Theorem lnconi 32065

Description: Lemma for lnopconi 32066 and lnfnconi 32087. (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 3152	. . 3 ⊢ (𝑇 ∈ 𝐶 → ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑆 · (normℎ‘𝑦)))
4		oveq1 7455	. . . . . 6 ⊢ (𝑥 = 𝑆 → (𝑥 · (normℎ‘𝑦)) = (𝑆 · (normℎ‘𝑦)))
5	4	breq2d 5178	. . . . 5 ⊢ (𝑥 = 𝑆 → ((𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)) ↔ (𝑁‘(𝑇‘𝑦)) ≤ (𝑆 · (normℎ‘𝑦))))
6	5	ralbidv 3184	. . . 4 ⊢ (𝑥 = 𝑆 → (∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)) ↔ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑆 · (normℎ‘𝑦))))
7	6	rspcev 3635	. . 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 583	. 2 ⊢ (𝑇 ∈ 𝐶 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)))
9		arch 12550	. . . . . 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 12300	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
12		simplll 774	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → 𝑥 ∈ ℝ)
13		simpllr 775	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → 𝑛 ∈ ℝ)
14		normcl 31157	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℋ → (normℎ‘𝑦) ∈ ℝ)
15	14	adantl 481	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (normℎ‘𝑦) ∈ ℝ)
16		normge0 31158	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℋ → 0 ≤ (normℎ‘𝑦))
17	16	adantl 481	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → 0 ≤ (normℎ‘𝑦))
18		ltle 11378	. . . . . . . . . . . . . 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 12234	. . . . . . . . . . 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 766	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) → 𝑥 ∈ ℝ)
25		remulcl 11269	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ (normℎ‘𝑦) ∈ ℝ) → (𝑥 · (normℎ‘𝑦)) ∈ ℝ)
26	24, 14, 25	syl2an 595	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (𝑥 · (normℎ‘𝑦)) ∈ ℝ)
27		simplr 768	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) → 𝑛 ∈ ℝ)
28		remulcl 11269	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℝ ∧ (normℎ‘𝑦) ∈ ℝ) → (𝑛 · (normℎ‘𝑦)) ∈ ℝ)
29	27, 14, 28	syl2an 595	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (𝑛 · (normℎ‘𝑦)) ∈ ℝ)
30		letr 11384	. . . . . . . . . . . 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 1371	. . . . . . . . . . 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 693	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → ((𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)) → (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))))
33	32	ralimdva 3173	. . . . . . . . 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 651	. . . . . . 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 592	. . . . . 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 3174	. . . . 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 3153	. . 3 ⊢ (∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)) → ∃𝑛 ∈ ℕ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦)))
40		simprr 772	. . . . . . . 8 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → 𝑧 ∈ ℝ+)
41		simpll 766	. . . . . . . . 9 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → 𝑛 ∈ ℕ)
42	41	nnrpd 13097	. . . . . . . 8 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → 𝑛 ∈ ℝ+)
43	40, 42	rpdivcld 13116	. . . . . . 7 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → (𝑧 / 𝑛) ∈ ℝ+)
44		simprr 772	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑤 ∈ ℋ)
45		simprll 778	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑥 ∈ ℋ)
46		hvsubcl 31049	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ ℋ ∧ 𝑥 ∈ ℋ) → (𝑤 −ℎ 𝑥) ∈ ℋ)
47	44, 45, 46	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑤 −ℎ 𝑥) ∈ ℋ)
48		2fveq3 6925	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑤 −ℎ 𝑥) → (𝑁‘(𝑇‘𝑦)) = (𝑁‘(𝑇‘(𝑤 −ℎ 𝑥))))
49		fveq2 6920	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑤 −ℎ 𝑥) → (normℎ‘𝑦) = (normℎ‘(𝑤 −ℎ 𝑥)))
50	49	oveq2d 7464	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑤 −ℎ 𝑥) → (𝑛 · (normℎ‘𝑦)) = (𝑛 · (normℎ‘(𝑤 −ℎ 𝑥))))
51	48, 50	breq12d 5179	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑤 −ℎ 𝑥) → ((𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦)) ↔ (𝑁‘(𝑇‘(𝑤 −ℎ 𝑥))) ≤ (𝑛 · (normℎ‘(𝑤 −ℎ 𝑥)))))
52	51	rspcva 3633	. . . . . . . . . . . . 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 579	. . . . . . . . . . . 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 651	. . . . . . . . . . 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 2829	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑤 −ℎ 𝑥) → ((𝑁‘(𝑇‘𝑦)) ∈ ℝ ↔ (𝑁‘(𝑇‘(𝑤 −ℎ 𝑥))) ∈ ℝ))
56	55, 22	vtoclga 3589	. . . . . . . . . . . . . 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 31157	. . . . . . . . . . . . . . 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 11269	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℝ ∧ (normℎ‘(𝑤 −ℎ 𝑥)) ∈ ℝ) → (𝑛 · (normℎ‘(𝑤 −ℎ 𝑥))) ∈ ℝ)
62	58, 60, 61	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑛 · (normℎ‘(𝑤 −ℎ 𝑥))) ∈ ℝ)
63		simprlr 779	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑧 ∈ ℝ+)
64	63	rpred 13099	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑧 ∈ ℝ)
65		lelttr 11380	. . . . . . . . . . . . 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 1371	. . . . . . . . . . . 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 714	. . . . . . . . . . 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 694	. . . . . . . . . 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 13068	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+)
70	69	rpregt0d 13105	. . . . . . . . . . . . 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 12169	. . . . . . . . . . . 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 1371	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → ((𝑛 · (normℎ‘(𝑤 −ℎ 𝑥))) < 𝑧 ↔ (normℎ‘(𝑤 −ℎ 𝑥)) < (𝑧 / 𝑛)))
74	73	adantlr 714	. . . . . . . . . 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 583	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑇‘(𝑤 −ℎ 𝑥)) = ((𝑇‘𝑤)𝑀(𝑇‘𝑥)))
77	76	adantlr 714	. . . . . . . . . . . 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 6924	. . . . . . . . . . 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 5176	. . . . . . . . . 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 3152	. . . . . . 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 5170	. . . . . . . 8 ⊢ (𝑦 = (𝑧 / 𝑛) → ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦 ↔ (normℎ‘(𝑤 −ℎ 𝑥)) < (𝑧 / 𝑛)))
84	83	rspceaimv 3641	. . . . . . 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 583	. . . . . 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 3208	. . . . 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 3153	. . . 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 2108  ∀wral 3067  ∃wrex 3076   class class class wbr 5166  ‘cfv 6573  (class class class)co 7448  ℝcr 11183  0cc0 11184   · cmul 11189   < clt 11324   ≤ cle 11325   / cdiv 11947  ℕcn 12293  ℝ⁺crp 13057   ℋchba 30951  norm_ℎcno 30955   −_ℎ cmv 30957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262  ax-hfvadd 31032  ax-hv0cl 31035  ax-hfvmul 31037  ax-hvmul0 31042  ax-hfi 31111  ax-his1 31114  ax-his3 31116  ax-his4 31117

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-sup 9511  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-z 12640  df-uz 12904  df-rp 13058  df-seq 14053  df-exp 14113  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-hnorm 31000  df-hvsub 31003

This theorem is referenced by:  lnopconi  32066  lnfnconi  32087