Theorem nmoleub2lem2 22724

Description: Lemma for nmoleub2a 22725 and similar theorems. (Contributed by Mario Carneiro, 19-Oct-2015.)

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+)
nmoleub2a.5	⊢ (𝜑 → ℚ ⊆ 𝐾)
nmoleub2lem2.6	⊢ (((𝐿‘𝑥) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((𝐿‘𝑥)𝑂𝑅 → (𝐿‘𝑥) ≤ 𝑅))
nmoleub2lem2.7	⊢ (((𝐿‘𝑥) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((𝐿‘𝑥) < 𝑅 → (𝐿‘𝑥)𝑂𝑅))

Assertion

Ref	Expression
nmoleub2lem2	⊢ (𝜑 → ((𝑁‘𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((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)   𝑂(𝑥)

Proof of Theorem nmoleub2lem2

Dummy variables 𝑦 𝑧 are mutually distinct and 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		lmghm 18852	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
13		eqid 2610	. . . . . . . . . 10 ⊢ (0g‘𝑆) = (0g‘𝑆)
14		eqid 2610	. . . . . . . . . 10 ⊢ (0g‘𝑇) = (0g‘𝑇)
15	13, 14	ghmid 17489	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
16	9, 12, 15	3syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
17	16	fveq2d 6107	. . . . . . 7 ⊢ (𝜑 → (𝑀‘(𝐹‘(0g‘𝑆))) = (𝑀‘(0g‘𝑇)))
18		inss1 3795	. . . . . . . . 9 ⊢ (NrmMod ∩ ℂMod) ⊆ NrmMod
19	18, 8	sseldi 3566	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ NrmMod)
20		nlmngp 22291	. . . . . . . 8 ⊢ (𝑇 ∈ NrmMod → 𝑇 ∈ NrmGrp)
21	4, 14	nm0 22243	. . . . . . . 8 ⊢ (𝑇 ∈ NrmGrp → (𝑀‘(0g‘𝑇)) = 0)
22	19, 20, 21	3syl 18	. . . . . . 7 ⊢ (𝜑 → (𝑀‘(0g‘𝑇)) = 0)
23	17, 22	eqtrd 2644	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝐹‘(0g‘𝑆))) = 0)
24	23	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → (𝑀‘(𝐹‘(0g‘𝑆))) = 0)
25	24	oveq1d 6564	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → ((𝑀‘(𝐹‘(0g‘𝑆))) / 𝑅) = (0 / 𝑅))
26	11	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → 𝑅 ∈ ℝ+)
27	26	rpcnd 11750	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → 𝑅 ∈ ℂ)
28	26	rpne0d 11753	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → 𝑅 ≠ 0)
29	27, 28	div0d 10679	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → (0 / 𝑅) = 0)
30	25, 29	eqtrd 2644	. . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → ((𝑀‘(𝐹‘(0g‘𝑆))) / 𝑅) = 0)
31	18, 7	sseldi 3566	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ NrmMod)
32		nlmngp 22291	. . . . . . 7 ⊢ (𝑆 ∈ NrmMod → 𝑆 ∈ NrmGrp)
33	31, 32	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ NrmGrp)
34		ngpgrp 22213	. . . . . 6 ⊢ (𝑆 ∈ NrmGrp → 𝑆 ∈ Grp)
35	2, 13	grpidcl 17273	. . . . . 6 ⊢ (𝑆 ∈ Grp → (0g‘𝑆) ∈ 𝑉)
36	33, 34, 35	3syl 18	. . . . 5 ⊢ (𝜑 → (0g‘𝑆) ∈ 𝑉)
37	36	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → (0g‘𝑆) ∈ 𝑉)
38	2, 3	nmcl 22230	. . . . . . . . 9 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑥 ∈ 𝑉) → (𝐿‘𝑥) ∈ ℝ)
39	33, 38	sylan 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐿‘𝑥) ∈ ℝ)
40	11	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑅 ∈ ℝ+)
41	40	rpred 11748	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑅 ∈ ℝ)
42		nmoleub2lem2.7	. . . . . . . 8 ⊢ (((𝐿‘𝑥) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((𝐿‘𝑥) < 𝑅 → (𝐿‘𝑥)𝑂𝑅))
43	39, 41, 42	syl2anc 691	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐿‘𝑥) < 𝑅 → (𝐿‘𝑥)𝑂𝑅))
44	43	imim1d 80	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴) → ((𝐿‘𝑥) < 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)))
45	44	ralimdva 2945	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴) → ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) < 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)))
46	45	imp 444	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) < 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴))
47	3, 13	nm0 22243	. . . . . . 7 ⊢ (𝑆 ∈ NrmGrp → (𝐿‘(0g‘𝑆)) = 0)
48	31, 32, 47	3syl 18	. . . . . 6 ⊢ (𝜑 → (𝐿‘(0g‘𝑆)) = 0)
49	48	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → (𝐿‘(0g‘𝑆)) = 0)
50	26	rpgt0d 11751	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → 0 < 𝑅)
51	49, 50	eqbrtrd 4605	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → (𝐿‘(0g‘𝑆)) < 𝑅)
52		fveq2 6103	. . . . . . 7 ⊢ (𝑥 = (0g‘𝑆) → (𝐿‘𝑥) = (𝐿‘(0g‘𝑆)))
53	52	breq1d 4593	. . . . . 6 ⊢ (𝑥 = (0g‘𝑆) → ((𝐿‘𝑥) < 𝑅 ↔ (𝐿‘(0g‘𝑆)) < 𝑅))
54		fveq2 6103	. . . . . . . . 9 ⊢ (𝑥 = (0g‘𝑆) → (𝐹‘𝑥) = (𝐹‘(0g‘𝑆)))
55	54	fveq2d 6107	. . . . . . . 8 ⊢ (𝑥 = (0g‘𝑆) → (𝑀‘(𝐹‘𝑥)) = (𝑀‘(𝐹‘(0g‘𝑆))))
56	55	oveq1d 6564	. . . . . . 7 ⊢ (𝑥 = (0g‘𝑆) → ((𝑀‘(𝐹‘𝑥)) / 𝑅) = ((𝑀‘(𝐹‘(0g‘𝑆))) / 𝑅))
57	56	breq1d 4593	. . . . . 6 ⊢ (𝑥 = (0g‘𝑆) → (((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴 ↔ ((𝑀‘(𝐹‘(0g‘𝑆))) / 𝑅) ≤ 𝐴))
58	53, 57	imbi12d 333	. . . . 5 ⊢ (𝑥 = (0g‘𝑆) → (((𝐿‘𝑥) < 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴) ↔ ((𝐿‘(0g‘𝑆)) < 𝑅 → ((𝑀‘(𝐹‘(0g‘𝑆))) / 𝑅) ≤ 𝐴)))
59	58	rspcv 3278	. . . 4 ⊢ ((0g‘𝑆) ∈ 𝑉 → (∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) < 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴) → ((𝐿‘(0g‘𝑆)) < 𝑅 → ((𝑀‘(𝐹‘(0g‘𝑆))) / 𝑅) ≤ 𝐴)))
60	37, 46, 51, 59	syl3c 64	. . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → ((𝑀‘(𝐹‘(0g‘𝑆))) / 𝑅) ≤ 𝐴)
61	30, 60	eqbrtrrd 4607	. 2 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → 0 ≤ 𝐴)
62		simp-4l 802	. . . . 5 ⊢ (((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) ∧ ¬ (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦))) → 𝜑)
63	62, 7	syl 17	. . . 4 ⊢ (((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) ∧ ¬ (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦))) → 𝑆 ∈ (NrmMod ∩ ℂMod))
64	62, 8	syl 17	. . . 4 ⊢ (((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) ∧ ¬ (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦))) → 𝑇 ∈ (NrmMod ∩ ℂMod))
65	62, 9	syl 17	. . . 4 ⊢ (((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) ∧ ¬ (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦))) → 𝐹 ∈ (𝑆 LMHom 𝑇))
66	62, 10	syl 17	. . . 4 ⊢ (((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) ∧ ¬ (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦))) → 𝐴 ∈ ℝ*)
67	62, 11	syl 17	. . . 4 ⊢ (((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) ∧ ¬ (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦))) → 𝑅 ∈ ℝ+)
68		nmoleub2a.5	. . . . 5 ⊢ (𝜑 → ℚ ⊆ 𝐾)
69	62, 68	syl 17	. . . 4 ⊢ (((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) ∧ ¬ (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦))) → ℚ ⊆ 𝐾)
70		eqid 2610	. . . 4 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
71		simpllr 795	. . . 4 ⊢ (((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) ∧ ¬ (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦))) → 𝐴 ∈ ℝ)
72	61	ad3antrrr 762	. . . 4 ⊢ (((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) ∧ ¬ (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦))) → 0 ≤ 𝐴)
73		simplrl 796	. . . 4 ⊢ (((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) ∧ ¬ (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦))) → 𝑦 ∈ 𝑉)
74		simplrr 797	. . . 4 ⊢ (((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) ∧ ¬ (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦))) → 𝑦 ≠ (0g‘𝑆))
75	46	ad3antrrr 762	. . . . 5 ⊢ (((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) ∧ ¬ (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦))) → ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) < 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴))
76		fveq2 6103	. . . . . . . 8 ⊢ (𝑥 = (𝑧( ·𝑠 ‘𝑆)𝑦) → (𝐿‘𝑥) = (𝐿‘(𝑧( ·𝑠 ‘𝑆)𝑦)))
77	76	breq1d 4593	. . . . . . 7 ⊢ (𝑥 = (𝑧( ·𝑠 ‘𝑆)𝑦) → ((𝐿‘𝑥) < 𝑅 ↔ (𝐿‘(𝑧( ·𝑠 ‘𝑆)𝑦)) < 𝑅))
78		fveq2 6103	. . . . . . . . . 10 ⊢ (𝑥 = (𝑧( ·𝑠 ‘𝑆)𝑦) → (𝐹‘𝑥) = (𝐹‘(𝑧( ·𝑠 ‘𝑆)𝑦)))
79	78	fveq2d 6107	. . . . . . . . 9 ⊢ (𝑥 = (𝑧( ·𝑠 ‘𝑆)𝑦) → (𝑀‘(𝐹‘𝑥)) = (𝑀‘(𝐹‘(𝑧( ·𝑠 ‘𝑆)𝑦))))
80	79	oveq1d 6564	. . . . . . . 8 ⊢ (𝑥 = (𝑧( ·𝑠 ‘𝑆)𝑦) → ((𝑀‘(𝐹‘𝑥)) / 𝑅) = ((𝑀‘(𝐹‘(𝑧( ·𝑠 ‘𝑆)𝑦))) / 𝑅))
81	80	breq1d 4593	. . . . . . 7 ⊢ (𝑥 = (𝑧( ·𝑠 ‘𝑆)𝑦) → (((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴 ↔ ((𝑀‘(𝐹‘(𝑧( ·𝑠 ‘𝑆)𝑦))) / 𝑅) ≤ 𝐴))
82	77, 81	imbi12d 333	. . . . . 6 ⊢ (𝑥 = (𝑧( ·𝑠 ‘𝑆)𝑦) → (((𝐿‘𝑥) < 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴) ↔ ((𝐿‘(𝑧( ·𝑠 ‘𝑆)𝑦)) < 𝑅 → ((𝑀‘(𝐹‘(𝑧( ·𝑠 ‘𝑆)𝑦))) / 𝑅) ≤ 𝐴)))
83	82	rspccv 3279	. . . . 5 ⊢ (∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) < 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴) → ((𝑧( ·𝑠 ‘𝑆)𝑦) ∈ 𝑉 → ((𝐿‘(𝑧( ·𝑠 ‘𝑆)𝑦)) < 𝑅 → ((𝑀‘(𝐹‘(𝑧( ·𝑠 ‘𝑆)𝑦))) / 𝑅) ≤ 𝐴)))
84	75, 83	syl 17	. . . 4 ⊢ (((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) ∧ ¬ (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦))) → ((𝑧( ·𝑠 ‘𝑆)𝑦) ∈ 𝑉 → ((𝐿‘(𝑧( ·𝑠 ‘𝑆)𝑦)) < 𝑅 → ((𝑀‘(𝐹‘(𝑧( ·𝑠 ‘𝑆)𝑦))) / 𝑅) ≤ 𝐴)))
85		simpr 476	. . . 4 ⊢ (((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) ∧ ¬ (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦))) → ¬ (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦)))
86	1, 2, 3, 4, 5, 6, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 84, 85	nmoleub2lem3 22723	. . 3 ⊢ ¬ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) ∧ ¬ (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦)))
87		iman 439	. . 3 ⊢ (((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦))) ↔ ¬ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) ∧ ¬ (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦))))
88	86, 87	mpbir 220	. 2 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦)))
89		nmoleub2lem2.6	. . 3 ⊢ (((𝐿‘𝑥) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((𝐿‘𝑥)𝑂𝑅 → (𝐿‘𝑥) ≤ 𝑅))
90	39, 41, 89	syl2anc 691	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐿‘𝑥)𝑂𝑅 → (𝐿‘𝑥) ≤ 𝑅))
91	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 61, 88, 90	nmoleub2lem 22722	1 ⊢ (𝜑 → ((𝑁‘𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896   ∩ cin 3539   ⊆ wss 3540   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815   · cmul 9820  ℝ^*cxr 9952   < clt 9953   ≤ cle 9954   / cdiv 10563  ℚcq 11664  ℝ⁺crp 11708  Basecbs 15695  Scalarcsca 15771   ·_𝑠 cvsca 15772  0_gc0g 15923  Grpcgrp 17245   GrpHom cghm 17480   LMHom clmhm 18840  normcnm 22191  NrmGrpcngp 22192  NrmModcnlm 22195   normOp cnmo 22319  ℂModcclm 22670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ico 12052  df-fz 12198  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-0g 15925  df-topgen 15927  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-subg 17414  df-ghm 17481  df-cmn 18018  df-mgp 18313  df-ring 18372  df-cring 18373  df-subrg 18601  df-lmod 18688  df-lmhm 18843  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-xms 21935  df-ms 21936  df-nm 22197  df-ngp 22198  df-nlm 22201  df-nmo 22322  df-nghm 22323  df-clm 22671

This theorem is referenced by:  nmoleub2a  22725  nmoleub2b  22726