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Theorem lnconi 30975
Description: Lemma for lnopconi 30976 and lnfnconi 30997. (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 3143 . . 3 (𝑇𝐶 → ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑆 · (norm𝑦)))
4 oveq1 7364 . . . . . 6 (𝑥 = 𝑆 → (𝑥 · (norm𝑦)) = (𝑆 · (norm𝑦)))
54breq2d 5117 . . . . 5 (𝑥 = 𝑆 → ((𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)) ↔ (𝑁‘(𝑇𝑦)) ≤ (𝑆 · (norm𝑦))))
65ralbidv 3174 . . . 4 (𝑥 = 𝑆 → (∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)) ↔ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑆 · (norm𝑦))))
76rspcev 3581 . . 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 12410 . . . . . 6 (𝑥 ∈ ℝ → ∃𝑛 ∈ ℕ 𝑥 < 𝑛)
109adantr 481 . . . . 5 ((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦))) → ∃𝑛 ∈ ℕ 𝑥 < 𝑛)
11 nnre 12160 . . . . . . 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 30067 . . . . . . . . . . . . 13 (𝑦 ∈ ℋ → (norm𝑦) ∈ ℝ)
1514adantl 482 . . . . . . . . . . . 12 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (norm𝑦) ∈ ℝ)
16 normge0 30068 . . . . . . . . . . . . 13 (𝑦 ∈ ℋ → 0 ≤ (norm𝑦))
1716adantl 482 . . . . . . . . . . . 12 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → 0 ≤ (norm𝑦))
18 ltle 11243 . . . . . . . . . . . . . 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 12094 . . . . . . . . . . 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 765 . . . . . . . . . . . . 13 (((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) → 𝑥 ∈ ℝ)
25 remulcl 11136 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ ∧ (norm𝑦) ∈ ℝ) → (𝑥 · (norm𝑦)) ∈ ℝ)
2624, 14, 25syl2an 596 . . . . . . . . . . . 12 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (𝑥 · (norm𝑦)) ∈ ℝ)
27 simplr 767 . . . . . . . . . . . . 13 (((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) → 𝑛 ∈ ℝ)
28 remulcl 11136 . . . . . . . . . . . . 13 ((𝑛 ∈ ℝ ∧ (norm𝑦) ∈ ℝ) → (𝑛 · (norm𝑦)) ∈ ℝ)
2927, 14, 28syl2an 596 . . . . . . . . . . . 12 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (𝑛 · (norm𝑦)) ∈ ℝ)
30 letr 11249 . . . . . . . . . . . 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 1371 . . . . . . . . . . 11 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (((𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)) ∧ (𝑥 · (norm𝑦)) ≤ (𝑛 · (norm𝑦))) → (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))))
3221, 31mpan2d 692 . . . . . . . . . 10 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → ((𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)) → (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))))
3332ralimdva 3164 . . . . . . . . 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 650 . . . . . . 7 (((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦))) ∧ 𝑛 ∈ ℝ) → (𝑥 < 𝑛 → ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))))
3611, 35sylan2 593 . . . . . 6 (((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦))) ∧ 𝑛 ∈ ℕ) → (𝑥 < 𝑛 → ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))))
3736reximdva 3165 . . . . 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 3144 . . 3 (∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)) → ∃𝑛 ∈ ℕ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦)))
40 simprr 771 . . . . . . . 8 (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → 𝑧 ∈ ℝ+)
41 simpll 765 . . . . . . . . 9 (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → 𝑛 ∈ ℕ)
4241nnrpd 12955 . . . . . . . 8 (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → 𝑛 ∈ ℝ+)
4340, 42rpdivcld 12974 . . . . . . 7 (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) ∧ (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 29959 . . . . . . . . . . . . . 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 6847 . . . . . . . . . . . . . . 15 (𝑦 = (𝑤 𝑥) → (𝑁‘(𝑇𝑦)) = (𝑁‘(𝑇‘(𝑤 𝑥))))
49 fveq2 6842 . . . . . . . . . . . . . . . 16 (𝑦 = (𝑤 𝑥) → (norm𝑦) = (norm‘(𝑤 𝑥)))
5049oveq2d 7373 . . . . . . . . . . . . . . 15 (𝑦 = (𝑤 𝑥) → (𝑛 · (norm𝑦)) = (𝑛 · (norm‘(𝑤 𝑥))))
5148, 50breq12d 5118 . . . . . . . . . . . . . 14 (𝑦 = (𝑤 𝑥) → ((𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦)) ↔ (𝑁‘(𝑇‘(𝑤 𝑥))) ≤ (𝑛 · (norm‘(𝑤 𝑥)))))
5251rspcva 3579 . . . . . . . . . . . . 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 650 . . . . . . . . . . 11 (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑁‘(𝑇‘(𝑤 𝑥))) ≤ (𝑛 · (norm‘(𝑤 𝑥))))
5548eleq1d 2822 . . . . . . . . . . . . . . 15 (𝑦 = (𝑤 𝑥) → ((𝑁‘(𝑇𝑦)) ∈ ℝ ↔ (𝑁‘(𝑇‘(𝑤 𝑥))) ∈ ℝ))
5655, 22vtoclga 3534 . . . . . . . . . . . . . 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 30067 . . . . . . . . . . . . . . 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 11136 . . . . . . . . . . . . . 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 778 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑧 ∈ ℝ+)
6463rpred 12957 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑧 ∈ ℝ)
65 lelttr 11245 . . . . . . . . . . . . 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 1371 . . . . . . . . . . . 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 713 . . . . . . . . . . 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 693 . . . . . . . . . 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 12926 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+)
7069rpregt0d 12963 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → (𝑛 ∈ ℝ ∧ 0 < 𝑛))
7170adantr 481 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑛 ∈ ℝ ∧ 0 < 𝑛))
72 ltmuldiv2 12029 . . . . . . . . . . . 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 1371 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → ((𝑛 · (norm‘(𝑤 𝑥))) < 𝑧 ↔ (norm‘(𝑤 𝑥)) < (𝑧 / 𝑛)))
7473adantlr 713 . . . . . . . . . 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 713 . . . . . . . . . . . 12 (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑇‘(𝑤 𝑥)) = ((𝑇𝑤)𝑀(𝑇𝑥)))
7877fveq2d 6846 . . . . . . . . . . 11 (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑁‘(𝑇‘(𝑤 𝑥))) = (𝑁‘((𝑇𝑤)𝑀(𝑇𝑥))))
7978breq1d 5115 . . . . . . . . . 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 292 . . . . . . . . 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 3143 . . . . . . 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 5109 . . . . . . . 8 (𝑦 = (𝑧 / 𝑛) → ((norm‘(𝑤 𝑥)) < 𝑦 ↔ (norm‘(𝑤 𝑥)) < (𝑧 / 𝑛)))
8483rspceaimv 3585 . . . . . . 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 3197 . . . . 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 3144 . . . 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 233 . . 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 208 1 (𝑇𝐶 ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064  wrex 3073   class class class wbr 5105  cfv 6496  (class class class)co 7357  cr 11050  0cc0 11051   · cmul 11056   < clt 11189  cle 11190   / cdiv 11812  cn 12153  +crp 12915  chba 29861  normcno 29865   cmv 29867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-hfvadd 29942  ax-hv0cl 29945  ax-hfvmul 29947  ax-hvmul0 29952  ax-hfi 30021  ax-his1 30024  ax-his3 30026  ax-his4 30027
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9378  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-hnorm 29910  df-hvsub 29913
This theorem is referenced by:  lnopconi  30976  lnfnconi  30997
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