Theorem nmoleub3 25108

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 482	. 2 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → 0 ≤ 𝐴)
14	9	ad3antrrr 737	. . . . . . . . 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 737	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → ℝ ⊆ 𝐾)
17	11	ad3antrrr 737	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → 𝑅 ∈ ℝ+)
18	7	elin1d 4136	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ∈ NrmMod)
19	18	ad3antrrr 737	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → 𝑆 ∈ NrmMod)
20		nlmngp 24664	. . . . . . . . . . . . . 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 777	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → 𝑦 ∈ 𝑉)
23		simprr 779	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → 𝑦 ≠ (0g‘𝑆))
24		eqid 2741	. . . . . . . . . . . . . 14 ⊢ (0g‘𝑆) = (0g‘𝑆)
25	2, 3, 24	nmrpcl 24607	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆)) → (𝐿‘𝑦) ∈ ℝ+)
26	21, 22, 23, 25	syl3anc 1380	. . . . . . . . . . . 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 12998	. . . . . . . . . . 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 12981	. . . . . . . . . 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 3918	. . . . . . . . 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 2741	. . . . . . . . . 10 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
31		eqid 2741	. . . . . . . . . 10 ⊢ ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑇)
32	5, 6, 2, 30, 31	lmhmlin 21029	. . . . . . . . 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 1380	. . . . . . . 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 6835	. . . . . . 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 4136	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ NrmMod)
36	35	ad3antrrr 737	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → 𝑇 ∈ NrmMod)
37		eqid 2741	. . . . . . . . . . . . 13 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
38	5, 37	lmhmsca 21024	. . . . . . . . . . . 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 6835	. . . . . . . . . 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 2794	. . . . . . . . 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 2844	. . . . . . . 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 2741	. . . . . . . . . . 11 ⊢ (Base‘𝑇) = (Base‘𝑇)
44	2, 43	lmhmf 21028	. . . . . . . . . 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 7030	. . . . . . . 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 2741	. . . . . . . . 9 ⊢ (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
48		eqid 2741	. . . . . . . . 9 ⊢ (norm‘(Scalar‘𝑇)) = (norm‘(Scalar‘𝑇))
49	43, 4, 31, 37, 47, 48	nmvs 24663	. . . . . . . 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 1380	. . . . . . 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 6835	. . . . . . . . . 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 6833	. . . . . . . . 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 4137	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ ℂMod)
54	53	ad3antrrr 737	. . . . . . . . . . 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 25072	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ ℂMod ∧ (𝑅 / (𝐿‘𝑦)) ∈ 𝐾) → (abs‘(𝑅 / (𝐿‘𝑦))) = ((norm‘𝐺)‘(𝑅 / (𝐿‘𝑦))))
56	54, 29, 55	syl2anc 591	. . . . . . . . . 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 12985	. . . . . . . . . . 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 15380	. . . . . . . . . 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 2778	. . . . . . . . 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 2776	. . . . . . . 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 7375	. . . . . . 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 2780	. . . . . 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 7375	. . . . 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 12983	. . . . . 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 24664	. . . . . . . . 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 24603	. . . . . . . 8 ⊢ ((𝑇 ∈ NrmGrp ∧ (𝐹‘𝑦) ∈ (Base‘𝑇)) → (𝑀‘(𝐹‘𝑦)) ∈ ℝ)
68	66, 46, 67	syl2anc 591	. . . . . . 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 11168	. . . . . 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 12983	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → 𝑅 ∈ ℂ)
71	17	rpne0d 12986	. . . . . 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 11961	. . . . 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 12983	. . . . . 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 12986	. . . . . 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 11955	. . . . 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 2780	. . . 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 2741	. . . . . . . 8 ⊢ (norm‘𝐺) = (norm‘𝐺)
78	2, 3, 30, 5, 6, 77	nmvs 24663	. . . . . . 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 1380	. . . . . 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 7375	. . . . . 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 11927	. . . . . 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 2780	. . . . 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 6840	. . . . . . 7 ⊢ (𝑥 = ((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦) → ((𝐿‘𝑥) = 𝑅 ↔ (𝐿‘((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦)) = 𝑅))
84		2fveq3 6836	. . . . . . . . 9 ⊢ (𝑥 = ((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦) → (𝑀‘(𝐹‘𝑥)) = (𝑀‘(𝐹‘((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦))))
85	84	oveq1d 7375	. . . . . . . 8 ⊢ (𝑥 = ((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦) → ((𝑀‘(𝐹‘𝑥)) / 𝑅) = ((𝑀‘(𝐹‘((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦))) / 𝑅))
86	85	breq1d 5085	. . . . . . 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 346	. . . . . 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 782	. . . . . 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 25077	. . . . . . 7 ⊢ ((𝑆 ∈ ℂMod ∧ (𝑅 / (𝐿‘𝑦)) ∈ 𝐾 ∧ 𝑦 ∈ 𝑉) → ((𝑅 / (𝐿‘𝑦))( ·𝑠 ‘𝑆)𝑦) ∈ 𝑉)
90	54, 29, 22, 89	syl3anc 1380	. . . . . 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 3563	. . . . 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 5099	. . 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 775	. . . 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 13035	. . 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 234	. 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 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑅 ∈ ℝ+)
98	97	rpred 12981	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑅 ∈ ℝ)
99	98	leidd 11711	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑅 ≤ 𝑅)
100		breq1 5078	. . 3 ⊢ ((𝐿‘𝑥) = 𝑅 → ((𝐿‘𝑥) ≤ 𝑅 ↔ 𝑅 ≤ 𝑅))
101	99, 100	syl5ibrcom 249	. 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 25103	1 ⊢ (𝜑 → ((𝑁‘𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121   ≠ wne 2936  ∀wral 3055   ∩ cin 3884   ⊆ wss 3885   class class class wbr 5075  ⟶wf 6485  ‘cfv 6489  (class class class)co 7360  ℝcr 11032  0cc0 11033   · cmul 11038  ℝ^*cxr 11173   ≤ cle 11175   / cdiv 11802  ℝ⁺crp 12937  abscabs 15191  Basecbs 17174  Scalarcsca 17218   ·_𝑠 cvsca 17219  0_gc0g 17397   LMHom clmhm 21013  normcnm 24563  NrmGrpcngp 24564  NrmModcnlm 24567   normOp cnmo 24692  ℂModcclm 25051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110  ax-pre-sup 11111  ax-addf 11112

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-sup 9349  df-inf 9350  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-div 11803  df-nn 12170  df-2 12239  df-3 12240  df-4 12241  df-5 12242  df-6 12243  df-7 12244  df-8 12245  df-9 12246  df-n0 12433  df-z 12520  df-dec 12640  df-uz 12784  df-q 12894  df-rp 12938  df-xneg 13058  df-xadd 13059  df-xmul 13060  df-ico 13299  df-fz 13457  df-seq 13959  df-exp 14019  df-cj 15056  df-re 15057  df-im 15058  df-sqrt 15192  df-abs 15193  df-struct 17112  df-sets 17129  df-slot 17147  df-ndx 17159  df-base 17175  df-ress 17196  df-plusg 17228  df-mulr 17229  df-starv 17230  df-tset 17234  df-ple 17235  df-ds 17237  df-unif 17238  df-0g 17399  df-topgen 17401  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-grp 18907  df-subg 19094  df-ghm 19183  df-cmn 19752  df-mgp 20117  df-ring 20211  df-cring 20212  df-subrg 20546  df-lmod 20856  df-lmhm 21016  df-psmet 21343  df-xmet 21344  df-met 21345  df-bl 21346  df-mopn 21347  df-cnfld 21352  df-top 22881  df-topon 22898  df-topsp 22920  df-bases 22933  df-xms 24307  df-ms 24308  df-nm 24569  df-ngp 24570  df-nlm 24573  df-nmo 24695  df-nghm 24696  df-clm 25052

This theorem is referenced by: (None)