Theorem lnconi 32543

Description: Lemma for lnopconi 32544 and lnfnconi 32565. (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 3154	. . 3 ⊢ (𝑇 ∈ 𝐶 → ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑆 · (normℎ‘𝑦)))
4		oveq1 7422	. . . . . 6 ⊢ (𝑥 = 𝑆 → (𝑥 · (normℎ‘𝑦)) = (𝑆 · (normℎ‘𝑦)))
5	4	breq2d 5115	. . . . 5 ⊢ (𝑥 = 𝑆 → ((𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)) ↔ (𝑁‘(𝑇‘𝑦)) ≤ (𝑆 · (normℎ‘𝑦))))
6	5	ralbidv 3185	. . . 4 ⊢ (𝑥 = 𝑆 → (∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)) ↔ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑆 · (normℎ‘𝑦))))
7	6	rspcev 3576	. . 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 596	. 2 ⊢ (𝑇 ∈ 𝐶 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)))
9		arch 12547	. . . . . 6 ⊢ (𝑥 ∈ ℝ → ∃𝑛 ∈ ℕ 𝑥 < 𝑛)
10	9	adantr 486	. . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦))) → ∃𝑛 ∈ ℕ 𝑥 < 𝑛)
11		nnre 12286	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
12		simplll 787	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → 𝑥 ∈ ℝ)
13		simpllr 788	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → 𝑛 ∈ ℝ)
14		normcl 31635	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℋ → (normℎ‘𝑦) ∈ ℝ)
15	14	adantl 487	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (normℎ‘𝑦) ∈ ℝ)
16		normge0 31636	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℋ → 0 ≤ (normℎ‘𝑦))
17	16	adantl 487	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → 0 ≤ (normℎ‘𝑦))
18		ltle 11344	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (𝑥 < 𝑛 → 𝑥 ≤ 𝑛))
19	18	imp 412	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) → 𝑥 ≤ 𝑛)
20	19	adantr 486	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → 𝑥 ≤ 𝑛)
21	12, 13, 15, 17, 20	lemul1ad 12200	. . . . . . . . . . 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 487	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (𝑁‘(𝑇‘𝑦)) ∈ ℝ)
24		simpll 779	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) → 𝑥 ∈ ℝ)
25		remulcl 11231	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ (normℎ‘𝑦) ∈ ℝ) → (𝑥 · (normℎ‘𝑦)) ∈ ℝ)
26	24, 14, 25	syl2an 608	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (𝑥 · (normℎ‘𝑦)) ∈ ℝ)
27		simplr 781	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) → 𝑛 ∈ ℝ)
28		remulcl 11231	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℝ ∧ (normℎ‘𝑦) ∈ ℝ) → (𝑛 · (normℎ‘𝑦)) ∈ ℝ)
29	27, 14, 28	syl2an 608	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (𝑛 · (normℎ‘𝑦)) ∈ ℝ)
30		letr 11350	. . . . . . . . . . . 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 1398	. . . . . . . . . . 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 707	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → ((𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)) → (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))))
33	32	ralimdva 3174	. . . . . . . . 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 457	. . . . . . . 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 665	. . . . . . 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 605	. . . . . 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 3175	. . . . 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 16	. . . 4 ⊢ ((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦))) → ∃𝑛 ∈ ℕ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦)))
39	38	rexlimiva 3155	. . 3 ⊢ (∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)) → ∃𝑛 ∈ ℕ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦)))
40		simprr 785	. . . . . . . 8 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → 𝑧 ∈ ℝ+)
41		simpll 779	. . . . . . . . 9 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → 𝑛 ∈ ℕ)
42	41	nnrpd 13106	. . . . . . . 8 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → 𝑛 ∈ ℝ+)
43	40, 42	rpdivcld 13125	. . . . . . 7 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → (𝑧 / 𝑛) ∈ ℝ+)
44		simprr 785	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑤 ∈ ℋ)
45		simprll 791	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑥 ∈ ℋ)
46		hvsubcl 31527	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ ℋ ∧ 𝑥 ∈ ℋ) → (𝑤 −ℎ 𝑥) ∈ ℋ)
47	44, 45, 46	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑤 −ℎ 𝑥) ∈ ℋ)
48		2fveq3 6885	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑤 −ℎ 𝑥) → (𝑁‘(𝑇‘𝑦)) = (𝑁‘(𝑇‘(𝑤 −ℎ 𝑥))))
49		fveq2 6880	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑤 −ℎ 𝑥) → (normℎ‘𝑦) = (normℎ‘(𝑤 −ℎ 𝑥)))
50	49	oveq2d 7431	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑤 −ℎ 𝑥) → (𝑛 · (normℎ‘𝑦)) = (𝑛 · (normℎ‘(𝑤 −ℎ 𝑥))))
51	48, 50	breq12d 5116	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑤 −ℎ 𝑥) → ((𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦)) ↔ (𝑁‘(𝑇‘(𝑤 −ℎ 𝑥))) ≤ (𝑛 · (normℎ‘(𝑤 −ℎ 𝑥)))))
52	51	rspcva 3574	. . . . . . . . . . . . 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 592	. . . . . . . . . . . 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 665	. . . . . . . . . . 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 2845	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑤 −ℎ 𝑥) → ((𝑁‘(𝑇‘𝑦)) ∈ ℝ ↔ (𝑁‘(𝑇‘(𝑤 −ℎ 𝑥))) ∈ ℝ))
56	55, 22	vtoclga 3536	. . . . . . . . . . . . . 14 ⊢ ((𝑤 −ℎ 𝑥) ∈ ℋ → (𝑁‘(𝑇‘(𝑤 −ℎ 𝑥))) ∈ ℝ)
57	47, 56	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑁‘(𝑇‘(𝑤 −ℎ 𝑥))) ∈ ℝ)
58	11	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑛 ∈ ℝ)
59		normcl 31635	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 −ℎ 𝑥) ∈ ℋ → (normℎ‘(𝑤 −ℎ 𝑥)) ∈ ℝ)
60	47, 59	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (normℎ‘(𝑤 −ℎ 𝑥)) ∈ ℝ)
61		remulcl 11231	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℝ ∧ (normℎ‘(𝑤 −ℎ 𝑥)) ∈ ℝ) → (𝑛 · (normℎ‘(𝑤 −ℎ 𝑥))) ∈ ℝ)
62	58, 60, 61	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑛 · (normℎ‘(𝑤 −ℎ 𝑥))) ∈ ℝ)
63		simprlr 792	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑧 ∈ ℝ+)
64	63	rpred 13108	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑧 ∈ ℝ)
65		lelttr 11346	. . . . . . . . . . . . 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 1398	. . . . . . . . . . . 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 728	. . . . . . . . . . 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 708	. . . . . . . . . 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 13076	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+)
70	69	rpregt0d 13114	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (𝑛 ∈ ℝ ∧ 0 < 𝑛))
71	70	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑛 ∈ ℝ ∧ 0 < 𝑛))
72		ltmuldiv2 12135	. . . . . . . . . . . 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 1398	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → ((𝑛 · (normℎ‘(𝑤 −ℎ 𝑥))) < 𝑧 ↔ (normℎ‘(𝑤 −ℎ 𝑥)) < (𝑧 / 𝑛)))
74	73	adantlr 728	. . . . . . . . . 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 596	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑇‘(𝑤 −ℎ 𝑥)) = ((𝑇‘𝑤)𝑀(𝑇‘𝑥)))
77	76	adantlr 728	. . . . . . . . . . . 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 6884	. . . . . . . . . . 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 5113	. . . . . . . . . 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 296	. . . . . . . . 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 473	. . . . . . . 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 3154	. . . . . . 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 5107	. . . . . . . 8 ⊢ (𝑦 = (𝑧 / 𝑛) → ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑦 ↔ (normℎ‘(𝑤 −ℎ 𝑥)) < (𝑧 / 𝑛)))
84	83	rspceaimv 3582	. . . . . . 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 596	. . . . . 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 3205	. . . . 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 3155	. . . 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 237	. . 3 ⊢ (∃𝑛 ∈ ℕ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑛 · (normℎ‘𝑦)) → 𝑇 ∈ 𝐶)
90	39, 89	syl 18	. 2 ⊢ (∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)) → 𝑇 ∈ 𝐶)
91	8, 90	impbii 212	1 ⊢ (𝑇 ∈ 𝐶 ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇‘𝑦)) ≤ (𝑥 · (normℎ‘𝑦)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086   class class class wbr 5103  ‘cfv 6534  (class class class)co 7415  ℝcr 11145  0cc0 11146   · cmul 11151   < clt 11289   ≤ cle 11290   / cdiv 11917  ℕcn 12279  ℝ⁺crp 13064   ℋchba 31429  norm_ℎcno 31433   −_ℎ cmv 31435

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7738  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223  ax-pre-sup 11224  ax-hfvadd 31510  ax-hv0cl 31513  ax-hfvmul 31515  ax-hvmul0 31520  ax-hfi 31589  ax-his1 31592  ax-his3 31594  ax-his4 31595

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6300  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7372  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7865  df-2nd 7989  df-frecs 8282  df-wrecs 8313  df-recs 8362  df-rdg 8401  df-er 8700  df-en 8957  df-dom 8958  df-sdom 8959  df-sup 9416  df-pnf 11291  df-mnf 11292  df-xr 11293  df-ltxr 11294  df-le 11295  df-sub 11489  df-neg 11490  df-div 11918  df-nn 12280  df-2 12349  df-3 12350  df-n0 12551  df-z 12638  df-uz 12910  df-rp 13065  df-seq 14088  df-exp 14148  df-cj 15208  df-re 15209  df-im 15210  df-sqrt 15344  df-hnorm 31478  df-hvsub 31481

This theorem is used by:  lnopconi  32544  lnfnconi  32565