nmoleub3 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > nmoleub3		Structured version Visualization version GIF version

Theorem nmoleub3 24866

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 726	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → 𝐹 ∈ (𝑆 LMHom 𝑇))
15		nmoleub3.6	. . . . . . . . . . 11 ⊢ (𝜑 → ℝ ⊆ 𝐾)
16	15	ad3antrrr 726	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → ℝ ⊆ 𝐾)
17	11	ad3antrrr 726	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → 𝑅 ∈ ℝ⁺)
18	7	elin1d 4197	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ∈ NrmMod)
19	18	ad3antrrr 726	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → 𝑆 ∈ NrmMod)
20		nlmngp 24414	. . . . . . . . . . . . . 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 ≠ (0_g‘𝑆))) → 𝑆 ∈ NrmGrp)
22		simprl 767	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → 𝑦 ∈ 𝑉)
23		simprr 769	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → 𝑦 ≠ (0_g‘𝑆))
24		eqid 2730	. . . . . . . . . . . . . 14 ⊢ (0_g‘𝑆) = (0_g‘𝑆)
25	2, 3, 24	nmrpcl 24349	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆)) → (𝐿‘𝑦) ∈ ℝ⁺)
26	21, 22, 23, 25	syl3anc 1369	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → (𝐿‘𝑦) ∈ ℝ⁺)
27	17, 26	rpdivcld 13037	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → (𝑅 / (𝐿‘𝑦)) ∈ ℝ⁺)
28	27	rpred 13020	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → (𝑅 / (𝐿‘𝑦)) ∈ ℝ)
29	16, 28	sseldd 3982	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → (𝑅 / (𝐿‘𝑦)) ∈ 𝐾)
30		eqid 2730	. . . . . . . . . 10 ⊢ ( ·_𝑠 ‘𝑆) = ( ·_𝑠 ‘𝑆)
31		eqid 2730	. . . . . . . . . 10 ⊢ ( ·_𝑠 ‘𝑇) = ( ·_𝑠 ‘𝑇)
32	5, 6, 2, 30, 31	lmhmlin 20790	. . . . . . . . 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 1369	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → (𝐹‘((𝑅 / (𝐿‘𝑦))( ·_𝑠 ‘𝑆)𝑦)) = ((𝑅 / (𝐿‘𝑦))( ·_𝑠 ‘𝑇)(𝐹‘𝑦)))
34	33	fveq2d 6894	. . . . . . 7 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → (𝑀‘(𝐹‘((𝑅 / (𝐿‘𝑦))( ·_𝑠 ‘𝑆)𝑦))) = (𝑀‘((𝑅 / (𝐿‘𝑦))( ·_𝑠 ‘𝑇)(𝐹‘𝑦))))
35	8	elin1d 4197	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ NrmMod)
36	35	ad3antrrr 726	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → 𝑇 ∈ NrmMod)
37		eqid 2730	. . . . . . . . . . . . 13 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
38	5, 37	lmhmsca 20785	. . . . . . . . . . . 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 ≠ (0_g‘𝑆))) → (Scalar‘𝑇) = 𝐺)
40	39	fveq2d 6894	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → (Base‘(Scalar‘𝑇)) = (Base‘𝐺))
41	40, 6	eqtr4di 2788	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → (Base‘(Scalar‘𝑇)) = 𝐾)
42	29, 41	eleqtrrd 2834	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → (𝑅 / (𝐿‘𝑦)) ∈ (Base‘(Scalar‘𝑇)))
43		eqid 2730	. . . . . . . . . . 11 ⊢ (Base‘𝑇) = (Base‘𝑇)
44	2, 43	lmhmf 20789	. . . . . . . . . 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 ≠ (0_g‘𝑆))) → 𝐹:𝑉⟶(Base‘𝑇))
46	45, 22	ffvelcdmd 7086	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → (𝐹‘𝑦) ∈ (Base‘𝑇))
47		eqid 2730	. . . . . . . . 9 ⊢ (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
48		eqid 2730	. . . . . . . . 9 ⊢ (norm‘(Scalar‘𝑇)) = (norm‘(Scalar‘𝑇))
49	43, 4, 31, 37, 47, 48	nmvs 24413	. . . . . . . 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 1369	. . . . . . 7 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → (𝑀‘((𝑅 / (𝐿‘𝑦))( ·_𝑠 ‘𝑇)(𝐹‘𝑦))) = (((norm‘(Scalar‘𝑇))‘(𝑅 / (𝐿‘𝑦))) · (𝑀‘(𝐹‘𝑦))))
51	39	fveq2d 6894	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → (norm‘(Scalar‘𝑇)) = (norm‘𝐺))
52	51	fveq1d 6892	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → ((norm‘(Scalar‘𝑇))‘(𝑅 / (𝐿‘𝑦))) = ((norm‘𝐺)‘(𝑅 / (𝐿‘𝑦))))
53	7	elin2d 4198	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ ℂMod)
54	53	ad3antrrr 726	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → 𝑆 ∈ ℂMod)
55	5, 6	clmabs 24830	. . . . . . . . . . 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 ≠ (0_g‘𝑆))) → (abs‘(𝑅 / (𝐿‘𝑦))) = ((norm‘𝐺)‘(𝑅 / (𝐿‘𝑦))))
57	27	rpge0d 13024	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → 0 ≤ (𝑅 / (𝐿‘𝑦)))
58	28, 57	absidd 15373	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → (abs‘(𝑅 / (𝐿‘𝑦))) = (𝑅 / (𝐿‘𝑦)))
59	56, 58	eqtr3d 2772	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → ((norm‘𝐺)‘(𝑅 / (𝐿‘𝑦))) = (𝑅 / (𝐿‘𝑦)))
60	52, 59	eqtrd 2770	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘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 ≠ (0_g‘𝑆))) → (((norm‘(Scalar‘𝑇))‘(𝑅 / (𝐿‘𝑦))) · (𝑀‘(𝐹‘𝑦))) = ((𝑅 / (𝐿‘𝑦)) · (𝑀‘(𝐹‘𝑦))))
62	34, 50, 61	3eqtrd 2774	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘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 ≠ (0_g‘𝑆))) → ((𝑀‘(𝐹‘((𝑅 / (𝐿‘𝑦))( ·_𝑠 ‘𝑆)𝑦))) / 𝑅) = (((𝑅 / (𝐿‘𝑦)) · (𝑀‘(𝐹‘𝑦))) / 𝑅))
64	27	rpcnd 13022	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → (𝑅 / (𝐿‘𝑦)) ∈ ℂ)
65		nlmngp 24414	. . . . . . . . 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 ≠ (0_g‘𝑆))) → 𝑇 ∈ NrmGrp)
67	43, 4	nmcl 24345	. . . . . . . 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 ≠ (0_g‘𝑆))) → (𝑀‘(𝐹‘𝑦)) ∈ ℝ)
69	68	recnd 11246	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → (𝑀‘(𝐹‘𝑦)) ∈ ℂ)
70	17	rpcnd 13022	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → 𝑅 ∈ ℂ)
71	17	rpne0d 13025	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → 𝑅 ≠ 0)
72	64, 69, 70, 71	divassd 12029	. . . . 5 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → (((𝑅 / (𝐿‘𝑦)) · (𝑀‘(𝐹‘𝑦))) / 𝑅) = ((𝑅 / (𝐿‘𝑦)) · ((𝑀‘(𝐹‘𝑦)) / 𝑅)))
73	26	rpcnd 13022	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → (𝐿‘𝑦) ∈ ℂ)
74	26	rpne0d 13025	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → (𝐿‘𝑦) ≠ 0)
75	69, 70, 73, 71, 74	dmdcand 12023	. . . . 5 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → ((𝑅 / (𝐿‘𝑦)) · ((𝑀‘(𝐹‘𝑦)) / 𝑅)) = ((𝑀‘(𝐹‘𝑦)) / (𝐿‘𝑦)))
76	63, 72, 75	3eqtrd 2774	. . . 4 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → ((𝑀‘(𝐹‘((𝑅 / (𝐿‘𝑦))( ·_𝑠 ‘𝑆)𝑦))) / 𝑅) = ((𝑀‘(𝐹‘𝑦)) / (𝐿‘𝑦)))
77		eqid 2730	. . . . . . . 8 ⊢ (norm‘𝐺) = (norm‘𝐺)
78	2, 3, 30, 5, 6, 77	nmvs 24413	. . . . . . 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 1369	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘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 ≠ (0_g‘𝑆))) → (((norm‘𝐺)‘(𝑅 / (𝐿‘𝑦))) · (𝐿‘𝑦)) = ((𝑅 / (𝐿‘𝑦)) · (𝐿‘𝑦)))
81	70, 73, 74	divcan1d 11995	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → ((𝑅 / (𝐿‘𝑦)) · (𝐿‘𝑦)) = 𝑅)
82	79, 80, 81	3eqtrd 2774	. . . . 5 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → (𝐿‘((𝑅 / (𝐿‘𝑦))( ·_𝑠 ‘𝑆)𝑦)) = 𝑅)
83		fveqeq2 6899	. . . . . . 7 ⊢ (𝑥 = ((𝑅 / (𝐿‘𝑦))( ·_𝑠 ‘𝑆)𝑦) → ((𝐿‘𝑥) = 𝑅 ↔ (𝐿‘((𝑅 / (𝐿‘𝑦))( ·_𝑠 ‘𝑆)𝑦)) = 𝑅))
84		2fveq3 6895	. . . . . . . . 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 5157	. . . . . . 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 772	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴))
89	2, 5, 30, 6	clmvscl 24835	. . . . . . 7 ⊢ ((𝑆 ∈ ℂMod ∧ (𝑅 / (𝐿‘𝑦)) ∈ 𝐾 ∧ 𝑦 ∈ 𝑉) → ((𝑅 / (𝐿‘𝑦))( ·_𝑠 ‘𝑆)𝑦) ∈ 𝑉)
90	54, 29, 22, 89	syl3anc 1369	. . . . . 6 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → ((𝑅 / (𝐿‘𝑦))( ·_𝑠 ‘𝑆)𝑦) ∈ 𝑉)
91	87, 88, 90	rspcdva 3612	. . . . 5 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘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 ≠ (0_g‘𝑆))) → ((𝑀‘(𝐹‘((𝑅 / (𝐿‘𝑦))( ·_𝑠 ‘𝑆)𝑦))) / 𝑅) ≤ 𝐴)
93	76, 92	eqbrtrrd 5171	. . 3 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → ((𝑀‘(𝐹‘𝑦)) / (𝐿‘𝑦)) ≤ 𝐴)
94		simplr 765	. . . 4 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘𝑆))) → 𝐴 ∈ ℝ)
95	68, 94, 26	ledivmul2d 13074	. . 3 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0_g‘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 ≠ (0_g‘𝑆))) → (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦)))
97	11	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑅 ∈ ℝ⁺)
98	97	rpred 13020	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑅 ∈ ℝ)
99	98	leidd 11784	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑅 ≤ 𝑅)
100		breq1 5150	. . 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 24861	1 ⊢ (𝜑 → ((𝑁‘𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 ∀wral 3059 ∩ cin 3946 ⊆ wss 3947 class class class wbr 5147 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 ℝcr 11111 0cc0 11112 · cmul 11117 ℝ^*cxr 11251 ≤ cle 11253 / cdiv 11875 ℝ⁺crp 12978 abscabs 15185 Basecbs 17148 Scalarcsca 17204 ·_𝑠 cvsca 17205 0_gc0g 17389 LMHom clmhm 20774 normcnm 24305 NrmGrpcngp 24306 NrmModcnlm 24309 normOp cnmo 24442 ℂModcclm 24809

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12979 df-xneg 13096 df-xadd 13097 df-xmul 13098 df-ico 13334 df-fz 13489 df-seq 13971 df-exp 14032 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-starv 17216 df-tset 17220 df-ple 17221 df-ds 17223 df-unif 17224 df-0g 17391 df-topgen 17393 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-grp 18858 df-subg 19039 df-ghm 19128 df-cmn 19691 df-mgp 20029 df-ring 20129 df-cring 20130 df-subrg 20459 df-lmod 20616 df-lmhm 20777 df-psmet 21136 df-xmet 21137 df-met 21138 df-bl 21139 df-mopn 21140 df-cnfld 21145 df-top 22616 df-topon 22633 df-topsp 22655 df-bases 22669 df-xms 24046 df-ms 24047 df-nm 24311 df-ngp 24312 df-nlm 24315 df-nmo 24445 df-nghm 24446 df-clm 24810

This theorem is referenced by: (None)

W3C validator