Theorem lnconi 32062

Description: Lemma for lnopconi 32063 and lnfnconi 32084. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
lncon.1	⊢ (𝑇 ∈ 𝐶 → 𝑆 ∈ ℝ)
lncon.2	⊢ ((𝑇 ∈ 𝐶 ∧ 𝑦 ∈ ℋ) → (𝑁‘(𝑇‘𝑦)) ≤ (𝑆 · (normℎ‘𝑦)))
lncon.3	⊢ (𝑇 ∈ 𝐶 ↔ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℝ+ ∃𝑦 ∈ ℝ+ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦 → (𝑁‘((𝑇‘𝑤)𝑀(𝑇‘𝑥))) < 𝑧))
lncon.4	⊢ (𝑦 ∈ ℋ → (𝑁‘(𝑇‘𝑦)) ∈ ℝ)
lncon.5	⊢ ((𝑤 ∈ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘(𝑤 −ℎ 𝑥)) = ((𝑇‘𝑤)𝑀(𝑇‘𝑥)))

Assertion

Ref	Expression
lnconi	⊢ (𝑇 ∈ 𝐶 ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)))

Distinct variable groups:   𝑥,𝑤,𝑦,𝑧,𝑁   𝑦,𝑀   𝑤,𝑇,𝑥,𝑦,𝑧   𝑥,𝑆,𝑦   𝑦,𝐶

Allowed substitution hints:   𝐶(𝑥,𝑧,𝑤)   𝑆(𝑧,𝑤)   𝑀(𝑥,𝑧,𝑤)

Proof of Theorem lnconi

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lncon.1	. . 3 ⊢ (𝑇 ∈ 𝐶 → 𝑆 ∈ ℝ)
2		lncon.2	. . . 4 ⊢ ((𝑇 ∈ 𝐶 ∧ 𝑦 ∈ ℋ) → (𝑁‘(𝑇‘𝑦)) ≤ (𝑆 · (normℎ‘𝑦)))
3	2	ralrimiva 3144	. . 3 ⊢ (𝑇 ∈ 𝐶 → ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑆 · (normℎ‘𝑦)))
4		oveq1 7438	. . . . . 6 ⊢ (𝑥 = 𝑆 → (𝑥 · (normℎ‘𝑦)) = (𝑆 · (normℎ‘𝑦)))
5	4	breq2d 5160	. . . . 5 ⊢ (𝑥 = 𝑆 → ((𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)) ↔ (𝑁‘(𝑇‘𝑦)) ≤ (𝑆 · (normℎ‘𝑦))))
6	5	ralbidv 3176	. . . 4 ⊢ (𝑥 = 𝑆 → (∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)) ↔ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑆 · (normℎ‘𝑦))))
7	6	rspcev 3622	. . 3 ⊢ ((𝑆 ∈ ℝ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑆 · (normℎ‘𝑦))) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)))
8	1, 3, 7	syl2anc 584	. 2 ⊢ (𝑇 ∈ 𝐶 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)))
9		arch 12521	. . . . . 6 ⊢ (𝑥 ∈ ℝ → ∃𝑛 ∈ ℕ 𝑥 < 𝑛)
10	9	adantr 480	. . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦))) → ∃𝑛 ∈ ℕ 𝑥 < 𝑛)
11		nnre 12271	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
12		simplll 775	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → 𝑥 ∈ ℝ)
13		simpllr 776	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → 𝑛 ∈ ℝ)
14		normcl 31154	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℋ → (normℎ‘𝑦) ∈ ℝ)
15	14	adantl 481	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (normℎ‘𝑦) ∈ ℝ)
16		normge0 31155	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℋ → 0 ≤ (normℎ‘𝑦))
17	16	adantl 481	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → 0 ≤ (normℎ‘𝑦))
18		ltle 11347	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (𝑥 < 𝑛 → 𝑥 ≤ 𝑛))
19	18	imp 406	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) → 𝑥 ≤ 𝑛)
20	19	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → 𝑥 ≤ 𝑛)
21	12, 13, 15, 17, 20	lemul1ad 12205	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (𝑥 · (normℎ‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦)))
22		lncon.4	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℋ → (𝑁‘(𝑇‘𝑦)) ∈ ℝ)
23	22	adantl 481	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (𝑁‘(𝑇‘𝑦)) ∈ ℝ)
24		simpll 767	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) → 𝑥 ∈ ℝ)
25		remulcl 11238	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ (normℎ‘𝑦) ∈ ℝ) → (𝑥 · (normℎ‘𝑦)) ∈ ℝ)
26	24, 14, 25	syl2an 596	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (𝑥 · (normℎ‘𝑦)) ∈ ℝ)
27		simplr 769	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) → 𝑛 ∈ ℝ)
28		remulcl 11238	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℝ ∧ (normℎ‘𝑦) ∈ ℝ) → (𝑛 · (normℎ‘𝑦)) ∈ ℝ)
29	27, 14, 28	syl2an 596	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (𝑛 · (normℎ‘𝑦)) ∈ ℝ)
30		letr 11353	. . . . . . . . . . . 12 ⊢ (((𝑁‘(𝑇‘𝑦)) ∈ ℝ ∧ (𝑥 · (normℎ‘𝑦)) ∈ ℝ ∧ (𝑛 · (normℎ‘𝑦)) ∈ ℝ) → (((𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)) ∧ (𝑥 · (normℎ‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))) → (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))))
31	23, 26, 29, 30	syl3anc 1370	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (((𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)) ∧ (𝑥 · (normℎ‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))) → (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))))
32	21, 31	mpan2d 694	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → ((𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)) → (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))))
33	32	ralimdva 3165	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) → (∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)) → ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))))
34	33	impancom 451	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦))) → (𝑥 < 𝑛 → ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))))
35	34	an32s 652	. . . . . . 7 ⊢ (((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦))) ∧ 𝑛 ∈ ℝ) → (𝑥 < 𝑛 → ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))))
36	11, 35	sylan2 593	. . . . . 6 ⊢ (((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦))) ∧ 𝑛 ∈ ℕ) → (𝑥 < 𝑛 → ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))))
37	36	reximdva 3166	. . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦))) → (∃𝑛 ∈ ℕ 𝑥 < 𝑛 → ∃𝑛 ∈ ℕ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))))
38	10, 37	mpd 15	. . . 4 ⊢ ((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦))) → ∃𝑛 ∈ ℕ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦)))
39	38	rexlimiva 3145	. . 3 ⊢ (∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)) → ∃𝑛 ∈ ℕ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦)))
40		simprr 773	. . . . . . . 8 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → 𝑧 ∈ ℝ+)
41		simpll 767	. . . . . . . . 9 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → 𝑛 ∈ ℕ)
42	41	nnrpd 13073	. . . . . . . 8 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → 𝑛 ∈ ℝ+)
43	40, 42	rpdivcld 13092	. . . . . . 7 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → (𝑧 / 𝑛) ∈ ℝ+)
44		simprr 773	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑤 ∈ ℋ)
45		simprll 779	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑥 ∈ ℋ)
46		hvsubcl 31046	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ ℋ ∧ 𝑥 ∈ ℋ) → (𝑤 −ℎ 𝑥) ∈ ℋ)
47	44, 45, 46	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑤 −ℎ 𝑥) ∈ ℋ)
48		2fveq3 6912	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑤 −ℎ 𝑥) → (𝑁‘(𝑇‘𝑦)) = (𝑁‘(𝑇‘(𝑤 −ℎ 𝑥))))
49		fveq2 6907	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑤 −ℎ 𝑥) → (normℎ‘𝑦) = (normℎ‘(𝑤 −ℎ 𝑥)))
50	49	oveq2d 7447	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑤 −ℎ 𝑥) → (𝑛 · (normℎ‘𝑦)) = (𝑛 · (normℎ‘(𝑤 −ℎ 𝑥))))
51	48, 50	breq12d 5161	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑤 −ℎ 𝑥) → ((𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦)) ↔ (𝑁‘(𝑇‘(𝑤 −ℎ 𝑥))) ≤ (𝑛 · (normℎ‘(𝑤 −ℎ 𝑥)))))
52	51	rspcva 3620	. . . . . . . . . . . . 13 ⊢ (((𝑤 −ℎ 𝑥) ∈ ℋ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))) → (𝑁‘(𝑇‘(𝑤 −ℎ 𝑥))) ≤ (𝑛 · (normℎ‘(𝑤 −ℎ 𝑥))))
53	47, 52	sylan 580	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))) → (𝑁‘(𝑇‘(𝑤 −ℎ 𝑥))) ≤ (𝑛 · (normℎ‘(𝑤 −ℎ 𝑥))))
54	53	an32s 652	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))) ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑁‘(𝑇‘(𝑤 −ℎ 𝑥))) ≤ (𝑛 · (normℎ‘(𝑤 −ℎ 𝑥))))
55	48	eleq1d 2824	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑤 −ℎ 𝑥) → ((𝑁‘(𝑇‘𝑦)) ∈ ℝ ↔ (𝑁‘(𝑇‘(𝑤 −ℎ 𝑥))) ∈ ℝ))
56	55, 22	vtoclga 3577	. . . . . . . . . . . . . 14 ⊢ ((𝑤 −ℎ 𝑥) ∈ ℋ → (𝑁‘(𝑇‘(𝑤 −ℎ 𝑥))) ∈ ℝ)
57	47, 56	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑁‘(𝑇‘(𝑤 −ℎ 𝑥))) ∈ ℝ)
58	11	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑛 ∈ ℝ)
59		normcl 31154	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 −ℎ 𝑥) ∈ ℋ → (normℎ‘(𝑤 −ℎ 𝑥)) ∈ ℝ)
60	47, 59	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (normℎ‘(𝑤 −ℎ 𝑥)) ∈ ℝ)
61		remulcl 11238	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℝ ∧ (normℎ‘(𝑤 −ℎ 𝑥)) ∈ ℝ) → (𝑛 · (normℎ‘(𝑤 −ℎ 𝑥))) ∈ ℝ)
62	58, 60, 61	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑛 · (normℎ‘(𝑤 −ℎ 𝑥))) ∈ ℝ)
63		simprlr 780	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑧 ∈ ℝ+)
64	63	rpred 13075	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑧 ∈ ℝ)
65		lelttr 11349	. . . . . . . . . . . . 13 ⊢ (((𝑁‘(𝑇‘(𝑤 −ℎ 𝑥))) ∈ ℝ ∧ (𝑛 · (normℎ‘(𝑤 −ℎ 𝑥))) ∈ ℝ ∧ 𝑧 ∈ ℝ) → (((𝑁‘(𝑇‘(𝑤 −ℎ 𝑥))) ≤ (𝑛 · (normℎ‘(𝑤 −ℎ 𝑥))) ∧ (𝑛 · (normℎ‘(𝑤 −ℎ 𝑥))) < 𝑧) → (𝑁‘(𝑇‘(𝑤 −ℎ 𝑥))) < 𝑧))
66	57, 62, 64, 65	syl3anc 1370	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (((𝑁‘(𝑇‘(𝑤 −ℎ 𝑥))) ≤ (𝑛 · (normℎ‘(𝑤 −ℎ 𝑥))) ∧ (𝑛 · (normℎ‘(𝑤 −ℎ 𝑥))) < 𝑧) → (𝑁‘(𝑇‘(𝑤 −ℎ 𝑥))) < 𝑧))
67	66	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))) ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (((𝑁‘(𝑇‘(𝑤 −ℎ 𝑥))) ≤ (𝑛 · (normℎ‘(𝑤 −ℎ 𝑥))) ∧ (𝑛 · (normℎ‘(𝑤 −ℎ 𝑥))) < 𝑧) → (𝑁‘(𝑇‘(𝑤 −ℎ 𝑥))) < 𝑧))
68	54, 67	mpand 695	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))) ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → ((𝑛 · (normℎ‘(𝑤 −ℎ 𝑥))) < 𝑧 → (𝑁‘(𝑇‘(𝑤 −ℎ 𝑥))) < 𝑧))
69		nnrp 13044	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+)
70	69	rpregt0d 13081	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (𝑛 ∈ ℝ ∧ 0 < 𝑛))
71	70	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑛 ∈ ℝ ∧ 0 < 𝑛))
72		ltmuldiv2 12140	. . . . . . . . . . . 12 ⊢ (((normℎ‘(𝑤 −ℎ 𝑥)) ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (𝑛 ∈ ℝ ∧ 0 < 𝑛)) → ((𝑛 · (normℎ‘(𝑤 −ℎ 𝑥))) < 𝑧 ↔ (normℎ‘(𝑤 −ℎ 𝑥)) < (𝑧 / 𝑛)))
73	60, 64, 71, 72	syl3anc 1370	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → ((𝑛 · (normℎ‘(𝑤 −ℎ 𝑥))) < 𝑧 ↔ (normℎ‘(𝑤 −ℎ 𝑥)) < (𝑧 / 𝑛)))
74	73	adantlr 715	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))) ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → ((𝑛 · (normℎ‘(𝑤 −ℎ 𝑥))) < 𝑧 ↔ (normℎ‘(𝑤 −ℎ 𝑥)) < (𝑧 / 𝑛)))
75		lncon.5	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘(𝑤 −ℎ 𝑥)) = ((𝑇‘𝑤)𝑀(𝑇‘𝑥)))
76	44, 45, 75	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑇‘(𝑤 −ℎ 𝑥)) = ((𝑇‘𝑤)𝑀(𝑇‘𝑥)))
77	76	adantlr 715	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))) ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑇‘(𝑤 −ℎ 𝑥)) = ((𝑇‘𝑤)𝑀(𝑇‘𝑥)))
78	77	fveq2d 6911	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))) ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑁‘(𝑇‘(𝑤 −ℎ 𝑥))) = (𝑁‘((𝑇‘𝑤)𝑀(𝑇‘𝑥))))
79	78	breq1d 5158	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))) ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → ((𝑁‘(𝑇‘(𝑤 −ℎ 𝑥))) < 𝑧 ↔ (𝑁‘((𝑇‘𝑤)𝑀(𝑇‘𝑥))) < 𝑧))
80	68, 74, 79	3imtr3d 293	. . . . . . . . 9 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))) ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → ((normℎ‘(𝑤 −ℎ 𝑥)) < (𝑧 / 𝑛) → (𝑁‘((𝑇‘𝑤)𝑀(𝑇‘𝑥))) < 𝑧))
81	80	anassrs 467	. . . . . . . 8 ⊢ ((((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) ∧ 𝑤 ∈ ℋ) → ((normℎ‘(𝑤 −ℎ 𝑥)) < (𝑧 / 𝑛) → (𝑁‘((𝑇‘𝑤)𝑀(𝑇‘𝑥))) < 𝑧))
82	81	ralrimiva 3144	. . . . . . 7 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < (𝑧 / 𝑛) → (𝑁‘((𝑇‘𝑤)𝑀(𝑇‘𝑥))) < 𝑧))
83		breq2 5152	. . . . . . . 8 ⊢ (𝑦 = (𝑧 / 𝑛) → ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦 ↔ (normℎ‘(𝑤 −ℎ 𝑥)) < (𝑧 / 𝑛)))
84	83	rspceaimv 3628	. . . . . . 7 ⊢ (((𝑧 / 𝑛) ∈ ℝ+ ∧ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < (𝑧 / 𝑛) → (𝑁‘((𝑇‘𝑤)𝑀(𝑇‘𝑥))) < 𝑧)) → ∃𝑦 ∈ ℝ+ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦 → (𝑁‘((𝑇‘𝑤)𝑀(𝑇‘𝑥))) < 𝑧))
85	43, 82, 84	syl2anc 584	. . . . . 6 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → ∃𝑦 ∈ ℝ+ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦 → (𝑁‘((𝑇‘𝑤)𝑀(𝑇‘𝑥))) < 𝑧))
86	85	ralrimivva 3200	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))) → ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℝ+ ∃𝑦 ∈ ℝ+ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦 → (𝑁‘((𝑇‘𝑤)𝑀(𝑇‘𝑥))) < 𝑧))
87	86	rexlimiva 3145	. . . 4 ⊢ (∃𝑛 ∈ ℕ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦)) → ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℝ+ ∃𝑦 ∈ ℝ+ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦 → (𝑁‘((𝑇‘𝑤)𝑀(𝑇‘𝑥))) < 𝑧))
88		lncon.3	. . . 4 ⊢ (𝑇 ∈ 𝐶 ↔ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℝ+ ∃𝑦 ∈ ℝ+ ∀𝑤 ∈ ℋ ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦 → (𝑁‘((𝑇‘𝑤)𝑀(𝑇‘𝑥))) < 𝑧))
89	87, 88	sylibr 234	. . 3 ⊢ (∃𝑛 ∈ ℕ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦)) → 𝑇 ∈ 𝐶)
90	39, 89	syl 17	. 2 ⊢ (∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)) → 𝑇 ∈ 𝐶)
91	8, 90	impbii 209	1 ⊢ (𝑇 ∈ 𝐶 ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068   class class class wbr 5148  ‘cfv 6563  (class class class)co 7431  ℝcr 11152  0cc0 11153   · cmul 11158   < clt 11293   ≤ cle 11294   / cdiv 11918  ℕcn 12264  ℝ⁺crp 13032   ℋchba 30948  norm_ℎcno 30952   −_ℎ cmv 30954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231  ax-hfvadd 31029  ax-hv0cl 31032  ax-hfvmul 31034  ax-hvmul0 31039  ax-hfi 31108  ax-his1 31111  ax-his3 31113  ax-his4 31114

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-sup 9480  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-rp 13033  df-seq 14040  df-exp 14100  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-hnorm 30997  df-hvsub 31000

This theorem is referenced by:  lnopconi  32063  lnfnconi  32084