Theorem nmoleub3 25134

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 479	. 2 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → 0 ≤ 𝐴)
14	9	ad3antrrr 728	. . . . . . . . 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 728	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → ℝ ⊆ 𝐾)
17	11	ad3antrrr 728	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → 𝑅 ∈ ℝ+)
18	7	elin1d 4196	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ∈ NrmMod)
19	18	ad3antrrr 728	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → 𝑆 ∈ NrmMod)
20		nlmngp 24682	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ NrmMod → 𝑆 ∈ NrmGrp)
21	19, 20	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → 𝑆 ∈ NrmGrp)
22		simprl 769	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → 𝑦 ∈ 𝑉)
23		simprr 771	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → 𝑦 ≠ (0g‘𝑆))
24		eqid 2726	. . . . . . . . . . . . . 14 ⊢ (0g‘𝑆) = (0g‘𝑆)
25	2, 3, 24	nmrpcl 24617	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆)) → (𝐿‘𝑦) ∈ ℝ+)
26	21, 22, 23, 25	syl3anc 1368	. . . . . . . . . . . 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 3979	. . . . . . . . 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 2726	. . . . . . . . . 10 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
31		eqid 2726	. . . . . . . . . 10 ⊢ ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑇)
32	5, 6, 2, 30, 31	lmhmlin 21009	. . . . . . . . 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 1368	. . . . . . . 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 6897	. . . . . . 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 4196	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ NrmMod)
36	35	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → 𝑇 ∈ NrmMod)
37		eqid 2726	. . . . . . . . . . . . 13 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
38	5, 37	lmhmsca 21004	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = 𝐺)
39	14, 38	syl 17	. . . . . . . . . . 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 6897	. . . . . . . . . 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 2784	. . . . . . . . 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 2829	. . . . . . . 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 2726	. . . . . . . . . . 11 ⊢ (Base‘𝑇) = (Base‘𝑇)
44	2, 43	lmhmf 21008	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:𝑉⟶(Base‘𝑇))
45	14, 44	syl 17	. . . . . . . . 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 7091	. . . . . . . 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 2726	. . . . . . . . 9 ⊢ (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
48		eqid 2726	. . . . . . . . 9 ⊢ (norm‘(Scalar‘𝑇)) = (norm‘(Scalar‘𝑇))
49	43, 4, 31, 37, 47, 48	nmvs 24681	. . . . . . . 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 1368	. . . . . . 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 6897	. . . . . . . . . 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 6895	. . . . . . . . 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 4197	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ ℂMod)
54	53	ad3antrrr 728	. . . . . . . . . . 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 25098	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ ℂMod ∧ (𝑅 / (𝐿‘𝑦)) ∈ 𝐾) → (abs‘(𝑅 / (𝐿‘𝑦))) = ((norm‘𝐺)‘(𝑅 / (𝐿‘𝑦))))
56	54, 29, 55	syl2anc 582	. . . . . . . . . 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 15422	. . . . . . . . . 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 2768	. . . . . . . . 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 2766	. . . . . . . 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 7431	. . . . . . 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 2770	. . . . . 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 7431	. . . . 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 24682	. . . . . . . . 9 ⊢ (𝑇 ∈ NrmMod → 𝑇 ∈ NrmGrp)
66	36, 65	syl 17	. . . . . . . 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 24613	. . . . . . . 8 ⊢ ((𝑇 ∈ NrmGrp ∧ (𝐹‘𝑦) ∈ (Base‘𝑇)) → (𝑀‘(𝐹‘𝑦)) ∈ ℝ)
68	66, 46, 67	syl2anc 582	. . . . . . 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 11283	. . . . . 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 12070	. . . . 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 12064	. . . . 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 2770	. . . 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 2726	. . . . . . . 8 ⊢ (norm‘𝐺) = (norm‘𝐺)
78	2, 3, 30, 5, 6, 77	nmvs 24681	. . . . . . 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 1368	. . . . . 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 7431	. . . . . 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 12036	. . . . . 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 2770	. . . . 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 6902	. . . . . . 7 ⊢ (𝑥 = ((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦) → ((𝐿‘𝑥) = 𝑅 ↔ (𝐿‘((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦)) = 𝑅))
84		2fveq3 6898	. . . . . . . . 9 ⊢ (𝑥 = ((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦) → (𝑀‘(𝐹‘𝑥)) = (𝑀‘(𝐹‘((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦))))
85	84	oveq1d 7431	. . . . . . . 8 ⊢ (𝑥 = ((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦) → ((𝑀‘(𝐹‘𝑥)) / 𝑅) = ((𝑀‘(𝐹‘((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦))) / 𝑅))
86	85	breq1d 5155	. . . . . . 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 343	. . . . . 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 774	. . . . . 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 25103	. . . . . . 7 ⊢ ((𝑆 ∈ ℂMod ∧ (𝑅 / (𝐿‘𝑦)) ∈ 𝐾 ∧ 𝑦 ∈ 𝑉) → ((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦) ∈ 𝑉)
90	54, 29, 22, 89	syl3anc 1368	. . . . . 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 3608	. . . . 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 15	. . . 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 5169	. . 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 767	. . . 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 231	. 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 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑅 ∈ ℝ+)
98	97	rpred 13064	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑅 ∈ ℝ)
99	98	leidd 11821	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑅 ≤ 𝑅)
100		breq1 5148	. . 3 ⊢ ((𝐿‘𝑥) = 𝑅 → ((𝐿‘𝑥) ≤ 𝑅 ↔ 𝑅 ≤ 𝑅))
101	99, 100	syl5ibrcom 246	. 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 25129	1 ⊢ (𝜑 → ((𝑁‘𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1534   ∈ wcel 2099   ≠ wne 2930  ∀wral 3051   ∩ cin 3945   ⊆ wss 3946   class class class wbr 5145  ⟶wf 6542  ‘cfv 6546  (class class class)co 7416  ℝcr 11148  0cc0 11149   · cmul 11154  ℝ^*cxr 11288   ≤ cle 11290   / cdiv 11912  ℝ⁺crp 13022  abscabs 15234  Basecbs 17208  Scalarcsca 17264   ·_𝑠 cvsca 17265  0_gc0g 17449   LMHom clmhm 20993  normcnm 24573  NrmGrpcngp 24574  NrmModcnlm 24577   normOp cnmo 24710  ℂModcclm 25077

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738  ax-cnex 11205  ax-resscn 11206  ax-1cn 11207  ax-icn 11208  ax-addcl 11209  ax-addrcl 11210  ax-mulcl 11211  ax-mulrcl 11212  ax-mulcom 11213  ax-addass 11214  ax-mulass 11215  ax-distr 11216  ax-i2m1 11217  ax-1ne0 11218  ax-1rid 11219  ax-rnegex 11220  ax-rrecex 11221  ax-cnre 11222  ax-pre-lttri 11223  ax-pre-lttrn 11224  ax-pre-ltadd 11225  ax-pre-mulgt0 11226  ax-pre-sup 11227  ax-addf 11228

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7995  df-2nd 7996  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-1o 8488  df-er 8726  df-map 8849  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-sup 9478  df-inf 9479  df-pnf 11291  df-mnf 11292  df-xr 11293  df-ltxr 11294  df-le 11295  df-sub 11487  df-neg 11488  df-div 11913  df-nn 12259  df-2 12321  df-3 12322  df-4 12323  df-5 12324  df-6 12325  df-7 12326  df-8 12327  df-9 12328  df-n0 12519  df-z 12605  df-dec 12724  df-uz 12869  df-q 12979  df-rp 13023  df-xneg 13140  df-xadd 13141  df-xmul 13142  df-ico 13378  df-fz 13533  df-seq 14016  df-exp 14076  df-cj 15099  df-re 15100  df-im 15101  df-sqrt 15235  df-abs 15236  df-struct 17144  df-sets 17161  df-slot 17179  df-ndx 17191  df-base 17209  df-ress 17238  df-plusg 17274  df-mulr 17275  df-starv 17276  df-tset 17280  df-ple 17281  df-ds 17283  df-unif 17284  df-0g 17451  df-topgen 17453  df-mgm 18628  df-sgrp 18707  df-mnd 18723  df-grp 18926  df-subg 19113  df-ghm 19203  df-cmn 19776  df-mgp 20114  df-ring 20214  df-cring 20215  df-subrg 20549  df-lmod 20834  df-lmhm 20996  df-psmet 21331  df-xmet 21332  df-met 21333  df-bl 21334  df-mopn 21335  df-cnfld 21340  df-top 22884  df-topon 22901  df-topsp 22923  df-bases 22937  df-xms 24314  df-ms 24315  df-nm 24579  df-ngp 24580  df-nlm 24583  df-nmo 24713  df-nghm 24714  df-clm 25078

This theorem is referenced by: (None)