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Theorem nmoleub 24767

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 726	. . . . . . . 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 2734	. . . . . . . . . . . 12 ⊢ (Base‘𝑇) = (Base‘𝑇)
6	4, 5	ghmf 19250	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝑉⟶(Base‘𝑇))
7	3, 6	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝑉⟶(Base‘𝑇))
8	7	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → 𝐹:𝑉⟶(Base‘𝑇))
9		simprl 771	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → 𝑥 ∈ 𝑉)
10		ffvelcdm 7100	. . . . . . . . 9 ⊢ ((𝐹:𝑉⟶(Base‘𝑇) ∧ 𝑥 ∈ 𝑉) → (𝐹‘𝑥) ∈ (Base‘𝑇))
11	8, 9, 10	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → (𝐹‘𝑥) ∈ (Base‘𝑇))
12		nmoi.4	. . . . . . . . 9 ⊢ 𝑀 = (norm‘𝑇)
13	5, 12	nmcl 24644	. . . . . . . 8 ⊢ ((𝑇 ∈ NrmGrp ∧ (𝐹‘𝑥) ∈ (Base‘𝑇)) → (𝑀‘(𝐹‘𝑥)) ∈ ℝ)
14	2, 11, 13	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → (𝑀‘(𝐹‘𝑥)) ∈ ℝ)
15		nmoleub.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ NrmGrp)
16	15	adantr 480	. . . . . . . 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 24648	. . . . . . . . 9 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 ) → (𝐿‘𝑥) ∈ ℝ⁺)
20	19	3expb 1119	. . . . . . . 8 ⊢ ((𝑆 ∈ NrmGrp ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → (𝐿‘𝑥) ∈ ℝ⁺)
21	16, 20	sylan 580	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → (𝐿‘𝑥) ∈ ℝ⁺)
22	14, 21	rerpdivcld 13105	. . . . . 6 ⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ∈ ℝ)
23	22	rexrd 11308	. . . . 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 24756	. . . . . . 7 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → (𝑁‘𝐹) ∈ ℝ^*)
26	15, 1, 3, 25	syl3anc 1370	. . . . . 6 ⊢ (𝜑 → (𝑁‘𝐹) ∈ ℝ^*)
27	26	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → (𝑁‘𝐹) ∈ ℝ^*)
28		nmoleub.4	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
29	28	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → 𝐴 ∈ ℝ^*)
30	15, 1, 3	3jca 1127	. . . . . . 7 ⊢ (𝜑 → (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)))
31	30	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) → (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)))
32	24, 4, 17, 12, 18	nmoi2 24766	. . . . . 6 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ (𝑁‘𝐹))
33	31, 32	sylan 580	. . . . 5 ⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ (𝑁‘𝐹))
34		simplr 769	. . . . 5 ⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → (𝑁‘𝐹) ≤ 𝐴)
35	23, 27, 29, 33, 34	xrletrd 13200	. . . 4 ⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴)
36	35	expr 456	. . 3 ⊢ (((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) ∧ 𝑥 ∈ 𝑉) → (𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴))
37	36	ralrimiva 3143	. 2 ⊢ ((𝜑 ∧ (𝑁‘𝐹) ≤ 𝐴) → ∀𝑥 ∈ 𝑉 (𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴))
38		0le0 12364	. . . . . . . . . . 11 ⊢ 0 ≤ 0
39		simpllr 776	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 = 0 ) → 𝐴 ∈ ℝ)
40	39	recnd 11286	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 = 0 ) → 𝐴 ∈ ℂ)
41	40	mul01d 11457	. . . . . . . . . . 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 6906	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → (𝐹‘𝑥) = (𝐹‘ 0 ))
44	3	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
45		eqid 2734	. . . . . . . . . . . . . . 15 ⊢ (0_g‘𝑇) = (0_g‘𝑇)
46	18, 45	ghmid 19252	. . . . . . . . . . . . . 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 2796	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 = 0 ) → (𝐹‘𝑥) = (0_g‘𝑇))
49	48	fveq2d 6910	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 = 0 ) → (𝑀‘(𝐹‘𝑥)) = (𝑀‘(0_g‘𝑇)))
50	1	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 = 0 ) → 𝑇 ∈ NrmGrp)
51	12, 45	nm0 24657	. . . . . . . . . . . 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 2774	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 = 0 ) → (𝑀‘(𝐹‘𝑥)) = 0)
54		fveq2 6906	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → (𝐿‘𝑥) = (𝐿‘ 0 ))
55	15	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) → 𝑆 ∈ NrmGrp)
56	17, 18	nm0 24657	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ NrmGrp → (𝐿‘ 0 ) = 0)
57	55, 56	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) → (𝐿‘ 0 ) = 0)
58	54, 57	sylan9eqr 2796	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 = 0 ) → (𝐿‘𝑥) = 0)
59	58	oveq2d 7446	. . . . . . . . . 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 484	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 ≠ 0 ) → 𝑥 ≠ 0 )
63	1	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) → 𝑇 ∈ NrmGrp)
64	7	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝐹:𝑉⟶(Base‘𝑇))
65	64, 10	sylan 580	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) → (𝐹‘𝑥) ∈ (Base‘𝑇))
66	63, 65, 13	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) → (𝑀‘(𝐹‘𝑥)) ∈ ℝ)
67	66	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 ≠ 0 ) → (𝑀‘(𝐹‘𝑥)) ∈ ℝ)
68		simpllr 776	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 ≠ 0 ) → 𝐴 ∈ ℝ)
69	15	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝑆 ∈ NrmGrp)
70	19	3expa 1117	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 ≠ 0 ) → (𝐿‘𝑥) ∈ ℝ⁺)
71	69, 70	sylanl1 680	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 ≠ 0 ) → (𝐿‘𝑥) ∈ ℝ⁺)
72	67, 68, 71	ledivmul2d 13128	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 ≠ 0 ) → (((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴 ↔ (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))))
73	72	biimpd 229	. . . . . . . . 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 3026	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ ℝ) ∧ 𝑥 ∈ 𝑉) → ((𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴) → (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))))
76	75	ralimdva 3164	. . . . . 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 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝑇 ∈ NrmGrp)
78	3	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
79		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝐴 ∈ ℝ)
80		nmoleub.5	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ 𝐴)
81	80	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 0 ≤ 𝐴)
82	24, 4, 17, 12	nmolb 24753	. . . . . . 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 1383	. . . . . 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 406	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ ℝ) ∧ ∀𝑥 ∈ 𝑉 (𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴)) → (𝑁‘𝐹) ≤ 𝐴)
86	85	an32s 652	. . 3 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝑉 (𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) → (𝑁‘𝐹) ≤ 𝐴)
87	26	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝑉 (𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴)) ∧ 𝐴 = +∞) → (𝑁‘𝐹) ∈ ℝ^*)
88		pnfge 13169	. . . . 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 484	. . . 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 13211	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) → 𝐴 ≠ -∞)
93	28, 80, 92	syl2anc 584	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ -∞)
94	28, 93	jca 511	. . . . 5 ⊢ (𝜑 → (𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞))
95		xrnemnf 13156	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞))
96	94, 95	sylib 218	. . . 4 ⊢ (𝜑 → (𝐴 ∈ ℝ ∨ 𝐴 = +∞))
97	96	adantr 480	. . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 (𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴)) → (𝐴 ∈ ℝ ∨ 𝐴 = +∞))
98	86, 91, 97	mpjaodan 960	. 2 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 (𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴)) → (𝑁‘𝐹) ≤ 𝐴)
99	37, 98	impbida 801	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 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∀wral 3058 class class class wbr 5147 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ℝcr 11151 0cc0 11152 · cmul 11157 +∞cpnf 11289 -∞cmnf 11290 ℝ^*cxr 11291 ≤ cle 11293 / cdiv 11917 ℝ⁺crp 13031 Basecbs 17244 0_gc0g 17485 GrpHom cghm 19242 normcnm 24604 NrmGrpcngp 24605 normOp cnmo 24741

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-sup 9479 df-inf 9480 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-n0 12524 df-z 12611 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ico 13389 df-0g 17487 df-topgen 17489 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-ghm 19243 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-xms 24345 df-ms 24346 df-nm 24610 df-ngp 24611 df-nmo 24744 df-nghm 24745

This theorem is referenced by: (None)

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