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Theorem lnconi 29726
Description: Lemma for lnopconi 29727 and lnfnconi 29748. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
lncon.1 (𝑇𝐶𝑆 ∈ ℝ)
lncon.2 ((𝑇𝐶𝑦 ∈ ℋ) → (𝑁‘(𝑇𝑦)) ≤ (𝑆 · (norm𝑦)))
lncon.3 (𝑇𝐶 ↔ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℝ+𝑦 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑦 → (𝑁‘((𝑇𝑤)𝑀(𝑇𝑥))) < 𝑧))
lncon.4 (𝑦 ∈ ℋ → (𝑁‘(𝑇𝑦)) ∈ ℝ)
lncon.5 ((𝑤 ∈ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘(𝑤 𝑥)) = ((𝑇𝑤)𝑀(𝑇𝑥)))
Assertion
Ref Expression
lnconi (𝑇𝐶 ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)))
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.
StepHypRef Expression
1 lncon.1 . . 3 (𝑇𝐶𝑆 ∈ ℝ)
2 lncon.2 . . . 4 ((𝑇𝐶𝑦 ∈ ℋ) → (𝑁‘(𝑇𝑦)) ≤ (𝑆 · (norm𝑦)))
32ralrimiva 3186 . . 3 (𝑇𝐶 → ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑆 · (norm𝑦)))
4 oveq1 7158 . . . . . 6 (𝑥 = 𝑆 → (𝑥 · (norm𝑦)) = (𝑆 · (norm𝑦)))
54breq2d 5074 . . . . 5 (𝑥 = 𝑆 → ((𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)) ↔ (𝑁‘(𝑇𝑦)) ≤ (𝑆 · (norm𝑦))))
65ralbidv 3201 . . . 4 (𝑥 = 𝑆 → (∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)) ↔ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑆 · (norm𝑦))))
76rspcev 3626 . . 3 ((𝑆 ∈ ℝ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑆 · (norm𝑦))) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)))
81, 3, 7syl2anc 584 . 2 (𝑇𝐶 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)))
9 arch 11886 . . . . . 6 (𝑥 ∈ ℝ → ∃𝑛 ∈ ℕ 𝑥 < 𝑛)
109adantr 481 . . . . 5 ((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦))) → ∃𝑛 ∈ ℕ 𝑥 < 𝑛)
11 nnre 11637 . . . . . . 7 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
12 simplll 771 . . . . . . . . . . . 12 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → 𝑥 ∈ ℝ)
13 simpllr 772 . . . . . . . . . . . 12 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → 𝑛 ∈ ℝ)
14 normcl 28818 . . . . . . . . . . . . 13 (𝑦 ∈ ℋ → (norm𝑦) ∈ ℝ)
1514adantl 482 . . . . . . . . . . . 12 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (norm𝑦) ∈ ℝ)
16 normge0 28819 . . . . . . . . . . . . 13 (𝑦 ∈ ℋ → 0 ≤ (norm𝑦))
1716adantl 482 . . . . . . . . . . . 12 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → 0 ≤ (norm𝑦))
18 ltle 10721 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (𝑥 < 𝑛𝑥𝑛))
1918imp 407 . . . . . . . . . . . . 13 (((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) → 𝑥𝑛)
2019adantr 481 . . . . . . . . . . . 12 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → 𝑥𝑛)
2112, 13, 15, 17, 20lemul1ad 11571 . . . . . . . . . . 11 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (𝑥 · (norm𝑦)) ≤ (𝑛 · (norm𝑦)))
22 lncon.4 . . . . . . . . . . . . 13 (𝑦 ∈ ℋ → (𝑁‘(𝑇𝑦)) ∈ ℝ)
2322adantl 482 . . . . . . . . . . . 12 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (𝑁‘(𝑇𝑦)) ∈ ℝ)
24 simpll 763 . . . . . . . . . . . . 13 (((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) → 𝑥 ∈ ℝ)
25 remulcl 10614 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ ∧ (norm𝑦) ∈ ℝ) → (𝑥 · (norm𝑦)) ∈ ℝ)
2624, 14, 25syl2an 595 . . . . . . . . . . . 12 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (𝑥 · (norm𝑦)) ∈ ℝ)
27 simplr 765 . . . . . . . . . . . . 13 (((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) → 𝑛 ∈ ℝ)
28 remulcl 10614 . . . . . . . . . . . . 13 ((𝑛 ∈ ℝ ∧ (norm𝑦) ∈ ℝ) → (𝑛 · (norm𝑦)) ∈ ℝ)
2927, 14, 28syl2an 595 . . . . . . . . . . . 12 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (𝑛 · (norm𝑦)) ∈ ℝ)
30 letr 10726 . . . . . . . . . . . 12 (((𝑁‘(𝑇𝑦)) ∈ ℝ ∧ (𝑥 · (norm𝑦)) ∈ ℝ ∧ (𝑛 · (norm𝑦)) ∈ ℝ) → (((𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)) ∧ (𝑥 · (norm𝑦)) ≤ (𝑛 · (norm𝑦))) → (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))))
3123, 26, 29, 30syl3anc 1365 . . . . . . . . . . 11 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (((𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)) ∧ (𝑥 · (norm𝑦)) ≤ (𝑛 · (norm𝑦))) → (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))))
3221, 31mpan2d 690 . . . . . . . . . 10 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → ((𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)) → (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))))
3332ralimdva 3181 . . . . . . . . 9 (((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) → (∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)) → ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))))
3433impancom 452 . . . . . . . 8 (((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦))) → (𝑥 < 𝑛 → ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))))
3534an32s 648 . . . . . . 7 (((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦))) ∧ 𝑛 ∈ ℝ) → (𝑥 < 𝑛 → ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))))
3611, 35sylan2 592 . . . . . 6 (((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦))) ∧ 𝑛 ∈ ℕ) → (𝑥 < 𝑛 → ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))))
3736reximdva 3278 . . . . 5 ((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦))) → (∃𝑛 ∈ ℕ 𝑥 < 𝑛 → ∃𝑛 ∈ ℕ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))))
3810, 37mpd 15 . . . 4 ((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦))) → ∃𝑛 ∈ ℕ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦)))
3938rexlimiva 3285 . . 3 (∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)) → ∃𝑛 ∈ ℕ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦)))
40 simprr 769 . . . . . . . 8 (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → 𝑧 ∈ ℝ+)
41 simpll 763 . . . . . . . . 9 (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → 𝑛 ∈ ℕ)
4241nnrpd 12422 . . . . . . . 8 (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → 𝑛 ∈ ℝ+)
4340, 42rpdivcld 12441 . . . . . . 7 (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → (𝑧 / 𝑛) ∈ ℝ+)
44 simprr 769 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑤 ∈ ℋ)
45 simprll 775 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑥 ∈ ℋ)
46 hvsubcl 28710 . . . . . . . . . . . . . 14 ((𝑤 ∈ ℋ ∧ 𝑥 ∈ ℋ) → (𝑤 𝑥) ∈ ℋ)
4744, 45, 46syl2anc 584 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑤 𝑥) ∈ ℋ)
48 2fveq3 6671 . . . . . . . . . . . . . . 15 (𝑦 = (𝑤 𝑥) → (𝑁‘(𝑇𝑦)) = (𝑁‘(𝑇‘(𝑤 𝑥))))
49 fveq2 6666 . . . . . . . . . . . . . . . 16 (𝑦 = (𝑤 𝑥) → (norm𝑦) = (norm‘(𝑤 𝑥)))
5049oveq2d 7167 . . . . . . . . . . . . . . 15 (𝑦 = (𝑤 𝑥) → (𝑛 · (norm𝑦)) = (𝑛 · (norm‘(𝑤 𝑥))))
5148, 50breq12d 5075 . . . . . . . . . . . . . 14 (𝑦 = (𝑤 𝑥) → ((𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦)) ↔ (𝑁‘(𝑇‘(𝑤 𝑥))) ≤ (𝑛 · (norm‘(𝑤 𝑥)))))
5251rspcva 3624 . . . . . . . . . . . . 13 (((𝑤 𝑥) ∈ ℋ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) → (𝑁‘(𝑇‘(𝑤 𝑥))) ≤ (𝑛 · (norm‘(𝑤 𝑥))))
5347, 52sylan 580 . . . . . . . . . . . 12 (((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) → (𝑁‘(𝑇‘(𝑤 𝑥))) ≤ (𝑛 · (norm‘(𝑤 𝑥))))
5453an32s 648 . . . . . . . . . . 11 (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑁‘(𝑇‘(𝑤 𝑥))) ≤ (𝑛 · (norm‘(𝑤 𝑥))))
5548eleq1d 2901 . . . . . . . . . . . . . . 15 (𝑦 = (𝑤 𝑥) → ((𝑁‘(𝑇𝑦)) ∈ ℝ ↔ (𝑁‘(𝑇‘(𝑤 𝑥))) ∈ ℝ))
5655, 22vtoclga 3578 . . . . . . . . . . . . . 14 ((𝑤 𝑥) ∈ ℋ → (𝑁‘(𝑇‘(𝑤 𝑥))) ∈ ℝ)
5747, 56syl 17 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑁‘(𝑇‘(𝑤 𝑥))) ∈ ℝ)
5811adantr 481 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑛 ∈ ℝ)
59 normcl 28818 . . . . . . . . . . . . . . 15 ((𝑤 𝑥) ∈ ℋ → (norm‘(𝑤 𝑥)) ∈ ℝ)
6047, 59syl 17 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (norm‘(𝑤 𝑥)) ∈ ℝ)
61 remulcl 10614 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℝ ∧ (norm‘(𝑤 𝑥)) ∈ ℝ) → (𝑛 · (norm‘(𝑤 𝑥))) ∈ ℝ)
6258, 60, 61syl2anc 584 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑛 · (norm‘(𝑤 𝑥))) ∈ ℝ)
63 simprlr 776 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑧 ∈ ℝ+)
6463rpred 12424 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑧 ∈ ℝ)
65 lelttr 10723 . . . . . . . . . . . . 13 (((𝑁‘(𝑇‘(𝑤 𝑥))) ∈ ℝ ∧ (𝑛 · (norm‘(𝑤 𝑥))) ∈ ℝ ∧ 𝑧 ∈ ℝ) → (((𝑁‘(𝑇‘(𝑤 𝑥))) ≤ (𝑛 · (norm‘(𝑤 𝑥))) ∧ (𝑛 · (norm‘(𝑤 𝑥))) < 𝑧) → (𝑁‘(𝑇‘(𝑤 𝑥))) < 𝑧))
6657, 62, 64, 65syl3anc 1365 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (((𝑁‘(𝑇‘(𝑤 𝑥))) ≤ (𝑛 · (norm‘(𝑤 𝑥))) ∧ (𝑛 · (norm‘(𝑤 𝑥))) < 𝑧) → (𝑁‘(𝑇‘(𝑤 𝑥))) < 𝑧))
6766adantlr 711 . . . . . . . . . . 11 (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (((𝑁‘(𝑇‘(𝑤 𝑥))) ≤ (𝑛 · (norm‘(𝑤 𝑥))) ∧ (𝑛 · (norm‘(𝑤 𝑥))) < 𝑧) → (𝑁‘(𝑇‘(𝑤 𝑥))) < 𝑧))
6854, 67mpand 691 . . . . . . . . . 10 (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → ((𝑛 · (norm‘(𝑤 𝑥))) < 𝑧 → (𝑁‘(𝑇‘(𝑤 𝑥))) < 𝑧))
69 nnrp 12393 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+)
7069rpregt0d 12430 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → (𝑛 ∈ ℝ ∧ 0 < 𝑛))
7170adantr 481 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑛 ∈ ℝ ∧ 0 < 𝑛))
72 ltmuldiv2 11506 . . . . . . . . . . . 12 (((norm‘(𝑤 𝑥)) ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (𝑛 ∈ ℝ ∧ 0 < 𝑛)) → ((𝑛 · (norm‘(𝑤 𝑥))) < 𝑧 ↔ (norm‘(𝑤 𝑥)) < (𝑧 / 𝑛)))
7360, 64, 71, 72syl3anc 1365 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → ((𝑛 · (norm‘(𝑤 𝑥))) < 𝑧 ↔ (norm‘(𝑤 𝑥)) < (𝑧 / 𝑛)))
7473adantlr 711 . . . . . . . . . 10 (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → ((𝑛 · (norm‘(𝑤 𝑥))) < 𝑧 ↔ (norm‘(𝑤 𝑥)) < (𝑧 / 𝑛)))
75 lncon.5 . . . . . . . . . . . . . 14 ((𝑤 ∈ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘(𝑤 𝑥)) = ((𝑇𝑤)𝑀(𝑇𝑥)))
7644, 45, 75syl2anc 584 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑇‘(𝑤 𝑥)) = ((𝑇𝑤)𝑀(𝑇𝑥)))
7776adantlr 711 . . . . . . . . . . . 12 (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑇‘(𝑤 𝑥)) = ((𝑇𝑤)𝑀(𝑇𝑥)))
7877fveq2d 6670 . . . . . . . . . . 11 (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑁‘(𝑇‘(𝑤 𝑥))) = (𝑁‘((𝑇𝑤)𝑀(𝑇𝑥))))
7978breq1d 5072 . . . . . . . . . 10 (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → ((𝑁‘(𝑇‘(𝑤 𝑥))) < 𝑧 ↔ (𝑁‘((𝑇𝑤)𝑀(𝑇𝑥))) < 𝑧))
8068, 74, 793imtr3d 294 . . . . . . . . 9 (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → ((norm‘(𝑤 𝑥)) < (𝑧 / 𝑛) → (𝑁‘((𝑇𝑤)𝑀(𝑇𝑥))) < 𝑧))
8180anassrs 468 . . . . . . . 8 ((((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) ∧ 𝑤 ∈ ℋ) → ((norm‘(𝑤 𝑥)) < (𝑧 / 𝑛) → (𝑁‘((𝑇𝑤)𝑀(𝑇𝑥))) < 𝑧))
8281ralrimiva 3186 . . . . . . 7 (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → ∀𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < (𝑧 / 𝑛) → (𝑁‘((𝑇𝑤)𝑀(𝑇𝑥))) < 𝑧))
83 breq2 5066 . . . . . . . 8 (𝑦 = (𝑧 / 𝑛) → ((norm‘(𝑤 𝑥)) < 𝑦 ↔ (norm‘(𝑤 𝑥)) < (𝑧 / 𝑛)))
8483rspceaimv 3631 . . . . . . 7 (((𝑧 / 𝑛) ∈ ℝ+ ∧ ∀𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < (𝑧 / 𝑛) → (𝑁‘((𝑇𝑤)𝑀(𝑇𝑥))) < 𝑧)) → ∃𝑦 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑦 → (𝑁‘((𝑇𝑤)𝑀(𝑇𝑥))) < 𝑧))
8543, 82, 84syl2anc 584 . . . . . 6 (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → ∃𝑦 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑦 → (𝑁‘((𝑇𝑤)𝑀(𝑇𝑥))) < 𝑧))
8685ralrimivva 3195 . . . . 5 ((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) → ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℝ+𝑦 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑦 → (𝑁‘((𝑇𝑤)𝑀(𝑇𝑥))) < 𝑧))
8786rexlimiva 3285 . . . 4 (∃𝑛 ∈ ℕ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦)) → ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℝ+𝑦 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑦 → (𝑁‘((𝑇𝑤)𝑀(𝑇𝑥))) < 𝑧))
88 lncon.3 . . . 4 (𝑇𝐶 ↔ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℝ+𝑦 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑦 → (𝑁‘((𝑇𝑤)𝑀(𝑇𝑥))) < 𝑧))
8987, 88sylibr 235 . . 3 (∃𝑛 ∈ ℕ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦)) → 𝑇𝐶)
9039, 89syl 17 . 2 (∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)) → 𝑇𝐶)
918, 90impbii 210 1 (𝑇𝐶 ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wral 3142  wrex 3143   class class class wbr 5062  cfv 6351  (class class class)co 7151  cr 10528  0cc0 10529   · cmul 10534   < clt 10667  cle 10668   / cdiv 11289  cn 11630  +crp 12382  chba 28612  normcno 28616   cmv 28618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607  ax-hfvadd 28693  ax-hv0cl 28696  ax-hfvmul 28698  ax-hvmul0 28703  ax-hfi 28772  ax-his1 28775  ax-his3 28777  ax-his4 28778
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8282  df-en 8502  df-dom 8503  df-sdom 8504  df-sup 8898  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-rp 12383  df-seq 13363  df-exp 13423  df-cj 14451  df-re 14452  df-im 14453  df-sqrt 14587  df-hnorm 28661  df-hvsub 28664
This theorem is referenced by:  lnopconi  29727  lnfnconi  29748
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