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Theorem lnconi 29371
Description: Lemma for lnopconi 29372 and lnfnconi 29393. (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 3113 . . 3 (𝑇𝐶 → ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑆 · (norm𝑦)))
4 oveq1 6853 . . . . . 6 (𝑥 = 𝑆 → (𝑥 · (norm𝑦)) = (𝑆 · (norm𝑦)))
54breq2d 4823 . . . . 5 (𝑥 = 𝑆 → ((𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)) ↔ (𝑁‘(𝑇𝑦)) ≤ (𝑆 · (norm𝑦))))
65ralbidv 3133 . . . 4 (𝑥 = 𝑆 → (∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)) ↔ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑆 · (norm𝑦))))
76rspcev 3462 . . 3 ((𝑆 ∈ ℝ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑆 · (norm𝑦))) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)))
81, 3, 7syl2anc 579 . 2 (𝑇𝐶 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)))
9 arch 11540 . . . . . 6 (𝑥 ∈ ℝ → ∃𝑛 ∈ ℕ 𝑥 < 𝑛)
109adantr 472 . . . . 5 ((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦))) → ∃𝑛 ∈ ℕ 𝑥 < 𝑛)
11 nnre 11287 . . . . . . 7 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
12 simplll 791 . . . . . . . . . . . 12 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → 𝑥 ∈ ℝ)
13 simpllr 793 . . . . . . . . . . . 12 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → 𝑛 ∈ ℝ)
14 normcl 28461 . . . . . . . . . . . . 13 (𝑦 ∈ ℋ → (norm𝑦) ∈ ℝ)
1514adantl 473 . . . . . . . . . . . 12 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (norm𝑦) ∈ ℝ)
16 normge0 28462 . . . . . . . . . . . . 13 (𝑦 ∈ ℋ → 0 ≤ (norm𝑦))
1716adantl 473 . . . . . . . . . . . 12 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → 0 ≤ (norm𝑦))
18 ltle 10385 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (𝑥 < 𝑛𝑥𝑛))
1918imp 395 . . . . . . . . . . . . 13 (((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) → 𝑥𝑛)
2019adantr 472 . . . . . . . . . . . 12 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → 𝑥𝑛)
2112, 13, 15, 17, 20lemul1ad 11222 . . . . . . . . . . 11 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (𝑥 · (norm𝑦)) ≤ (𝑛 · (norm𝑦)))
22 lncon.4 . . . . . . . . . . . . 13 (𝑦 ∈ ℋ → (𝑁‘(𝑇𝑦)) ∈ ℝ)
2322adantl 473 . . . . . . . . . . . 12 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (𝑁‘(𝑇𝑦)) ∈ ℝ)
24 simpll 783 . . . . . . . . . . . . 13 (((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) → 𝑥 ∈ ℝ)
25 remulcl 10278 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ ∧ (norm𝑦) ∈ ℝ) → (𝑥 · (norm𝑦)) ∈ ℝ)
2624, 14, 25syl2an 589 . . . . . . . . . . . 12 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (𝑥 · (norm𝑦)) ∈ ℝ)
27 simplr 785 . . . . . . . . . . . . 13 (((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) → 𝑛 ∈ ℝ)
28 remulcl 10278 . . . . . . . . . . . . 13 ((𝑛 ∈ ℝ ∧ (norm𝑦) ∈ ℝ) → (𝑛 · (norm𝑦)) ∈ ℝ)
2927, 14, 28syl2an 589 . . . . . . . . . . . 12 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (𝑛 · (norm𝑦)) ∈ ℝ)
30 letr 10390 . . . . . . . . . . . 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 1490 . . . . . . . . . . 11 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (((𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)) ∧ (𝑥 · (norm𝑦)) ≤ (𝑛 · (norm𝑦))) → (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))))
3221, 31mpan2d 685 . . . . . . . . . 10 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → ((𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)) → (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))))
3332ralimdva 3109 . . . . . . . . 9 (((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) → (∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)) → ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))))
3433impancom 443 . . . . . . . 8 (((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦))) → (𝑥 < 𝑛 → ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))))
3534an32s 642 . . . . . . 7 (((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦))) ∧ 𝑛 ∈ ℝ) → (𝑥 < 𝑛 → ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))))
3611, 35sylan2 586 . . . . . 6 (((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦))) ∧ 𝑛 ∈ ℕ) → (𝑥 < 𝑛 → ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))))
3736reximdva 3163 . . . . 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 3175 . . 3 (∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)) → ∃𝑛 ∈ ℕ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦)))
40 simprr 789 . . . . . . . 8 (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → 𝑧 ∈ ℝ+)
41 simpll 783 . . . . . . . . 9 (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → 𝑛 ∈ ℕ)
4241nnrpd 12075 . . . . . . . 8 (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → 𝑛 ∈ ℝ+)
4340, 42rpdivcld 12094 . . . . . . 7 (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → (𝑧 / 𝑛) ∈ ℝ+)
44 simprr 789 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑤 ∈ ℋ)
45 simprll 797 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑥 ∈ ℋ)
46 hvsubcl 28353 . . . . . . . . . . . . . 14 ((𝑤 ∈ ℋ ∧ 𝑥 ∈ ℋ) → (𝑤 𝑥) ∈ ℋ)
4744, 45, 46syl2anc 579 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑤 𝑥) ∈ ℋ)
48 2fveq3 6384 . . . . . . . . . . . . . . 15 (𝑦 = (𝑤 𝑥) → (𝑁‘(𝑇𝑦)) = (𝑁‘(𝑇‘(𝑤 𝑥))))
49 fveq2 6379 . . . . . . . . . . . . . . . 16 (𝑦 = (𝑤 𝑥) → (norm𝑦) = (norm‘(𝑤 𝑥)))
5049oveq2d 6862 . . . . . . . . . . . . . . 15 (𝑦 = (𝑤 𝑥) → (𝑛 · (norm𝑦)) = (𝑛 · (norm‘(𝑤 𝑥))))
5148, 50breq12d 4824 . . . . . . . . . . . . . 14 (𝑦 = (𝑤 𝑥) → ((𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦)) ↔ (𝑁‘(𝑇‘(𝑤 𝑥))) ≤ (𝑛 · (norm‘(𝑤 𝑥)))))
5251rspcva 3460 . . . . . . . . . . . . 13 (((𝑤 𝑥) ∈ ℋ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) → (𝑁‘(𝑇‘(𝑤 𝑥))) ≤ (𝑛 · (norm‘(𝑤 𝑥))))
5347, 52sylan 575 . . . . . . . . . . . 12 (((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) → (𝑁‘(𝑇‘(𝑤 𝑥))) ≤ (𝑛 · (norm‘(𝑤 𝑥))))
5453an32s 642 . . . . . . . . . . 11 (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑁‘(𝑇‘(𝑤 𝑥))) ≤ (𝑛 · (norm‘(𝑤 𝑥))))
5548eleq1d 2829 . . . . . . . . . . . . . . 15 (𝑦 = (𝑤 𝑥) → ((𝑁‘(𝑇𝑦)) ∈ ℝ ↔ (𝑁‘(𝑇‘(𝑤 𝑥))) ∈ ℝ))
5655, 22vtoclga 3425 . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑛 ∈ ℝ)
59 normcl 28461 . . . . . . . . . . . . . . 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 10278 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℝ ∧ (norm‘(𝑤 𝑥)) ∈ ℝ) → (𝑛 · (norm‘(𝑤 𝑥))) ∈ ℝ)
6258, 60, 61syl2anc 579 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑛 · (norm‘(𝑤 𝑥))) ∈ ℝ)
63 simprlr 798 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑧 ∈ ℝ+)
6463rpred 12077 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑧 ∈ ℝ)
65 lelttr 10387 . . . . . . . . . . . . 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 1490 . . . . . . . . . . . 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 706 . . . . . . . . . . 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 686 . . . . . . . . . 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 12048 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+)
7069rpregt0d 12083 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → (𝑛 ∈ ℝ ∧ 0 < 𝑛))
7170adantr 472 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑛 ∈ ℝ ∧ 0 < 𝑛))
72 ltmuldiv2 11156 . . . . . . . . . . . 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 1490 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → ((𝑛 · (norm‘(𝑤 𝑥))) < 𝑧 ↔ (norm‘(𝑤 𝑥)) < (𝑧 / 𝑛)))
7473adantlr 706 . . . . . . . . . 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 579 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑇‘(𝑤 𝑥)) = ((𝑇𝑤)𝑀(𝑇𝑥)))
7776adantlr 706 . . . . . . . . . . . 12 (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑇‘(𝑤 𝑥)) = ((𝑇𝑤)𝑀(𝑇𝑥)))
7877fveq2d 6383 . . . . . . . . . . 11 (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑁‘(𝑇‘(𝑤 𝑥))) = (𝑁‘((𝑇𝑤)𝑀(𝑇𝑥))))
7978breq1d 4821 . . . . . . . . . 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 284 . . . . . . . . 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 459 . . . . . . . 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 3113 . . . . . . 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 4815 . . . . . . . 8 (𝑦 = (𝑧 / 𝑛) → ((norm‘(𝑤 𝑥)) < 𝑦 ↔ (norm‘(𝑤 𝑥)) < (𝑧 / 𝑛)))
8483rspceaimv 3470 . . . . . . 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 579 . . . . . 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 3118 . . . . 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 3175 . . . 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 225 . . 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 200 1 (𝑇𝐶 ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wral 3055  wrex 3056   class class class wbr 4811  cfv 6070  (class class class)co 6846  cr 10192  0cc0 10193   · cmul 10198   < clt 10332  cle 10333   / cdiv 10943  cn 11279  +crp 12035  chba 28255  normcno 28259   cmv 28261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-cnex 10249  ax-resscn 10250  ax-1cn 10251  ax-icn 10252  ax-addcl 10253  ax-addrcl 10254  ax-mulcl 10255  ax-mulrcl 10256  ax-mulcom 10257  ax-addass 10258  ax-mulass 10259  ax-distr 10260  ax-i2m1 10261  ax-1ne0 10262  ax-1rid 10263  ax-rnegex 10264  ax-rrecex 10265  ax-cnre 10266  ax-pre-lttri 10267  ax-pre-lttrn 10268  ax-pre-ltadd 10269  ax-pre-mulgt0 10270  ax-pre-sup 10271  ax-hfvadd 28336  ax-hv0cl 28339  ax-hfvmul 28341  ax-hvmul0 28346  ax-hfi 28415  ax-his1 28418  ax-his3 28420  ax-his4 28421
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-om 7268  df-2nd 7371  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-er 7951  df-en 8165  df-dom 8166  df-sdom 8167  df-sup 8559  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-sub 10527  df-neg 10528  df-div 10944  df-nn 11280  df-2 11340  df-3 11341  df-n0 11544  df-z 11630  df-uz 11894  df-rp 12036  df-seq 13016  df-exp 13075  df-cj 14140  df-re 14141  df-im 14142  df-sqrt 14276  df-hnorm 28304  df-hvsub 28307
This theorem is referenced by:  lnopconi  29372  lnfnconi  29393
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