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Theorem lnconi 29020
Description: Lemma for lnopconi 29021 and lnfnconi 29042. (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 2995 . . 3 (𝑇𝐶 → ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑆 · (norm𝑦)))
4 oveq1 6697 . . . . . 6 (𝑥 = 𝑆 → (𝑥 · (norm𝑦)) = (𝑆 · (norm𝑦)))
54breq2d 4697 . . . . 5 (𝑥 = 𝑆 → ((𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)) ↔ (𝑁‘(𝑇𝑦)) ≤ (𝑆 · (norm𝑦))))
65ralbidv 3015 . . . 4 (𝑥 = 𝑆 → (∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)) ↔ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑆 · (norm𝑦))))
76rspcev 3340 . . 3 ((𝑆 ∈ ℝ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑆 · (norm𝑦))) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)))
81, 3, 7syl2anc 694 . 2 (𝑇𝐶 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)))
9 arch 11327 . . . . . 6 (𝑥 ∈ ℝ → ∃𝑛 ∈ ℕ 𝑥 < 𝑛)
109adantr 480 . . . . 5 ((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦))) → ∃𝑛 ∈ ℕ 𝑥 < 𝑛)
11 nnre 11065 . . . . . . 7 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
12 simplll 813 . . . . . . . . . . . 12 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → 𝑥 ∈ ℝ)
13 simpllr 815 . . . . . . . . . . . 12 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → 𝑛 ∈ ℝ)
14 normcl 28110 . . . . . . . . . . . . 13 (𝑦 ∈ ℋ → (norm𝑦) ∈ ℝ)
1514adantl 481 . . . . . . . . . . . 12 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (norm𝑦) ∈ ℝ)
16 normge0 28111 . . . . . . . . . . . . 13 (𝑦 ∈ ℋ → 0 ≤ (norm𝑦))
1716adantl 481 . . . . . . . . . . . 12 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → 0 ≤ (norm𝑦))
18 ltle 10164 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (𝑥 < 𝑛𝑥𝑛))
1918imp 444 . . . . . . . . . . . . 13 (((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) → 𝑥𝑛)
2019adantr 480 . . . . . . . . . . . 12 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → 𝑥𝑛)
2112, 13, 15, 17, 20lemul1ad 11001 . . . . . . . . . . 11 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (𝑥 · (norm𝑦)) ≤ (𝑛 · (norm𝑦)))
22 lncon.4 . . . . . . . . . . . . 13 (𝑦 ∈ ℋ → (𝑁‘(𝑇𝑦)) ∈ ℝ)
2322adantl 481 . . . . . . . . . . . 12 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (𝑁‘(𝑇𝑦)) ∈ ℝ)
24 simpll 805 . . . . . . . . . . . . 13 (((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) → 𝑥 ∈ ℝ)
25 remulcl 10059 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ ∧ (norm𝑦) ∈ ℝ) → (𝑥 · (norm𝑦)) ∈ ℝ)
2624, 14, 25syl2an 493 . . . . . . . . . . . 12 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (𝑥 · (norm𝑦)) ∈ ℝ)
27 simplr 807 . . . . . . . . . . . . 13 (((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) → 𝑛 ∈ ℝ)
28 remulcl 10059 . . . . . . . . . . . . 13 ((𝑛 ∈ ℝ ∧ (norm𝑦) ∈ ℝ) → (𝑛 · (norm𝑦)) ∈ ℝ)
2927, 14, 28syl2an 493 . . . . . . . . . . . 12 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (𝑛 · (norm𝑦)) ∈ ℝ)
30 letr 10169 . . . . . . . . . . . 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 1366 . . . . . . . . . . 11 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → (((𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)) ∧ (𝑥 · (norm𝑦)) ≤ (𝑛 · (norm𝑦))) → (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))))
3221, 31mpan2d 710 . . . . . . . . . 10 ((((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) ∧ 𝑦 ∈ ℋ) → ((𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)) → (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))))
3332ralimdva 2991 . . . . . . . . 9 (((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ 𝑥 < 𝑛) → (∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)) → ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))))
3433impancom 455 . . . . . . . 8 (((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦))) → (𝑥 < 𝑛 → ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))))
3534an32s 863 . . . . . . 7 (((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦))) ∧ 𝑛 ∈ ℝ) → (𝑥 < 𝑛 → ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))))
3611, 35sylan2 490 . . . . . 6 (((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦))) ∧ 𝑛 ∈ ℕ) → (𝑥 < 𝑛 → ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))))
3736reximdva 3046 . . . . 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 3057 . . 3 (∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)) → ∃𝑛 ∈ ℕ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦)))
40 simprr 811 . . . . . . . 8 (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → 𝑧 ∈ ℝ+)
41 simpll 805 . . . . . . . . 9 (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → 𝑛 ∈ ℕ)
4241nnrpd 11908 . . . . . . . 8 (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → 𝑛 ∈ ℝ+)
4340, 42rpdivcld 11927 . . . . . . 7 (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) ∧ (𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+)) → (𝑧 / 𝑛) ∈ ℝ+)
44 simprr 811 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑤 ∈ ℋ)
45 simprll 819 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑥 ∈ ℋ)
46 hvsubcl 28002 . . . . . . . . . . . . . 14 ((𝑤 ∈ ℋ ∧ 𝑥 ∈ ℋ) → (𝑤 𝑥) ∈ ℋ)
4744, 45, 46syl2anc 694 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑤 𝑥) ∈ ℋ)
48 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑦 = (𝑤 𝑥) → (𝑇𝑦) = (𝑇‘(𝑤 𝑥)))
4948fveq2d 6233 . . . . . . . . . . . . . . 15 (𝑦 = (𝑤 𝑥) → (𝑁‘(𝑇𝑦)) = (𝑁‘(𝑇‘(𝑤 𝑥))))
50 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑦 = (𝑤 𝑥) → (norm𝑦) = (norm‘(𝑤 𝑥)))
5150oveq2d 6706 . . . . . . . . . . . . . . 15 (𝑦 = (𝑤 𝑥) → (𝑛 · (norm𝑦)) = (𝑛 · (norm‘(𝑤 𝑥))))
5249, 51breq12d 4698 . . . . . . . . . . . . . 14 (𝑦 = (𝑤 𝑥) → ((𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦)) ↔ (𝑁‘(𝑇‘(𝑤 𝑥))) ≤ (𝑛 · (norm‘(𝑤 𝑥)))))
5352rspcva 3338 . . . . . . . . . . . . 13 (((𝑤 𝑥) ∈ ℋ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) → (𝑁‘(𝑇‘(𝑤 𝑥))) ≤ (𝑛 · (norm‘(𝑤 𝑥))))
5447, 53sylan 487 . . . . . . . . . . . 12 (((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) → (𝑁‘(𝑇‘(𝑤 𝑥))) ≤ (𝑛 · (norm‘(𝑤 𝑥))))
5554an32s 863 . . . . . . . . . . 11 (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑁‘(𝑇‘(𝑤 𝑥))) ≤ (𝑛 · (norm‘(𝑤 𝑥))))
5649eleq1d 2715 . . . . . . . . . . . . . . 15 (𝑦 = (𝑤 𝑥) → ((𝑁‘(𝑇𝑦)) ∈ ℝ ↔ (𝑁‘(𝑇‘(𝑤 𝑥))) ∈ ℝ))
5756, 22vtoclga 3303 . . . . . . . . . . . . . 14 ((𝑤 𝑥) ∈ ℋ → (𝑁‘(𝑇‘(𝑤 𝑥))) ∈ ℝ)
5847, 57syl 17 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑁‘(𝑇‘(𝑤 𝑥))) ∈ ℝ)
5911adantr 480 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑛 ∈ ℝ)
60 normcl 28110 . . . . . . . . . . . . . . 15 ((𝑤 𝑥) ∈ ℋ → (norm‘(𝑤 𝑥)) ∈ ℝ)
6147, 60syl 17 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (norm‘(𝑤 𝑥)) ∈ ℝ)
62 remulcl 10059 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℝ ∧ (norm‘(𝑤 𝑥)) ∈ ℝ) → (𝑛 · (norm‘(𝑤 𝑥))) ∈ ℝ)
6359, 61, 62syl2anc 694 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑛 · (norm‘(𝑤 𝑥))) ∈ ℝ)
64 simprlr 820 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑧 ∈ ℝ+)
6564rpred 11910 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → 𝑧 ∈ ℝ)
66 lelttr 10166 . . . . . . . . . . . . 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))
6758, 63, 65, 66syl3anc 1366 . . . . . . . . . . . 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))
6867adantlr 751 . . . . . . . . . . 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))
6955, 68mpand 711 . . . . . . . . . 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))
70 nnrp 11880 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+)
7170rpregt0d 11916 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → (𝑛 ∈ ℝ ∧ 0 < 𝑛))
7271adantr 480 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑛 ∈ ℝ ∧ 0 < 𝑛))
73 ltmuldiv2 10935 . . . . . . . . . . . 12 (((norm‘(𝑤 𝑥)) ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ (𝑛 ∈ ℝ ∧ 0 < 𝑛)) → ((𝑛 · (norm‘(𝑤 𝑥))) < 𝑧 ↔ (norm‘(𝑤 𝑥)) < (𝑧 / 𝑛)))
7461, 65, 72, 73syl3anc 1366 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → ((𝑛 · (norm‘(𝑤 𝑥))) < 𝑧 ↔ (norm‘(𝑤 𝑥)) < (𝑧 / 𝑛)))
7574adantlr 751 . . . . . . . . . 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)))
76 lncon.5 . . . . . . . . . . . . . 14 ((𝑤 ∈ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘(𝑤 𝑥)) = ((𝑇𝑤)𝑀(𝑇𝑥)))
7744, 45, 76syl2anc 694 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑇‘(𝑤 𝑥)) = ((𝑇𝑤)𝑀(𝑇𝑥)))
7877adantlr 751 . . . . . . . . . . . 12 (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑇‘(𝑤 𝑥)) = ((𝑇𝑤)𝑀(𝑇𝑥)))
7978fveq2d 6233 . . . . . . . . . . 11 (((𝑛 ∈ ℕ ∧ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦))) ∧ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℋ)) → (𝑁‘(𝑇‘(𝑤 𝑥))) = (𝑁‘((𝑇𝑤)𝑀(𝑇𝑥))))
8079breq1d 4695 . . . . . . . . . 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))
8169, 75, 803imtr3d 282 . . . . . . . . 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))
8281anassrs 681 . . . . . . . 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))
8382ralrimiva 2995 . . . . . . 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))
84 breq2 4689 . . . . . . . . . 10 (𝑦 = (𝑧 / 𝑛) → ((norm‘(𝑤 𝑥)) < 𝑦 ↔ (norm‘(𝑤 𝑥)) < (𝑧 / 𝑛)))
8584imbi1d 330 . . . . . . . . 9 (𝑦 = (𝑧 / 𝑛) → (((norm‘(𝑤 𝑥)) < 𝑦 → (𝑁‘((𝑇𝑤)𝑀(𝑇𝑥))) < 𝑧) ↔ ((norm‘(𝑤 𝑥)) < (𝑧 / 𝑛) → (𝑁‘((𝑇𝑤)𝑀(𝑇𝑥))) < 𝑧)))
8685ralbidv 3015 . . . . . . . 8 (𝑦 = (𝑧 / 𝑛) → (∀𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑦 → (𝑁‘((𝑇𝑤)𝑀(𝑇𝑥))) < 𝑧) ↔ ∀𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < (𝑧 / 𝑛) → (𝑁‘((𝑇𝑤)𝑀(𝑇𝑥))) < 𝑧)))
8786rspcev 3340 . . . . . . 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))
8843, 83, 87syl2anc 694 . . . . . 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))
8988ralrimivva 3000 . . . . 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))
9089rexlimiva 3057 . . . 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))
91 lncon.3 . . . 4 (𝑇𝐶 ↔ ∀𝑥 ∈ ℋ ∀𝑧 ∈ ℝ+𝑦 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑦 → (𝑁‘((𝑇𝑤)𝑀(𝑇𝑥))) < 𝑧))
9290, 91sylibr 224 . . 3 (∃𝑛 ∈ ℕ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑛 · (norm𝑦)) → 𝑇𝐶)
9339, 92syl 17 . 2 (∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)) → 𝑇𝐶)
948, 93impbii 199 1 (𝑇𝐶 ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℋ (𝑁‘(𝑇𝑦)) ≤ (𝑥 · (norm𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942   class class class wbr 4685  cfv 5926  (class class class)co 6690  cr 9973  0cc0 9974   · cmul 9979   < clt 10112  cle 10113   / cdiv 10722  cn 11058  +crp 11870  chil 27904  normcno 27908   cmv 27910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-hfvadd 27985  ax-hv0cl 27988  ax-hfvmul 27990  ax-hvmul0 27995  ax-hfi 28064  ax-his1 28067  ax-his3 28069  ax-his4 28070
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-sup 8389  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-seq 12842  df-exp 12901  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-hnorm 27953  df-hvsub 27956
This theorem is referenced by:  lnopconi  29021  lnfnconi  29042
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