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Theorem nmoleub 24468

Description: The operator norm, defined as an infimum of upper bounds, can also be defined as a supremum of norms of 𝐹(𝑥) away from zero. (Contributed by Mario Carneiro, 18-Oct-2015.)

**Hypotheses**
Ref	Expression
nmofval.1	⊢ 𝑁 = (𝑆 normOp 𝑇)
nmoi.2	⊢ 𝑉 = (Base‘𝑆)
nmoi.3	⊢ 𝐿 = (norm‘𝑆)
nmoi.4	⊢ 𝑀 = (norm‘𝑇)
nmoi2.z	⊢ 0 = (0_g‘𝑆)
nmoleub.1	⊢ (𝜑 → 𝑆 ∈ NrmGrp)
nmoleub.2	⊢ (𝜑 → 𝑇 ∈ NrmGrp)
nmoleub.3	⊢ (𝜑 → 𝐹 ∈ (𝑆 GrpHom 𝑇))
nmoleub.4	⊢ (𝜑 → 𝐴 ∈ ℝ^*)
nmoleub.5	⊢ (𝜑 → 0 ≤ 𝐴)

**Assertion**
Ref	Expression
nmoleub	⊢ (𝜑 → ((𝑁‘𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑉 (𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴)))

Distinct variable groups: 𝑥,𝐿 𝑥,𝑀 𝑥,𝑆 𝑥,𝑇 𝑥,𝐴 𝑥,𝐹 𝜑,𝑥 𝑥,𝑉 𝑥,𝑁 Allowed substitution hint: 0 (𝑥)

**Proof of Theorem nmoleub**
Step	Hyp	Ref	Expression
1		nmoleub.2	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ NrmGrp)
2	1	ad2antrr 722	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → 𝑇 ∈ NrmGrp)
3		nmoleub.3	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (𝑆 GrpHom 𝑇))
4		nmoi.2	. . . . . . . . . . . 12 ⊢ 𝑉 = (Base‘𝑆)
5		eqid 2730	. . . . . . . . . . . 12 ⊢ (Base‘𝑇) = (Base‘𝑇)
6	4, 5	ghmf 19134	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝑉⟶(Base‘𝑇))
7	3, 6	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝑉⟶(Base‘𝑇))
8	7	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → 𝐹:𝑉⟶(Base‘𝑇))
9		simprl 767	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → 𝑥 ∈ 𝑉)
10		ffvelcdm 7082	. . . . . . . . 9 ⊢ ((𝐹:𝑉⟶(Base‘𝑇) ∧ 𝑥 ∈ 𝑉) → (𝐹‘𝑥) ∈ (Base‘𝑇))
11	8, 9, 10	syl2anc 582	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → (𝐹‘𝑥) ∈ (Base‘𝑇))
12		nmoi.4	. . . . . . . . 9 ⊢ 𝑀 = (norm‘𝑇)
13	5, 12	nmcl 24345	. . . . . . . 8 ⊢ ((𝑇 ∈ NrmGrp ∧ (𝐹‘𝑥) ∈ (Base‘𝑇)) → (𝑀‘(𝐹‘𝑥)) ∈ ℝ)
14	2, 11, 13	syl2anc 582	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → (𝑀‘(𝐹‘𝑥)) ∈ ℝ)
15		nmoleub.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ NrmGrp)
16	15	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) → 𝑆 ∈ NrmGrp)
17		nmoi.3	. . . . . . . . . 10 ⊢ 𝐿 = (norm‘𝑆)
18		nmoi2.z	. . . . . . . . . 10 ⊢ 0 = (0_g‘𝑆)
19	4, 17, 18	nmrpcl 24349	. . . . . . . . 9 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 ) → (𝐿‘𝑥) ∈ ℝ⁺)
20	19	3expb 1118	. . . . . . . 8 ⊢ ((𝑆 ∈ NrmGrp ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → (𝐿‘𝑥) ∈ ℝ⁺)
21	16, 20	sylan 578	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → (𝐿‘𝑥) ∈ ℝ⁺)
22	14, 21	rerpdivcld 13051	. . . . . 6 ⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ∈ ℝ)
23	22	rexrd 11268	. . . . 5 ⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ∈ ℝ^*)
24		nmofval.1	. . . . . . . 8 ⊢ 𝑁 = (𝑆 normOp 𝑇)
25	24	nmocl 24457	. . . . . . 7 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → (𝑁‘𝐹) ∈ ℝ^*)
26	15, 1, 3, 25	syl3anc 1369	. . . . . 6 ⊢ (𝜑 → (𝑁‘𝐹) ∈ ℝ^*)
27	26	ad2antrr 722	. . . . 5 ⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → (𝑁‘𝐹) ∈ ℝ^*)
28		nmoleub.4	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
29	28	ad2antrr 722	. . . . 5 ⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → 𝐴 ∈ ℝ^*)
30	15, 1, 3	3jca 1126	. . . . . . 7 ⊢ (𝜑 → (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)))
31	30	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) → (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)))
32	24, 4, 17, 12, 18	nmoi2 24467	. . . . . 6 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ (𝑁‘𝐹))
33	31, 32	sylan 578	. . . . 5 ⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ (𝑁‘𝐹))
34		simplr 765	. . . . 5 ⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → (𝑁‘𝐹) ≤ 𝐴)
35	23, 27, 29, 33, 34	xrletrd 13145	. . . 4 ⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴)
36	35	expr 455	. . 3 ⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ 𝑥 ∈ 𝑉) → (𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴))
37	36	ralrimiva 3144	. 2 ⊢ ((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) → ∀𝑥 ∈ 𝑉 (𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴))
38		0le0 12317	. . . . . . . . . . 11 ⊢ 0 ≤ 0
39		simpllr 772	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 = 0 ) → 𝐴 ∈ ℝ)
40	39	recnd 11246	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 = 0 ) → 𝐴 ∈ ℂ)
41	40	mul01d 11417	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 = 0 ) → (𝐴 · 0) = 0)
42	38, 41	breqtrrid 5185	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 = 0 ) → 0 ≤ (𝐴 · 0))
43		fveq2 6890	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → (𝐹‘𝑥) = (𝐹‘ 0 ))
44	3	ad2antrr 722	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
45		eqid 2730	. . . . . . . . . . . . . . 15 ⊢ (0_g‘𝑇) = (0_g‘𝑇)
46	18, 45	ghmid 19136	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘ 0 ) = (0_g‘𝑇))
47	44, 46	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) → (𝐹‘ 0 ) = (0_g‘𝑇))
48	43, 47	sylan9eqr 2792	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 = 0 ) → (𝐹‘𝑥) = (0_g‘𝑇))
49	48	fveq2d 6894	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 = 0 ) → (𝑀‘(𝐹‘𝑥)) = (𝑀‘(0_g‘𝑇)))
50	1	ad3antrrr 726	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 = 0 ) → 𝑇 ∈ NrmGrp)
51	12, 45	nm0 24358	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ NrmGrp → (𝑀‘(0_g‘𝑇)) = 0)
52	50, 51	syl 17	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 = 0 ) → (𝑀‘(0_g‘𝑇)) = 0)
53	49, 52	eqtrd 2770	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 = 0 ) → (𝑀‘(𝐹‘𝑥)) = 0)
54		fveq2 6890	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → (𝐿‘𝑥) = (𝐿‘ 0 ))
55	15	ad2antrr 722	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) → 𝑆 ∈ NrmGrp)
56	17, 18	nm0 24358	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ NrmGrp → (𝐿‘ 0 ) = 0)
57	55, 56	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) → (𝐿‘ 0 ) = 0)
58	54, 57	sylan9eqr 2792	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 = 0 ) → (𝐿‘𝑥) = 0)
59	58	oveq2d 7427	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 = 0 ) → (𝐴 · (𝐿‘𝑥)) = (𝐴 · 0))
60	42, 53, 59	3brtr4d 5179	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 = 0 ) → (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥)))
61	60	a1d 25	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 = 0 ) → ((𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴) → (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))))
62		simpr 483	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 ≠ 0 ) → 𝑥 ≠ 0 )
63	1	ad2antrr 722	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) → 𝑇 ∈ NrmGrp)
64	7	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝐹:𝑉⟶(Base‘𝑇))
65	64, 10	sylan 578	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) → (𝐹‘𝑥) ∈ (Base‘𝑇))
66	63, 65, 13	syl2anc 582	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) → (𝑀‘(𝐹‘𝑥)) ∈ ℝ)
67	66	adantr 479	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 ≠ 0 ) → (𝑀‘(𝐹‘𝑥)) ∈ ℝ)
68		simpllr 772	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 ≠ 0 ) → 𝐴 ∈ ℝ)
69	15	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝑆 ∈ NrmGrp)
70	19	3expa 1116	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 ≠ 0 ) → (𝐿‘𝑥) ∈ ℝ⁺)
71	69, 70	sylanl1 676	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 ≠ 0 ) → (𝐿‘𝑥) ∈ ℝ⁺)
72	67, 68, 71	ledivmul2d 13074	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 ≠ 0 ) → (((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴 ↔ (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))))
73	72	biimpd 228	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 ≠ 0 ) → (((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴 → (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))))
74	62, 73	embantd 59	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 ≠ 0 ) → ((𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴) → (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))))
75	61, 74	pm2.61dane 3027	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) → ((𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴) → (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))))
76	75	ralimdva 3165	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → (∀𝑥 ∈ 𝑉 (𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴) → ∀𝑥 ∈ 𝑉 (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))))
77	1	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝑇 ∈ NrmGrp)
78	3	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
79		simpr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝐴 ∈ ℝ)
80		nmoleub.5	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ 𝐴)
81	80	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 0 ≤ 𝐴)
82	24, 4, 17, 12	nmolb 24454	. . . . . . 7 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (∀𝑥 ∈ 𝑉 (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥)) → (𝑁‘𝐹) ≤ 𝐴))
83	69, 77, 78, 79, 81, 82	syl311anc 1382	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → (∀𝑥 ∈ 𝑉 (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥)) → (𝑁‘𝐹) ≤ 𝐴))
84	76, 83	syld 47	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → (∀𝑥 ∈ 𝑉 (𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴) → (𝑁‘𝐹) ≤ 𝐴))
85	84	imp 405	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ ℝ) ∧ ∀𝑥 ∈ 𝑉 (𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴)) → (𝑁‘𝐹) ≤ 𝐴)
86	85	an32s 648	. . 3 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝑉 (𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) → (𝑁‘𝐹) ≤ 𝐴)
87	26	ad2antrr 722	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝑉 (𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴)) ∧ 𝐴 = +∞) → (𝑁‘𝐹) ∈ ℝ^*)
88		pnfge 13114	. . . . 5 ⊢ ((𝑁‘𝐹) ∈ ℝ^* → (𝑁‘𝐹) ≤ +∞)
89	87, 88	syl 17	. . . 4 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝑉 (𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴)) ∧ 𝐴 = +∞) → (𝑁‘𝐹) ≤ +∞)
90		simpr 483	. . . 4 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝑉 (𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴)) ∧ 𝐴 = +∞) → 𝐴 = +∞)
91	89, 90	breqtrrd 5175	. . 3 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝑉 (𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴)) ∧ 𝐴 = +∞) → (𝑁‘𝐹) ≤ 𝐴)
92		ge0nemnf 13156	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) → 𝐴 ≠ -∞)
93	28, 80, 92	syl2anc 582	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ -∞)
94	28, 93	jca 510	. . . . 5 ⊢ (𝜑 → (𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞))
95		xrnemnf 13101	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞))
96	94, 95	sylib 217	. . . 4 ⊢ (𝜑 → (𝐴 ∈ ℝ ∨ 𝐴 = +∞))
97	96	adantr 479	. . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 (𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴)) → (𝐴 ∈ ℝ ∨ 𝐴 = +∞))
98	86, 91, 97	mpjaodan 955	. 2 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 (𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴)) → (𝑁‘𝐹) ≤ 𝐴)
99	37, 98	impbida 797	1 ⊢ (𝜑 → ((𝑁‘𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑉 (𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 843 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 ∀wral 3059 class class class wbr 5147 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 ℝcr 11111 0cc0 11112 · cmul 11117 +∞cpnf 11249 -∞cmnf 11250 ℝ^*cxr 11251 ≤ cle 11253 / cdiv 11875 ℝ⁺crp 12978 Basecbs 17148 0_gc0g 17389 GrpHom cghm 19127 normcnm 24305 NrmGrpcngp 24306 normOp cnmo 24442

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

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-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-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 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-n0 12477 df-z 12563 df-uz 12827 df-q 12937 df-rp 12979 df-xneg 13096 df-xadd 13097 df-xmul 13098 df-ico 13334 df-0g 17391 df-topgen 17393 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-grp 18858 df-ghm 19128 df-psmet 21136 df-xmet 21137 df-met 21138 df-bl 21139 df-mopn 21140 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-nmo 24445 df-nghm 24446

This theorem is referenced by: (None)

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