Theorem lnconi 32123

Description: Lemma for lnopconi 32124 and lnfnconi 32145. (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 3131	. . 3 ⊢ (𝑇 ∈ 𝐶 → ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑆 · (normℎ‘𝑦)))
4		oveq1 7364	. . . . . 6 ⊢ (𝑥 = 𝑆 → (𝑥 · (normℎ‘𝑦)) = (𝑆 · (normℎ‘𝑦)))
5	4	breq2d 5085	. . . . 5 ⊢ (𝑥 = 𝑆 → ((𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)) ↔ (𝑁‘(𝑇‘𝑦)) ≤ (𝑆 · (normℎ‘𝑦))))
6	5	ralbidv 3162	. . . 4 ⊢ (𝑥 = 𝑆 → (∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)) ↔ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑆 · (normℎ‘𝑦))))
7	6	rspcev 3560	. . 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 590	. 2 ⊢ (𝑇 ∈ 𝐶 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)))
9		arch 12426	. . . . . 6 ⊢ (𝑥 ∈ ℝ → ∃𝑛 ∈ ℕ 𝑥 < 𝑛)
10	9	adantr 481	. . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦))) → ∃𝑛 ∈ ℕ 𝑥 < 𝑛)
11		nnre 12173	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
12		simplll 780	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → 𝑥 ∈ ℝ)
13		simpllr 781	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → 𝑛 ∈ ℝ)
14		normcl 31215	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℋ → (normℎ‘𝑦) ∈ ℝ)
15	14	adantl 482	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (normℎ‘𝑦) ∈ ℝ)
16		normge0 31216	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℋ → 0 ≤ (normℎ‘𝑦))
17	16	adantl 482	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → 0 ≤ (normℎ‘𝑦))
18		ltle 11226	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (𝑥 < 𝑛 → 𝑥 ≤ 𝑛))
19	18	imp 407	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) → 𝑥 ≤ 𝑛)
20	19	adantr 481	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → 𝑥 ≤ 𝑛)
21	12, 13, 15, 17, 20	lemul1ad 12087	. . . . . . . . . . 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 482	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (𝑁‘(𝑇‘𝑦)) ∈ ℝ)
24		simpll 772	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) → 𝑥 ∈ ℝ)
25		remulcl 11115	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ (normℎ‘𝑦) ∈ ℝ) → (𝑥 · (normℎ‘𝑦)) ∈ ℝ)
26	24, 14, 25	syl2an 602	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (𝑥 · (normℎ‘𝑦)) ∈ ℝ)
27		simplr 774	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) → 𝑛 ∈ ℝ)
28		remulcl 11115	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℝ ∧ (normℎ‘𝑦) ∈ ℝ) → (𝑛 · (normℎ‘𝑦)) ∈ ℝ)
29	27, 14, 28	syl2an 602	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (𝑛 · (normℎ‘𝑦)) ∈ ℝ)
30		letr 11232	. . . . . . . . . . . 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 1379	. . . . . . . . . . 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 700	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → ((𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)) → (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))))
33	32	ralimdva 3151	. . . . . . . . 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 452	. . . . . . . 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 658	. . . . . . 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 599	. . . . . 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 3152	. . . . 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 3132	. . 3 ⊢ (∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)) → ∃𝑛 ∈ ℕ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦)))
40		simprr 778	. . . . . . . 8 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → 𝑧 ∈ ℝ+)
41		simpll 772	. . . . . . . . 9 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → 𝑛 ∈ ℕ)
42	41	nnrpd 12976	. . . . . . . 8 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → 𝑛 ∈ ℝ+)
43	40, 42	rpdivcld 12995	. . . . . . 7 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → (𝑧 / 𝑛) ∈ ℝ+)
44		simprr 778	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑤 ∈ ℋ)
45		simprll 784	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑥 ∈ ℋ)
46		hvsubcl 31107	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ ℋ ∧ 𝑥 ∈ ℋ) → (𝑤 −ℎ 𝑥) ∈ ℋ)
47	44, 45, 46	syl2anc 590	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑤 −ℎ 𝑥) ∈ ℋ)
48		2fveq3 6833	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑤 −ℎ 𝑥) → (𝑁‘(𝑇‘𝑦)) = (𝑁‘(𝑇‘(𝑤 −ℎ 𝑥))))
49		fveq2 6828	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑤 −ℎ 𝑥) → (normℎ‘𝑦) = (normℎ‘(𝑤 −ℎ 𝑥)))
50	49	oveq2d 7373	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑤 −ℎ 𝑥) → (𝑛 · (normℎ‘𝑦)) = (𝑛 · (normℎ‘(𝑤 −ℎ 𝑥))))
51	48, 50	breq12d 5086	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑤 −ℎ 𝑥) → ((𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦)) ↔ (𝑁‘(𝑇‘(𝑤 −ℎ 𝑥))) ≤ (𝑛 · (normℎ‘(𝑤 −ℎ 𝑥)))))
52	51	rspcva 3558	. . . . . . . . . . . . 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 586	. . . . . . . . . . . 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 658	. . . . . . . . . . 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 3520	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑛 ∈ ℝ)
59		normcl 31215	. . . . . . . . . . . . . . 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 11115	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℝ ∧ (normℎ‘(𝑤 −ℎ 𝑥)) ∈ ℝ) → (𝑛 · (normℎ‘(𝑤 −ℎ 𝑥))) ∈ ℝ)
62	58, 60, 61	syl2anc 590	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑛 · (normℎ‘(𝑤 −ℎ 𝑥))) ∈ ℝ)
63		simprlr 785	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑧 ∈ ℝ+)
64	63	rpred 12978	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑧 ∈ ℝ)
65		lelttr 11228	. . . . . . . . . . . . 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 1379	. . . . . . . . . . . 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 721	. . . . . . . . . . 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 701	. . . . . . . . . 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 12946	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+)
70	69	rpregt0d 12984	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (𝑛 ∈ ℝ ∧ 0 < 𝑛))
71	70	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑛 ∈ ℝ ∧ 0 < 𝑛))
72		ltmuldiv2 12022	. . . . . . . . . . . 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 1379	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → ((𝑛 · (normℎ‘(𝑤 −ℎ 𝑥))) < 𝑧 ↔ (normℎ‘(𝑤 −ℎ 𝑥)) < (𝑧 / 𝑛)))
74	73	adantlr 721	. . . . . . . . . 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 590	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑇‘(𝑤 −ℎ 𝑥)) = ((𝑇‘𝑤)𝑀(𝑇‘𝑥)))
77	76	adantlr 721	. . . . . . . . . . . 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 6832	. . . . . . . . . . 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 5083	. . . . . . . . . 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 294	. . . . . . . . 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 468	. . . . . . . 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 3131	. . . . . . 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 5077	. . . . . . . 8 ⊢ (𝑦 = (𝑧 / 𝑛) → ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦 ↔ (normℎ‘(𝑤 −ℎ 𝑥)) < (𝑧 / 𝑛)))
84	83	rspceaimv 3566	. . . . . . 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 590	. . . . . 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 3182	. . . . 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 3132	. . . 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 235	. . 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 210	1 ⊢ (𝑇 ∈ 𝐶 ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053  ∃wrex 3063   class class class wbr 5073  ‘cfv 6486  (class class class)co 7357  ℝcr 11029  0cc0 11030   · cmul 11035   < clt 11171   ≤ cle 11172   / cdiv 11799  ℕcn 12166  ℝ⁺crp 12934   ℋchba 31009  norm_ℎcno 31013   −_ℎ cmv 31015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107  ax-pre-sup 11108  ax-hfvadd 31090  ax-hv0cl 31093  ax-hfvmul 31095  ax-hvmul0 31100  ax-hfi 31169  ax-his1 31172  ax-his3 31174  ax-his4 31175

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-tr 5181  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7314  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7808  df-2nd 7933  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-er 8634  df-en 8885  df-dom 8886  df-sdom 8887  df-sup 9346  df-pnf 11173  df-mnf 11174  df-xr 11175  df-ltxr 11176  df-le 11177  df-sub 11371  df-neg 11372  df-div 11800  df-nn 12167  df-2 12236  df-3 12237  df-n0 12430  df-z 12517  df-uz 12781  df-rp 12935  df-seq 13956  df-exp 14016  df-cj 15053  df-re 15054  df-im 15055  df-sqrt 15189  df-hnorm 31058  df-hvsub 31061

This theorem is referenced by:  lnopconi  32124  lnfnconi  32145