘𝑇))‘(𝑅 / (𝐿‘𝑦))) · (𝑀‘(𝐹‘𝑦))))
51	39	fveq2d 6886	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → (norm‘(Scalar‘𝑇)) = (norm‘𝐺))
52	51	fveq1d 6884	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → ((norm‘(Scalar‘𝑇))‘(𝑅 / (𝐿‘𝑦))) = ((norm‘𝐺)‘(𝑅 / (𝐿‘𝑦))))
53	7	elin2d 4166	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ ℂMod)
54	53	ad3antrrr 742	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → 𝑆 ∈ ℂMod)
55	5, 6	clmabs 25211	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ ℂMod ∧ (𝑅 / (𝐿‘𝑦)) ∈ 𝐾) → (abs‘(𝑅 / (𝐿‘𝑦))) = ((norm‘𝐺)‘(𝑅 / (𝐿‘𝑦))))
56	54, 29, 55	syl2anc 595	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → (abs‘(𝑅 / (𝐿‘𝑦))) = ((norm‘𝐺)‘(𝑅 / (𝐿‘𝑦))))
57	27	rpge0d 13064	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → 0 ≤ (𝑅 / (𝐿‘𝑦)))
58	28, 57	absidd 15474	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → (abs‘(𝑅 / (𝐿‘𝑦))) = (𝑅 / (𝐿‘𝑦)))
59	56, 58	eqtr3d 2806	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → ((norm‘𝐺)‘(𝑅 / (𝐿‘𝑦))) = (𝑅 / (𝐿‘𝑦)))
60	52, 59	eqtrd 2804	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → ((norm‘(Scalar‘𝑇))‘(𝑅 / (𝐿‘𝑦))) = (𝑅 / (𝐿‘𝑦)))
61	60	oveq1d 7426	. . . . . . 7 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → (((norm‘(Scalar‘𝑇))‘(𝑅 / (𝐿‘𝑦))) · (𝑀‘(𝐹‘𝑦))) = ((𝑅 / (𝐿‘𝑦)) · (𝑀‘(𝐹‘𝑦))))
62	34, 50, 61	3eqtrd 2808	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → (𝑀‘(𝐹‘((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦))) = ((𝑅 / (𝐿‘𝑦)) · (𝑀‘(𝐹‘𝑦))))
63	62	oveq1d 7426	. . . . 5 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → ((𝑀‘(𝐹‘((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦))) / 𝑅) = (((𝑅 / (𝐿‘𝑦)) · (𝑀‘(𝐹‘𝑦))) / 𝑅))
64	27	rpcnd 13062	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → (𝑅 / (𝐿‘𝑦)) ∈ ℂ)
65		nlmngp 24803	. . . . . . . . 9 ⊢ (𝑇 ∈ NrmMod → 𝑇 ∈ NrmGrp)
66	36, 65	syl 18	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → 𝑇 ∈ NrmGrp)
67	43, 4	nmcl 24742	. . . . . . . 8 ⊢ ((𝑇 ∈ NrmGrp ∧ (𝐹‘𝑦) ∈ (Base‘𝑇)) → (𝑀‘(𝐹‘𝑦)) ∈ ℝ)
68	66, 46, 67	syl2anc 595	. . . . . . 7 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → (𝑀‘(𝐹‘𝑦)) ∈ ℝ)
69	68	recnd 11237	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → (𝑀‘(𝐹‘𝑦)) ∈ ℂ)
70	17	rpcnd 13062	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → 𝑅 ∈ ℂ)
71	17	rpne0d 13065	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → 𝑅 ≠ 0)
72	64, 69, 70, 71	divassd 12026	. . . . 5 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → (((𝑅 / (𝐿‘𝑦)) · (𝑀‘(𝐹‘𝑦))) / 𝑅) = ((𝑅 / (𝐿‘𝑦)) · ((𝑀‘(𝐹‘𝑦)) / 𝑅)))
73	26	rpcnd 13062	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → (𝐿‘𝑦) ∈ ℂ)
74	26	rpne0d 13065	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → (𝐿‘𝑦) ≠ 0)
75	69, 70, 73, 71, 74	dmdcand 12020	. . . . 5 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → ((𝑅 / (𝐿‘𝑦)) · ((𝑀‘(𝐹‘𝑦)) / 𝑅)) = ((𝑀‘(𝐹‘𝑦)) / (𝐿‘𝑦)))
76	63, 72, 75	3eqtrd 2808	. . . 4 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → ((𝑀‘(𝐹‘((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦))) / 𝑅) = ((𝑀‘(𝐹‘𝑦)) / (𝐿‘𝑦)))
77		eqid 2769	. . . . . . . 8 ⊢ (norm‘𝐺) = (norm‘𝐺)
78	2, 3, 30, 5, 6, 77	nmvs 24802	. . . . . . 7 ⊢ ((𝑆 ∈ NrmMod ∧ (𝑅 / (𝐿‘𝑦)) ∈ 𝐾 ∧ 𝑦 ∈ 𝑉) → (𝐿‘((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦)) = (((norm‘𝐺)‘(𝑅 / (𝐿‘𝑦))) · (𝐿‘𝑦)))
79	19, 29, 22, 78	syl3anc 1396	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → (𝐿‘((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦)) = (((norm‘𝐺)‘(𝑅 / (𝐿‘𝑦))) · (𝐿‘𝑦)))
80	59	oveq1d 7426	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → (((norm‘𝐺)‘(𝑅 / (𝐿‘𝑦))) · (𝐿‘𝑦)) = ((𝑅 / (𝐿‘𝑦)) · (𝐿‘𝑦)))
81	70, 73, 74	divcan1d 11992	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → ((𝑅 / (𝐿‘𝑦)) · (𝐿‘𝑦)) = 𝑅)
82	79, 80, 81	3eqtrd 2808	. . . . 5 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → (𝐿‘((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦)) = 𝑅)
83		fveqeq2 6891	. . . . . . 7 ⊢ (𝑥 = ((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦) → ((𝐿‘𝑥) = 𝑅 ↔ (𝐿‘((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦)) = 𝑅))
84		2fveq3 6887	. . . . . . . . 9 ⊢ (𝑥 = ((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦) → (𝑀‘(𝐹‘𝑥)) = (𝑀‘(𝐹‘((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦))))
85	84	oveq1d 7426	. . . . . . . 8 ⊢ (𝑥 = ((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦) → ((𝑀‘(𝐹‘𝑥)) / 𝑅) = ((𝑀‘(𝐹‘((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦))) / 𝑅))
86	85	breq1d 5123	. . . . . . 7 ⊢ (𝑥 = ((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦) → (((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴 ↔ ((𝑀‘(𝐹‘((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦))) / 𝑅) ≤ 𝐴))
87	83, 86	imbi12d 347	. . . . . 6 ⊢ (𝑥 = ((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦) → (((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴) ↔ ((𝐿‘((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦)) = 𝑅 → ((𝑀‘(𝐹‘((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦))) / 𝑅) ≤ 𝐴)))
88		simpllr 787	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴))
89	2, 5, 30, 6	clmvscl 25216	. . . . . . 7 ⊢ ((𝑆 ∈ ℂMod ∧ (𝑅 / (𝐿‘𝑦)) ∈ 𝐾 ∧ 𝑦 ∈ 𝑉) → ((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦) ∈ 𝑉)
90	54, 29, 22, 89	syl3anc 1396	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → ((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦) ∈ 𝑉)
91	87, 88, 90	rspcdva 3591	. . . . 5 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → ((𝐿‘((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦)) = 𝑅 → ((𝑀‘(𝐹‘((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦))) / 𝑅) ≤ 𝐴))
92	82, 91	mpd 16	. . . 4 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → ((𝑀‘(𝐹‘((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦))) / 𝑅) ≤ 𝐴)
93	76, 92	eqbrtrrd 5139	. . 3 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → ((𝑀‘(𝐹‘𝑦)) / (𝐿‘𝑦)) ≤ 𝐴)
94		simplr 780	. . . 4 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → 𝐴 ∈ ℝ)
95	68, 94, 26	ledivmul2d 13114	. . 3 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → (((𝑀‘(𝐹‘𝑦)) / (𝐿‘𝑦)) ≤ 𝐴 ↔ (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦))))
96	93, 95	mpbid 235	. 2 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦)))
97	11	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑅 ∈ ℝ+)
98	97	rpred 13060	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑅 ∈ ℝ)
99	98	leidd 11780	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑅 ≤ 𝑅)
100		breq1 5116	. . 3 ⊢ ((𝐿‘𝑥) = 𝑅 → ((𝐿‘𝑥) ≤ 𝑅 ↔ 𝑅 ≤ 𝑅))
101	99, 100	syl5ibrcom 250	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐿‘𝑥) = 𝑅 → (𝐿‘𝑥) ≤ 𝑅))
102	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 96, 101	nmoleub2lem 25242	1 ⊢ (𝜑 → ((𝑁‘𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085   ∩ cin 3912   ⊆ wss 3913   class class class wbr 5113  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411  ℝcr 11099  0cc0 11100   · cmul 11105  ℝ^*cxr 11242   ≤ cle 11244   / cdiv 11871  ℝ⁺crp 13016  abscabs 15285  Basecbs 17269  Scalarcsca 17313   ·_𝑠 cvsca 17314  0_gc0g 17492   LMHom clmhm 21118  normcnm 24702  NrmGrpcngp 24703  NrmModcnlm 24706   normOp cnmo 24831  ℂModcclm 25190

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177  ax-pre-sup 11178  ax-addf 11179

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-er 8694  df-map 8826  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-sup 9402  df-inf 9403  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-div 11872  df-nn 12234  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12505  df-z 12592  df-dec 12712  df-uz 12863  df-q 12973  df-rp 13017  df-xneg 13137  df-xadd 13138  df-xmul 13139  df-ico 13378  df-fz 13536  df-seq 14038  df-exp 14098  df-cj 15150  df-re 15151  df-im 15152  df-sqrt 15286  df-abs 15287  df-struct 17207  df-sets 17224  df-slot 17242  df-ndx 17254  df-base 17270  df-ress 17291  df-plusg 17323  df-mulr 17324  df-starv 17325  df-tset 17329  df-ple 17330  df-ds 17332  df-unif 17333  df-0g 17494  df-topgen 17496  df-mgm 18698  df-sgrp 18777  df-mnd 18793  df-grp 19003  df-subg 19189  df-ghm 19284  df-cmn 19852  df-mgp 20217  df-ring 20317  df-cring 20318  df-subrg 20655  df-lmod 20961  df-lmhm 21121  df-psmet 21483  df-xmet 21484  df-met 21485  df-bl 21486  df-mopn 21487  df-cnfld 21492  df-top 23020  df-topon 23037  df-topsp 23059  df-bases 23072  df-xms 24446  df-ms 24447  df-nm 24708  df-ngp 24709  df-nlm 24712  df-nmo 24834  df-nghm 24835  df-clm 25191

This theorem is referenced by: (None)