Theorem nmoleub3 25287

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 4157	. . . . . . . . . . . . . . 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 24843	. . . . . . . . . . . . . 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 2763	. . . . . . . . . . . . . 14 ⊢ (0g‘𝑆) = (0g‘𝑆)
25	2, 3, 24	nmrpcl 24786	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆)) → (𝐿‘𝑦) ∈ ℝ+)
26	21, 22, 23, 25	syl3anc 1398	. . . . . . . . . . . 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 13081	. . . . . . . . . . 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 13064	. . . . . . . . . 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 3938	. . . . . . . . 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 2763	. . . . . . . . . 10 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
31		eqid 2763	. . . . . . . . . 10 ⊢ ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑇)
32	5, 6, 2, 30, 31	lmhmlin 21165	. . . . . . . . 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 1398	. . . . . . . 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 6885	. . . . . . 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 4157	. . . . . . . . 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 2763	. . . . . . . . . . . . 13 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
38	5, 37	lmhmsca 21160	. . . . . . . . . . . 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 6885	. . . . . . . . . 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 2816	. . . . . . . . 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 2866	. . . . . . . 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 2763	. . . . . . . . . . 11 ⊢ (Base‘𝑇) = (Base‘𝑇)
44	2, 43	lmhmf 21164	. . . . . . . . . 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 7080	. . . . . . . 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 2763	. . . . . . . . 9 ⊢ (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
48		eqid 2763	. . . . . . . . 9 ⊢ (norm‘(Scalar‘𝑇)) = (norm‘(Scalar‘𝑇))
49	43, 4, 31, 37, 47, 48	nmvs 24842	. . . . . . . 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 1398	. . . . . . 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 6885	. . . . . . . . . 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 6883	. . . . . . . . 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 4158	. . . . . . . . . . . 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 25251	. . . . . . . . . . 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 13068	. . . . . . . . . . 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 15479	. . . . . . . . . 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 2800	. . . . . . . . 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 2798	. . . . . . . 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 7425	. . . . . . 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 2802	. . . . . 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 7425	. . . . 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 13066	. . . . . 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 24843	. . . . . . . . 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 24782	. . . . . . . 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 11241	. . . . . 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 13066	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → 𝑅 ∈ ℂ)
71	17	rpne0d 13069	. . . . . 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 12030	. . . . 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 13066	. . . . . 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 13069	. . . . . 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 12024	. . . . 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 2802	. . . 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 2763	. . . . . . . 8 ⊢ (norm‘𝐺) = (norm‘𝐺)
78	2, 3, 30, 5, 6, 77	nmvs 24842	. . . . . . 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 1398	. . . . . 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 7425	. . . . . 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 11996	. . . . . 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 2802	. . . . 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 6890	. . . . . . 7 ⊢ (𝑥 = ((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦) → ((𝐿‘𝑥) = 𝑅 ↔ (𝐿‘((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦)) = 𝑅))
84		2fveq3 6886	. . . . . . . . 9 ⊢ (𝑥 = ((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦) → (𝑀‘(𝐹‘𝑥)) = (𝑀‘(𝐹‘((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦))))
85	84	oveq1d 7425	. . . . . . . 8 ⊢ (𝑥 = ((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦) → ((𝑀‘(𝐹‘𝑥)) / 𝑅) = ((𝑀‘(𝐹‘((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦))) / 𝑅))
86	85	breq1d 5119	. . . . . . 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 25256	. . . . . . 7 ⊢ ((𝑆 ∈ ℂMod ∧ (𝑅 / (𝐿‘𝑦)) ∈ 𝐾 ∧ 𝑦 ∈ 𝑉) → ((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦) ∈ 𝑉)
90	54, 29, 22, 89	syl3anc 1398	. . . . . 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 3582	. . . . 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 5135	. . 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 13118	. . 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 13064	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑅 ∈ ℝ)
99	98	leidd 11784	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑅 ≤ 𝑅)
100		breq1 5112	. . 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 25282	1 ⊢ (𝜑 → ((𝑁‘𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079   ∩ cin 3904   ⊆ wss 3905   class class class wbr 5109  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  ℝcr 11103  0cc0 11104   · cmul 11109  ℝ^*cxr 11246   ≤ cle 11248   / cdiv 11875  ℝ⁺crp 13020  abscabs 15290  Basecbs 17273  Scalarcsca 17317   ·_𝑠 cvsca 17318  0_gc0g 17496   LMHom clmhm 21149  normcnm 24742  NrmGrpcngp 24743  NrmModcnlm 24746   normOp cnmo 24871  ℂModcclm 25230

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11160  ax-resscn 11161  ax-1cn 11162  ax-icn 11163  ax-addcl 11164  ax-addrcl 11165  ax-mulcl 11166  ax-mulrcl 11167  ax-mulcom 11168  ax-addass 11169  ax-mulass 11170  ax-distr 11171  ax-i2m1 11172  ax-1ne0 11173  ax-1rid 11174  ax-rnegex 11175  ax-rrecex 11176  ax-cnre 11177  ax-pre-lttri 11178  ax-pre-lttrn 11179  ax-pre-ltadd 11180  ax-pre-mulgt0 11181  ax-pre-sup 11182  ax-addf 11183

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9398  df-inf 9399  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11447  df-neg 11448  df-div 11876  df-nn 12238  df-2 12307  df-3 12308  df-4 12309  df-5 12310  df-6 12311  df-7 12312  df-8 12313  df-9 12314  df-n0 12509  df-z 12596  df-dec 12716  df-uz 12867  df-q 12977  df-rp 13021  df-xneg 13141  df-xadd 13142  df-xmul 13143  df-ico 13382  df-fz 13540  df-seq 14043  df-exp 14103  df-cj 15155  df-re 15156  df-im 15157  df-sqrt 15291  df-abs 15292  df-struct 17211  df-sets 17228  df-slot 17246  df-ndx 17258  df-base 17274  df-ress 17295  df-plusg 17327  df-mulr 17328  df-starv 17329  df-tset 17333  df-ple 17334  df-ds 17336  df-unif 17337  df-0g 17498  df-topgen 17500  df-mgm 18702  df-sgrp 18781  df-mnd 18797  df-grp 19007  df-subg 19193  df-ghm 19288  df-cmn 19856  df-mgp 20221  df-ring 20321  df-cring 20322  df-subrg 20678  df-lmod 20992  df-lmhm 21152  df-psmet 21523  df-xmet 21524  df-met 21525  df-bl 21526  df-mopn 21527  df-cnfld 21532  df-top 23060  df-topon 23077  df-topsp 23099  df-bases 23112  df-xms 24486  df-ms 24487  df-nm 24748  df-ngp 24749  df-nlm 24752  df-nmo 24874  df-nghm 24875  df-clm 25231

This theorem is used by: (None)