Theorem lnconi 29809

Description: Lemma for lnopconi 29810 and lnfnconi 29831. (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 3182	. . 3 ⊢ (𝑇 ∈ 𝐶 → ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑆 · (normℎ‘𝑦)))
4		oveq1 7162	. . . . . 6 ⊢ (𝑥 = 𝑆 → (𝑥 · (normℎ‘𝑦)) = (𝑆 · (normℎ‘𝑦)))
5	4	breq2d 5077	. . . . 5 ⊢ (𝑥 = 𝑆 → ((𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)) ↔ (𝑁‘(𝑇‘𝑦)) ≤ (𝑆 · (normℎ‘𝑦))))
6	5	ralbidv 3197	. . . 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 586	. 2 ⊢ (𝑇 ∈ 𝐶 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)))
9		arch 11893	. . . . . 6 ⊢ (𝑥 ∈ ℝ → ∃𝑛 ∈ ℕ 𝑥 < 𝑛)
10	9	adantr 483	. . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦))) → ∃𝑛 ∈ ℕ 𝑥 < 𝑛)
11		nnre 11644	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
12		simplll 773	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → 𝑥 ∈ ℝ)
13		simpllr 774	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → 𝑛 ∈ ℝ)
14		normcl 28901	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℋ → (normℎ‘𝑦) ∈ ℝ)
15	14	adantl 484	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (normℎ‘𝑦) ∈ ℝ)
16		normge0 28902	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℋ → 0 ≤ (normℎ‘𝑦))
17	16	adantl 484	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → 0 ≤ (normℎ‘𝑦))
18		ltle 10728	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (𝑥 < 𝑛 → 𝑥 ≤ 𝑛))
19	18	imp 409	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) → 𝑥 ≤ 𝑛)
20	19	adantr 483	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → 𝑥 ≤ 𝑛)
21	12, 13, 15, 17, 20	lemul1ad 11578	. . . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (𝑁‘(𝑇‘𝑦)) ∈ ℝ)
24		simpll 765	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) → 𝑥 ∈ ℝ)
25		remulcl 10621	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ (normℎ‘𝑦) ∈ ℝ) → (𝑥 · (normℎ‘𝑦)) ∈ ℝ)
26	24, 14, 25	syl2an 597	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (𝑥 · (normℎ‘𝑦)) ∈ ℝ)
27		simplr 767	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) → 𝑛 ∈ ℝ)
28		remulcl 10621	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℝ ∧ (normℎ‘𝑦) ∈ ℝ) → (𝑛 · (normℎ‘𝑦)) ∈ ℝ)
29	27, 14, 28	syl2an 597	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (𝑛 · (normℎ‘𝑦)) ∈ ℝ)
30		letr 10733	. . . . . . . . . . . 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 1367	. . . . . . . . . . 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 692	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → ((𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)) → (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))))
33	32	ralimdva 3177	. . . . . . . . 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 454	. . . . . . . 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 650	. . . . . . 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 594	. . . . . 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 3274	. . . . 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 3281	. . 3 ⊢ (∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)) → ∃𝑛 ∈ ℕ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦)))
40		simprr 771	. . . . . . . 8 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → 𝑧 ∈ ℝ+)
41		simpll 765	. . . . . . . . 9 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → 𝑛 ∈ ℕ)
42	41	nnrpd 12428	. . . . . . . 8 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → 𝑛 ∈ ℝ+)
43	40, 42	rpdivcld 12447	. . . . . . 7 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → (𝑧 / 𝑛) ∈ ℝ+)
44		simprr 771	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑤 ∈ ℋ)
45		simprll 777	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑥 ∈ ℋ)
46		hvsubcl 28793	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ ℋ ∧ 𝑥 ∈ ℋ) → (𝑤 −ℎ 𝑥) ∈ ℋ)
47	44, 45, 46	syl2anc 586	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑤 −ℎ 𝑥) ∈ ℋ)
48		2fveq3 6674	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑤 −ℎ 𝑥) → (𝑁‘(𝑇‘𝑦)) = (𝑁‘(𝑇‘(𝑤 −ℎ 𝑥))))
49		fveq2 6669	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑤 −ℎ 𝑥) → (normℎ‘𝑦) = (normℎ‘(𝑤 −ℎ 𝑥)))
50	49	oveq2d 7171	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑤 −ℎ 𝑥) → (𝑛 · (normℎ‘𝑦)) = (𝑛 · (normℎ‘(𝑤 −ℎ 𝑥))))
51	48, 50	breq12d 5078	. . . . . . . . . . . . . 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 582	. . . . . . . . . . . 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 650	. . . . . . . . . . 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 2897	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑤 −ℎ 𝑥) → ((𝑁‘(𝑇‘𝑦)) ∈ ℝ ↔ (𝑁‘(𝑇‘(𝑤 −ℎ 𝑥))) ∈ ℝ))
56	55, 22	vtoclga 3573	. . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑛 ∈ ℝ)
59		normcl 28901	. . . . . . . . . . . . . . 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 10621	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℝ ∧ (normℎ‘(𝑤 −ℎ 𝑥)) ∈ ℝ) → (𝑛 · (normℎ‘(𝑤 −ℎ 𝑥))) ∈ ℝ)
62	58, 60, 61	syl2anc 586	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑛 · (normℎ‘(𝑤 −ℎ 𝑥))) ∈ ℝ)
63		simprlr 778	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑧 ∈ ℝ+)
64	63	rpred 12430	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑧 ∈ ℝ)
65		lelttr 10730	. . . . . . . . . . . . 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 1367	. . . . . . . . . . . 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 713	. . . . . . . . . . 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 693	. . . . . . . . . 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 12399	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+)
70	69	rpregt0d 12436	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (𝑛 ∈ ℝ ∧ 0 < 𝑛))
71	70	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑛 ∈ ℝ ∧ 0 < 𝑛))
72		ltmuldiv2 11513	. . . . . . . . . . . 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 1367	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → ((𝑛 · (normℎ‘(𝑤 −ℎ 𝑥))) < 𝑧 ↔ (normℎ‘(𝑤 −ℎ 𝑥)) < (𝑧 / 𝑛)))
74	73	adantlr 713	. . . . . . . . . 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 586	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑇‘(𝑤 −ℎ 𝑥)) = ((𝑇‘𝑤)𝑀(𝑇‘𝑥)))
77	76	adantlr 713	. . . . . . . . . . . 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 6673	. . . . . . . . . . 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 5075	. . . . . . . . . 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 295	. . . . . . . . 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 470	. . . . . . . 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 3182	. . . . . . 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 5069	. . . . . . . 8 ⊢ (𝑦 = (𝑧 / 𝑛) → ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦 ↔ (normℎ‘(𝑤 −ℎ 𝑥)) < (𝑧 / 𝑛)))
84	83	rspceaimv 3627	. . . . . . 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 586	. . . . . 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 3191	. . . . 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 3281	. . . 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 236	. . 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 211	1 ⊢ (𝑇 ∈ 𝐶 ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139   class class class wbr 5065  ‘cfv 6354  (class class class)co 7155  ℝcr 10535  0cc0 10536   · cmul 10541   < clt 10674   ≤ cle 10675   / cdiv 11296  ℕcn 11637  ℝ⁺crp 12388   ℋchba 28695  norm_ℎcno 28699   −_ℎ cmv 28701

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613  ax-pre-sup 10614  ax-hfvadd 28776  ax-hv0cl 28779  ax-hfvmul 28781  ax-hvmul0 28786  ax-hfi 28855  ax-his1 28858  ax-his3 28860  ax-his4 28861

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7580  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-er 8288  df-en 8509  df-dom 8510  df-sdom 8511  df-sup 8905  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-div 11297  df-nn 11638  df-2 11699  df-3 11700  df-n0 11897  df-z 11981  df-uz 12243  df-rp 12389  df-seq 13369  df-exp 13429  df-cj 14457  df-re 14458  df-im 14459  df-sqrt 14593  df-hnorm 28744  df-hvsub 28747

This theorem is referenced by:  lnopconi  29810  lnfnconi  29831