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Theorem nmcexi 29730
Description: Lemma for nmcopexi 29731 and nmcfnexi 29755. The norm of a continuous linear Hilbert space operator or functional exists. Theorem 3.5(i) of [Beran] p. 99. (Contributed by Mario Carneiro, 17-Nov-2013.) (Proof shortened by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
nmcex.1 𝑦 ∈ ℝ+𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1)
nmcex.2 (𝑆𝑇) = sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}, ℝ*, < )
nmcex.3 (𝑥 ∈ ℋ → (𝑁‘(𝑇𝑥)) ∈ ℝ)
nmcex.4 (𝑁‘(𝑇‘0)) = 0
nmcex.5 (((𝑦 / 2) ∈ ℝ+𝑥 ∈ ℋ) → ((𝑦 / 2) · (𝑁‘(𝑇𝑥))) = (𝑁‘(𝑇‘((𝑦 / 2) · 𝑥))))
Assertion
Ref Expression
nmcexi (𝑆𝑇) ∈ ℝ
Distinct variable groups:   𝑥,𝑚,𝑦,𝑧,𝑁   𝑇,𝑚,𝑥,𝑦,𝑧
Allowed substitution hints:   𝑆(𝑥,𝑦,𝑧,𝑚)

Proof of Theorem nmcexi
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 nmcex.2 . . 3 (𝑆𝑇) = sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}, ℝ*, < )
2 nmcex.3 . . . . . . . . 9 (𝑥 ∈ ℋ → (𝑁‘(𝑇𝑥)) ∈ ℝ)
3 eleq1 2897 . . . . . . . . 9 (𝑚 = (𝑁‘(𝑇𝑥)) → (𝑚 ∈ ℝ ↔ (𝑁‘(𝑇𝑥)) ∈ ℝ))
42, 3syl5ibrcom 248 . . . . . . . 8 (𝑥 ∈ ℋ → (𝑚 = (𝑁‘(𝑇𝑥)) → 𝑚 ∈ ℝ))
54imp 407 . . . . . . 7 ((𝑥 ∈ ℋ ∧ 𝑚 = (𝑁‘(𝑇𝑥))) → 𝑚 ∈ ℝ)
65adantrl 712 . . . . . 6 ((𝑥 ∈ ℋ ∧ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))) → 𝑚 ∈ ℝ)
76rexlimiva 3278 . . . . 5 (∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥))) → 𝑚 ∈ ℝ)
87abssi 4043 . . . 4 {𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))} ⊆ ℝ
9 ax-hv0cl 28707 . . . . . . 7 0 ∈ ℋ
10 norm0 28832 . . . . . . . . 9 (norm‘0) = 0
11 0le1 11151 . . . . . . . . 9 0 ≤ 1
1210, 11eqbrtri 5078 . . . . . . . 8 (norm‘0) ≤ 1
13 nmcex.4 . . . . . . . . 9 (𝑁‘(𝑇‘0)) = 0
1413eqcomi 2827 . . . . . . . 8 0 = (𝑁‘(𝑇‘0))
1512, 14pm3.2i 471 . . . . . . 7 ((norm‘0) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘0)))
16 fveq2 6663 . . . . . . . . . 10 (𝑥 = 0 → (norm𝑥) = (norm‘0))
1716breq1d 5067 . . . . . . . . 9 (𝑥 = 0 → ((norm𝑥) ≤ 1 ↔ (norm‘0) ≤ 1))
18 2fveq3 6668 . . . . . . . . . 10 (𝑥 = 0 → (𝑁‘(𝑇𝑥)) = (𝑁‘(𝑇‘0)))
1918eqeq2d 2829 . . . . . . . . 9 (𝑥 = 0 → (0 = (𝑁‘(𝑇𝑥)) ↔ 0 = (𝑁‘(𝑇‘0))))
2017, 19anbi12d 630 . . . . . . . 8 (𝑥 = 0 → (((norm𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇𝑥))) ↔ ((norm‘0) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘0)))))
2120rspcev 3620 . . . . . . 7 ((0 ∈ ℋ ∧ ((norm‘0) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘0)))) → ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇𝑥))))
229, 15, 21mp2an 688 . . . . . 6 𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇𝑥)))
23 c0ex 10623 . . . . . . 7 0 ∈ V
24 eqeq1 2822 . . . . . . . . 9 (𝑚 = 0 → (𝑚 = (𝑁‘(𝑇𝑥)) ↔ 0 = (𝑁‘(𝑇𝑥))))
2524anbi2d 628 . . . . . . . 8 (𝑚 = 0 → (((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥))) ↔ ((norm𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇𝑥)))))
2625rexbidv 3294 . . . . . . 7 (𝑚 = 0 → (∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥))) ↔ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇𝑥)))))
2723, 26elab 3664 . . . . . 6 (0 ∈ {𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))} ↔ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇𝑥))))
2822, 27mpbir 232 . . . . 5 0 ∈ {𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}
2928ne0ii 4300 . . . 4 {𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))} ≠ ∅
30 nmcex.1 . . . . 5 𝑦 ∈ ℝ+𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1)
31 2rp 12382 . . . . . . . . . 10 2 ∈ ℝ+
32 rpdivcl 12402 . . . . . . . . . 10 ((2 ∈ ℝ+𝑦 ∈ ℝ+) → (2 / 𝑦) ∈ ℝ+)
3331, 32mpan 686 . . . . . . . . 9 (𝑦 ∈ ℝ+ → (2 / 𝑦) ∈ ℝ+)
3433rpred 12419 . . . . . . . 8 (𝑦 ∈ ℝ+ → (2 / 𝑦) ∈ ℝ)
3534adantr 481 . . . . . . 7 ((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1)) → (2 / 𝑦) ∈ ℝ)
36 rpre 12385 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ ℝ+𝑦 ∈ ℝ)
3736adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → 𝑦 ∈ ℝ)
3837rehalfcld 11872 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (𝑦 / 2) ∈ ℝ)
3938recnd 10657 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (𝑦 / 2) ∈ ℂ)
40 simprl 767 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → 𝑥 ∈ ℋ)
41 hvmulcl 28717 . . . . . . . . . . . . . . . . . . 19 (((𝑦 / 2) ∈ ℂ ∧ 𝑥 ∈ ℋ) → ((𝑦 / 2) · 𝑥) ∈ ℋ)
4239, 40, 41syl2anc 584 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → ((𝑦 / 2) · 𝑥) ∈ ℋ)
43 normcl 28829 . . . . . . . . . . . . . . . . . 18 (((𝑦 / 2) · 𝑥) ∈ ℋ → (norm‘((𝑦 / 2) · 𝑥)) ∈ ℝ)
4442, 43syl 17 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (norm‘((𝑦 / 2) · 𝑥)) ∈ ℝ)
45 simprr 769 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (norm𝑥) ≤ 1)
46 normcl 28829 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ ℋ → (norm𝑥) ∈ ℝ)
4746ad2antrl 724 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (norm𝑥) ∈ ℝ)
48 1red 10630 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → 1 ∈ ℝ)
49 rphalfcl 12404 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ ℝ+ → (𝑦 / 2) ∈ ℝ+)
5049adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (𝑦 / 2) ∈ ℝ+)
5147, 48, 50lemul2d 12463 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → ((norm𝑥) ≤ 1 ↔ ((𝑦 / 2) · (norm𝑥)) ≤ ((𝑦 / 2) · 1)))
5245, 51mpbid 233 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → ((𝑦 / 2) · (norm𝑥)) ≤ ((𝑦 / 2) · 1))
53 rpcn 12387 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 / 2) ∈ ℝ+ → (𝑦 / 2) ∈ ℂ)
54 norm-iii 28844 . . . . . . . . . . . . . . . . . . . . 21 (((𝑦 / 2) ∈ ℂ ∧ 𝑥 ∈ ℋ) → (norm‘((𝑦 / 2) · 𝑥)) = ((abs‘(𝑦 / 2)) · (norm𝑥)))
5553, 54sylan 580 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 / 2) ∈ ℝ+𝑥 ∈ ℋ) → (norm‘((𝑦 / 2) · 𝑥)) = ((abs‘(𝑦 / 2)) · (norm𝑥)))
56 rpre 12385 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 / 2) ∈ ℝ+ → (𝑦 / 2) ∈ ℝ)
57 rpge0 12390 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 / 2) ∈ ℝ+ → 0 ≤ (𝑦 / 2))
5856, 57absidd 14770 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 / 2) ∈ ℝ+ → (abs‘(𝑦 / 2)) = (𝑦 / 2))
5958oveq1d 7160 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 / 2) ∈ ℝ+ → ((abs‘(𝑦 / 2)) · (norm𝑥)) = ((𝑦 / 2) · (norm𝑥)))
6059adantr 481 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 / 2) ∈ ℝ+𝑥 ∈ ℋ) → ((abs‘(𝑦 / 2)) · (norm𝑥)) = ((𝑦 / 2) · (norm𝑥)))
6155, 60eqtr2d 2854 . . . . . . . . . . . . . . . . . . 19 (((𝑦 / 2) ∈ ℝ+𝑥 ∈ ℋ) → ((𝑦 / 2) · (norm𝑥)) = (norm‘((𝑦 / 2) · 𝑥)))
6250, 40, 61syl2anc 584 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → ((𝑦 / 2) · (norm𝑥)) = (norm‘((𝑦 / 2) · 𝑥)))
6339mulid1d 10646 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → ((𝑦 / 2) · 1) = (𝑦 / 2))
6452, 62, 633brtr3d 5088 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (norm‘((𝑦 / 2) · 𝑥)) ≤ (𝑦 / 2))
65 rphalflt 12406 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ℝ+ → (𝑦 / 2) < 𝑦)
6665adantr 481 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (𝑦 / 2) < 𝑦)
6744, 38, 37, 64, 66lelttrd 10786 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (norm‘((𝑦 / 2) · 𝑥)) < 𝑦)
68 fveq2 6663 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = ((𝑦 / 2) · 𝑥) → (norm𝑧) = (norm‘((𝑦 / 2) · 𝑥)))
6968breq1d 5067 . . . . . . . . . . . . . . . . . . 19 (𝑧 = ((𝑦 / 2) · 𝑥) → ((norm𝑧) < 𝑦 ↔ (norm‘((𝑦 / 2) · 𝑥)) < 𝑦))
70 2fveq3 6668 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = ((𝑦 / 2) · 𝑥) → (𝑁‘(𝑇𝑧)) = (𝑁‘(𝑇‘((𝑦 / 2) · 𝑥))))
7170breq1d 5067 . . . . . . . . . . . . . . . . . . 19 (𝑧 = ((𝑦 / 2) · 𝑥) → ((𝑁‘(𝑇𝑧)) < 1 ↔ (𝑁‘(𝑇‘((𝑦 / 2) · 𝑥))) < 1))
7269, 71imbi12d 346 . . . . . . . . . . . . . . . . . 18 (𝑧 = ((𝑦 / 2) · 𝑥) → (((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1) ↔ ((norm‘((𝑦 / 2) · 𝑥)) < 𝑦 → (𝑁‘(𝑇‘((𝑦 / 2) · 𝑥))) < 1)))
7372rspcv 3615 . . . . . . . . . . . . . . . . 17 (((𝑦 / 2) · 𝑥) ∈ ℋ → (∀𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1) → ((norm‘((𝑦 / 2) · 𝑥)) < 𝑦 → (𝑁‘(𝑇‘((𝑦 / 2) · 𝑥))) < 1)))
7442, 73syl 17 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (∀𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1) → ((norm‘((𝑦 / 2) · 𝑥)) < 𝑦 → (𝑁‘(𝑇‘((𝑦 / 2) · 𝑥))) < 1)))
7567, 74mpid 44 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (∀𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1) → (𝑁‘(𝑇‘((𝑦 / 2) · 𝑥))) < 1))
762ad2antrl 724 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (𝑁‘(𝑇𝑥)) ∈ ℝ)
7776, 48, 50ltmuldiv2d 12467 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (((𝑦 / 2) · (𝑁‘(𝑇𝑥))) < 1 ↔ (𝑁‘(𝑇𝑥)) < (1 / (𝑦 / 2))))
7850rprecred 12430 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (1 / (𝑦 / 2)) ∈ ℝ)
79 ltle 10717 . . . . . . . . . . . . . . . . . 18 (((𝑁‘(𝑇𝑥)) ∈ ℝ ∧ (1 / (𝑦 / 2)) ∈ ℝ) → ((𝑁‘(𝑇𝑥)) < (1 / (𝑦 / 2)) → (𝑁‘(𝑇𝑥)) ≤ (1 / (𝑦 / 2))))
8076, 78, 79syl2anc 584 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → ((𝑁‘(𝑇𝑥)) < (1 / (𝑦 / 2)) → (𝑁‘(𝑇𝑥)) ≤ (1 / (𝑦 / 2))))
8177, 80sylbid 241 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (((𝑦 / 2) · (𝑁‘(𝑇𝑥))) < 1 → (𝑁‘(𝑇𝑥)) ≤ (1 / (𝑦 / 2))))
82 nmcex.5 . . . . . . . . . . . . . . . . . 18 (((𝑦 / 2) ∈ ℝ+𝑥 ∈ ℋ) → ((𝑦 / 2) · (𝑁‘(𝑇𝑥))) = (𝑁‘(𝑇‘((𝑦 / 2) · 𝑥))))
8350, 40, 82syl2anc 584 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → ((𝑦 / 2) · (𝑁‘(𝑇𝑥))) = (𝑁‘(𝑇‘((𝑦 / 2) · 𝑥))))
8483breq1d 5067 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (((𝑦 / 2) · (𝑁‘(𝑇𝑥))) < 1 ↔ (𝑁‘(𝑇‘((𝑦 / 2) · 𝑥))) < 1))
85 rpcn 12387 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ℝ+𝑦 ∈ ℂ)
86 rpne0 12393 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ℝ+𝑦 ≠ 0)
87 2cn 11700 . . . . . . . . . . . . . . . . . . . 20 2 ∈ ℂ
88 2ne0 11729 . . . . . . . . . . . . . . . . . . . 20 2 ≠ 0
89 recdiv 11334 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → (1 / (𝑦 / 2)) = (2 / 𝑦))
9087, 88, 89mpanr12 701 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) → (1 / (𝑦 / 2)) = (2 / 𝑦))
9185, 86, 90syl2anc 584 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ℝ+ → (1 / (𝑦 / 2)) = (2 / 𝑦))
9291adantr 481 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (1 / (𝑦 / 2)) = (2 / 𝑦))
9392breq2d 5069 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → ((𝑁‘(𝑇𝑥)) ≤ (1 / (𝑦 / 2)) ↔ (𝑁‘(𝑇𝑥)) ≤ (2 / 𝑦)))
9481, 84, 933imtr3d 294 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → ((𝑁‘(𝑇‘((𝑦 / 2) · 𝑥))) < 1 → (𝑁‘(𝑇𝑥)) ≤ (2 / 𝑦)))
9575, 94syld 47 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (∀𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1) → (𝑁‘(𝑇𝑥)) ≤ (2 / 𝑦)))
9695imp 407 . . . . . . . . . . . . 13 (((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) ∧ ∀𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1)) → (𝑁‘(𝑇𝑥)) ≤ (2 / 𝑦))
9796an32s 648 . . . . . . . . . . . 12 (((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1)) ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (𝑁‘(𝑇𝑥)) ≤ (2 / 𝑦))
9897anassrs 468 . . . . . . . . . . 11 ((((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1)) ∧ 𝑥 ∈ ℋ) ∧ (norm𝑥) ≤ 1) → (𝑁‘(𝑇𝑥)) ≤ (2 / 𝑦))
99 breq1 5060 . . . . . . . . . . 11 (𝑛 = (𝑁‘(𝑇𝑥)) → (𝑛 ≤ (2 / 𝑦) ↔ (𝑁‘(𝑇𝑥)) ≤ (2 / 𝑦)))
10098, 99syl5ibrcom 248 . . . . . . . . . 10 ((((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1)) ∧ 𝑥 ∈ ℋ) ∧ (norm𝑥) ≤ 1) → (𝑛 = (𝑁‘(𝑇𝑥)) → 𝑛 ≤ (2 / 𝑦)))
101100expimpd 454 . . . . . . . . 9 (((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1)) ∧ 𝑥 ∈ ℋ) → (((norm𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇𝑥))) → 𝑛 ≤ (2 / 𝑦)))
102101rexlimdva 3281 . . . . . . . 8 ((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1)) → (∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇𝑥))) → 𝑛 ≤ (2 / 𝑦)))
103102alrimiv 1919 . . . . . . 7 ((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1)) → ∀𝑛(∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇𝑥))) → 𝑛 ≤ (2 / 𝑦)))
104 eqeq1 2822 . . . . . . . . . . . 12 (𝑚 = 𝑛 → (𝑚 = (𝑁‘(𝑇𝑥)) ↔ 𝑛 = (𝑁‘(𝑇𝑥))))
105104anbi2d 628 . . . . . . . . . . 11 (𝑚 = 𝑛 → (((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥))) ↔ ((norm𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇𝑥)))))
106105rexbidv 3294 . . . . . . . . . 10 (𝑚 = 𝑛 → (∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥))) ↔ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇𝑥)))))
107106ralab 3681 . . . . . . . . 9 (∀𝑛 ∈ {𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}𝑛𝑧 ↔ ∀𝑛(∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇𝑥))) → 𝑛𝑧))
108 breq2 5061 . . . . . . . . . . 11 (𝑧 = (2 / 𝑦) → (𝑛𝑧𝑛 ≤ (2 / 𝑦)))
109108imbi2d 342 . . . . . . . . . 10 (𝑧 = (2 / 𝑦) → ((∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇𝑥))) → 𝑛𝑧) ↔ (∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇𝑥))) → 𝑛 ≤ (2 / 𝑦))))
110109albidv 1912 . . . . . . . . 9 (𝑧 = (2 / 𝑦) → (∀𝑛(∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇𝑥))) → 𝑛𝑧) ↔ ∀𝑛(∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇𝑥))) → 𝑛 ≤ (2 / 𝑦))))
111107, 110syl5bb 284 . . . . . . . 8 (𝑧 = (2 / 𝑦) → (∀𝑛 ∈ {𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}𝑛𝑧 ↔ ∀𝑛(∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇𝑥))) → 𝑛 ≤ (2 / 𝑦))))
112111rspcev 3620 . . . . . . 7 (((2 / 𝑦) ∈ ℝ ∧ ∀𝑛(∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇𝑥))) → 𝑛 ≤ (2 / 𝑦))) → ∃𝑧 ∈ ℝ ∀𝑛 ∈ {𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}𝑛𝑧)
11335, 103, 112syl2anc 584 . . . . . 6 ((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1)) → ∃𝑧 ∈ ℝ ∀𝑛 ∈ {𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}𝑛𝑧)
114113rexlimiva 3278 . . . . 5 (∃𝑦 ∈ ℝ+𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1) → ∃𝑧 ∈ ℝ ∀𝑛 ∈ {𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}𝑛𝑧)
11530, 114ax-mp 5 . . . 4 𝑧 ∈ ℝ ∀𝑛 ∈ {𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}𝑛𝑧
116 supxrre 12708 . . . 4 (({𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))} ⊆ ℝ ∧ {𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))} ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑛 ∈ {𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}𝑛𝑧) → sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}, ℝ*, < ) = sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}, ℝ, < ))
1178, 29, 115, 116mp3an 1452 . . 3 sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}, ℝ*, < ) = sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}, ℝ, < )
1181, 117eqtri 2841 . 2 (𝑆𝑇) = sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}, ℝ, < )
119 suprcl 11589 . . 3 (({𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))} ⊆ ℝ ∧ {𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))} ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑛 ∈ {𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}𝑛𝑧) → sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}, ℝ, < ) ∈ ℝ)
1208, 29, 115, 119mp3an 1452 . 2 sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}, ℝ, < ) ∈ ℝ
121118, 120eqeltri 2906 1 (𝑆𝑇) ∈ ℝ
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1526   = wceq 1528  wcel 2105  {cab 2796  wne 3013  wral 3135  wrex 3136  wss 3933  c0 4288   class class class wbr 5057  cfv 6348  (class class class)co 7145  supcsup 8892  cc 10523  cr 10524  0cc0 10525  1c1 10526   · cmul 10530  *cxr 10662   < clt 10663  cle 10664   / cdiv 11285  2c2 11680  +crp 12377  abscabs 14581  chba 28623   · csm 28625  normcno 28627  0c0v 28628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603  ax-hv0cl 28707  ax-hfvmul 28709  ax-hvmul0 28714  ax-hfi 28783  ax-his1 28786  ax-his3 28788  ax-his4 28789
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-sup 8894  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-seq 13358  df-exp 13418  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-hnorm 28672
This theorem is referenced by:  nmcopexi  29731  nmcfnexi  29755
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