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Theorem nmcexi 28731
Description: Lemma for nmcopexi 28732 and nmcfnexi 28756. 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 2686 . . . . . . . . 9 (𝑚 = (𝑁‘(𝑇𝑥)) → (𝑚 ∈ ℝ ↔ (𝑁‘(𝑇𝑥)) ∈ ℝ))
42, 3syl5ibrcom 237 . . . . . . . 8 (𝑥 ∈ ℋ → (𝑚 = (𝑁‘(𝑇𝑥)) → 𝑚 ∈ ℝ))
54imp 445 . . . . . . 7 ((𝑥 ∈ ℋ ∧ 𝑚 = (𝑁‘(𝑇𝑥))) → 𝑚 ∈ ℝ)
65adantrl 751 . . . . . 6 ((𝑥 ∈ ℋ ∧ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))) → 𝑚 ∈ ℝ)
76rexlimiva 3021 . . . . 5 (∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥))) → 𝑚 ∈ ℝ)
87abssi 3656 . . . 4 {𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))} ⊆ ℝ
9 ax-hv0cl 27706 . . . . . . 7 0 ∈ ℋ
10 norm0 27831 . . . . . . . . 9 (norm‘0) = 0
11 0le1 10495 . . . . . . . . 9 0 ≤ 1
1210, 11eqbrtri 4634 . . . . . . . 8 (norm‘0) ≤ 1
13 nmcex.4 . . . . . . . . 9 (𝑁‘(𝑇‘0)) = 0
1413eqcomi 2630 . . . . . . . 8 0 = (𝑁‘(𝑇‘0))
1512, 14pm3.2i 471 . . . . . . 7 ((norm‘0) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘0)))
16 fveq2 6148 . . . . . . . . . 10 (𝑥 = 0 → (norm𝑥) = (norm‘0))
1716breq1d 4623 . . . . . . . . 9 (𝑥 = 0 → ((norm𝑥) ≤ 1 ↔ (norm‘0) ≤ 1))
18 fveq2 6148 . . . . . . . . . . 11 (𝑥 = 0 → (𝑇𝑥) = (𝑇‘0))
1918fveq2d 6152 . . . . . . . . . 10 (𝑥 = 0 → (𝑁‘(𝑇𝑥)) = (𝑁‘(𝑇‘0)))
2019eqeq2d 2631 . . . . . . . . 9 (𝑥 = 0 → (0 = (𝑁‘(𝑇𝑥)) ↔ 0 = (𝑁‘(𝑇‘0))))
2117, 20anbi12d 746 . . . . . . . 8 (𝑥 = 0 → (((norm𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇𝑥))) ↔ ((norm‘0) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘0)))))
2221rspcev 3295 . . . . . . 7 ((0 ∈ ℋ ∧ ((norm‘0) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘0)))) → ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇𝑥))))
239, 15, 22mp2an 707 . . . . . 6 𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇𝑥)))
24 c0ex 9978 . . . . . . 7 0 ∈ V
25 eqeq1 2625 . . . . . . . . 9 (𝑚 = 0 → (𝑚 = (𝑁‘(𝑇𝑥)) ↔ 0 = (𝑁‘(𝑇𝑥))))
2625anbi2d 739 . . . . . . . 8 (𝑚 = 0 → (((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥))) ↔ ((norm𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇𝑥)))))
2726rexbidv 3045 . . . . . . 7 (𝑚 = 0 → (∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥))) ↔ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇𝑥)))))
2824, 27elab 3333 . . . . . 6 (0 ∈ {𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))} ↔ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇𝑥))))
2923, 28mpbir 221 . . . . 5 0 ∈ {𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}
3029ne0ii 3899 . . . 4 {𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))} ≠ ∅
31 nmcex.1 . . . . 5 𝑦 ∈ ℝ+𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1)
32 2rp 11781 . . . . . . . . . 10 2 ∈ ℝ+
33 rpdivcl 11800 . . . . . . . . . 10 ((2 ∈ ℝ+𝑦 ∈ ℝ+) → (2 / 𝑦) ∈ ℝ+)
3432, 33mpan 705 . . . . . . . . 9 (𝑦 ∈ ℝ+ → (2 / 𝑦) ∈ ℝ+)
3534rpred 11816 . . . . . . . 8 (𝑦 ∈ ℝ+ → (2 / 𝑦) ∈ ℝ)
3635adantr 481 . . . . . . 7 ((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1)) → (2 / 𝑦) ∈ ℝ)
37 rpre 11783 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ ℝ+𝑦 ∈ ℝ)
3837adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → 𝑦 ∈ ℝ)
3938rehalfcld 11223 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (𝑦 / 2) ∈ ℝ)
4039recnd 10012 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (𝑦 / 2) ∈ ℂ)
41 simprl 793 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → 𝑥 ∈ ℋ)
42 hvmulcl 27716 . . . . . . . . . . . . . . . . . . 19 (((𝑦 / 2) ∈ ℂ ∧ 𝑥 ∈ ℋ) → ((𝑦 / 2) · 𝑥) ∈ ℋ)
4340, 41, 42syl2anc 692 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → ((𝑦 / 2) · 𝑥) ∈ ℋ)
44 normcl 27828 . . . . . . . . . . . . . . . . . 18 (((𝑦 / 2) · 𝑥) ∈ ℋ → (norm‘((𝑦 / 2) · 𝑥)) ∈ ℝ)
4543, 44syl 17 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (norm‘((𝑦 / 2) · 𝑥)) ∈ ℝ)
46 simprr 795 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (norm𝑥) ≤ 1)
47 normcl 27828 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ ℋ → (norm𝑥) ∈ ℝ)
4847ad2antrl 763 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (norm𝑥) ∈ ℝ)
49 1red 9999 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → 1 ∈ ℝ)
50 rphalfcl 11802 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ ℝ+ → (𝑦 / 2) ∈ ℝ+)
5150adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (𝑦 / 2) ∈ ℝ+)
5248, 49, 51lemul2d 11860 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → ((norm𝑥) ≤ 1 ↔ ((𝑦 / 2) · (norm𝑥)) ≤ ((𝑦 / 2) · 1)))
5346, 52mpbid 222 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → ((𝑦 / 2) · (norm𝑥)) ≤ ((𝑦 / 2) · 1))
54 rpcn 11785 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 / 2) ∈ ℝ+ → (𝑦 / 2) ∈ ℂ)
55 norm-iii 27843 . . . . . . . . . . . . . . . . . . . . 21 (((𝑦 / 2) ∈ ℂ ∧ 𝑥 ∈ ℋ) → (norm‘((𝑦 / 2) · 𝑥)) = ((abs‘(𝑦 / 2)) · (norm𝑥)))
5654, 55sylan 488 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 / 2) ∈ ℝ+𝑥 ∈ ℋ) → (norm‘((𝑦 / 2) · 𝑥)) = ((abs‘(𝑦 / 2)) · (norm𝑥)))
57 rpre 11783 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 / 2) ∈ ℝ+ → (𝑦 / 2) ∈ ℝ)
58 rpge0 11789 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 / 2) ∈ ℝ+ → 0 ≤ (𝑦 / 2))
5957, 58absidd 14095 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 / 2) ∈ ℝ+ → (abs‘(𝑦 / 2)) = (𝑦 / 2))
6059oveq1d 6619 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 / 2) ∈ ℝ+ → ((abs‘(𝑦 / 2)) · (norm𝑥)) = ((𝑦 / 2) · (norm𝑥)))
6160adantr 481 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 / 2) ∈ ℝ+𝑥 ∈ ℋ) → ((abs‘(𝑦 / 2)) · (norm𝑥)) = ((𝑦 / 2) · (norm𝑥)))
6256, 61eqtr2d 2656 . . . . . . . . . . . . . . . . . . 19 (((𝑦 / 2) ∈ ℝ+𝑥 ∈ ℋ) → ((𝑦 / 2) · (norm𝑥)) = (norm‘((𝑦 / 2) · 𝑥)))
6351, 41, 62syl2anc 692 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → ((𝑦 / 2) · (norm𝑥)) = (norm‘((𝑦 / 2) · 𝑥)))
6440mulid1d 10001 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → ((𝑦 / 2) · 1) = (𝑦 / 2))
6553, 63, 643brtr3d 4644 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (norm‘((𝑦 / 2) · 𝑥)) ≤ (𝑦 / 2))
66 rphalflt 11804 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ℝ+ → (𝑦 / 2) < 𝑦)
6766adantr 481 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (𝑦 / 2) < 𝑦)
6845, 39, 38, 65, 67lelttrd 10139 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (norm‘((𝑦 / 2) · 𝑥)) < 𝑦)
69 fveq2 6148 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = ((𝑦 / 2) · 𝑥) → (norm𝑧) = (norm‘((𝑦 / 2) · 𝑥)))
7069breq1d 4623 . . . . . . . . . . . . . . . . . . 19 (𝑧 = ((𝑦 / 2) · 𝑥) → ((norm𝑧) < 𝑦 ↔ (norm‘((𝑦 / 2) · 𝑥)) < 𝑦))
71 fveq2 6148 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = ((𝑦 / 2) · 𝑥) → (𝑇𝑧) = (𝑇‘((𝑦 / 2) · 𝑥)))
7271fveq2d 6152 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = ((𝑦 / 2) · 𝑥) → (𝑁‘(𝑇𝑧)) = (𝑁‘(𝑇‘((𝑦 / 2) · 𝑥))))
7372breq1d 4623 . . . . . . . . . . . . . . . . . . 19 (𝑧 = ((𝑦 / 2) · 𝑥) → ((𝑁‘(𝑇𝑧)) < 1 ↔ (𝑁‘(𝑇‘((𝑦 / 2) · 𝑥))) < 1))
7470, 73imbi12d 334 . . . . . . . . . . . . . . . . . 18 (𝑧 = ((𝑦 / 2) · 𝑥) → (((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1) ↔ ((norm‘((𝑦 / 2) · 𝑥)) < 𝑦 → (𝑁‘(𝑇‘((𝑦 / 2) · 𝑥))) < 1)))
7574rspcv 3291 . . . . . . . . . . . . . . . . 17 (((𝑦 / 2) · 𝑥) ∈ ℋ → (∀𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1) → ((norm‘((𝑦 / 2) · 𝑥)) < 𝑦 → (𝑁‘(𝑇‘((𝑦 / 2) · 𝑥))) < 1)))
7643, 75syl 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)))
7768, 76mpid 44 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (∀𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1) → (𝑁‘(𝑇‘((𝑦 / 2) · 𝑥))) < 1))
782ad2antrl 763 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (𝑁‘(𝑇𝑥)) ∈ ℝ)
7978, 49, 51ltmuldiv2d 11864 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (((𝑦 / 2) · (𝑁‘(𝑇𝑥))) < 1 ↔ (𝑁‘(𝑇𝑥)) < (1 / (𝑦 / 2))))
8051rprecred 11827 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (1 / (𝑦 / 2)) ∈ ℝ)
81 ltle 10070 . . . . . . . . . . . . . . . . . 18 (((𝑁‘(𝑇𝑥)) ∈ ℝ ∧ (1 / (𝑦 / 2)) ∈ ℝ) → ((𝑁‘(𝑇𝑥)) < (1 / (𝑦 / 2)) → (𝑁‘(𝑇𝑥)) ≤ (1 / (𝑦 / 2))))
8278, 80, 81syl2anc 692 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → ((𝑁‘(𝑇𝑥)) < (1 / (𝑦 / 2)) → (𝑁‘(𝑇𝑥)) ≤ (1 / (𝑦 / 2))))
8379, 82sylbid 230 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (((𝑦 / 2) · (𝑁‘(𝑇𝑥))) < 1 → (𝑁‘(𝑇𝑥)) ≤ (1 / (𝑦 / 2))))
84 nmcex.5 . . . . . . . . . . . . . . . . . 18 (((𝑦 / 2) ∈ ℝ+𝑥 ∈ ℋ) → ((𝑦 / 2) · (𝑁‘(𝑇𝑥))) = (𝑁‘(𝑇‘((𝑦 / 2) · 𝑥))))
8551, 41, 84syl2anc 692 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → ((𝑦 / 2) · (𝑁‘(𝑇𝑥))) = (𝑁‘(𝑇‘((𝑦 / 2) · 𝑥))))
8685breq1d 4623 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (((𝑦 / 2) · (𝑁‘(𝑇𝑥))) < 1 ↔ (𝑁‘(𝑇‘((𝑦 / 2) · 𝑥))) < 1))
87 rpcn 11785 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ℝ+𝑦 ∈ ℂ)
88 rpne0 11792 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ℝ+𝑦 ≠ 0)
89 2cn 11035 . . . . . . . . . . . . . . . . . . . 20 2 ∈ ℂ
90 2ne0 11057 . . . . . . . . . . . . . . . . . . . 20 2 ≠ 0
91 recdiv 10675 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → (1 / (𝑦 / 2)) = (2 / 𝑦))
9289, 90, 91mpanr12 720 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) → (1 / (𝑦 / 2)) = (2 / 𝑦))
9387, 88, 92syl2anc 692 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ℝ+ → (1 / (𝑦 / 2)) = (2 / 𝑦))
9493adantr 481 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (1 / (𝑦 / 2)) = (2 / 𝑦))
9594breq2d 4625 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → ((𝑁‘(𝑇𝑥)) ≤ (1 / (𝑦 / 2)) ↔ (𝑁‘(𝑇𝑥)) ≤ (2 / 𝑦)))
9683, 86, 953imtr3d 282 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → ((𝑁‘(𝑇‘((𝑦 / 2) · 𝑥))) < 1 → (𝑁‘(𝑇𝑥)) ≤ (2 / 𝑦)))
9777, 96syld 47 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (∀𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1) → (𝑁‘(𝑇𝑥)) ≤ (2 / 𝑦)))
9897imp 445 . . . . . . . . . . . . 13 (((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) ∧ ∀𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1)) → (𝑁‘(𝑇𝑥)) ≤ (2 / 𝑦))
9998an32s 845 . . . . . . . . . . . 12 (((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1)) ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (𝑁‘(𝑇𝑥)) ≤ (2 / 𝑦))
10099anassrs 679 . . . . . . . . . . 11 ((((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1)) ∧ 𝑥 ∈ ℋ) ∧ (norm𝑥) ≤ 1) → (𝑁‘(𝑇𝑥)) ≤ (2 / 𝑦))
101 breq1 4616 . . . . . . . . . . 11 (𝑛 = (𝑁‘(𝑇𝑥)) → (𝑛 ≤ (2 / 𝑦) ↔ (𝑁‘(𝑇𝑥)) ≤ (2 / 𝑦)))
102100, 101syl5ibrcom 237 . . . . . . . . . 10 ((((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1)) ∧ 𝑥 ∈ ℋ) ∧ (norm𝑥) ≤ 1) → (𝑛 = (𝑁‘(𝑇𝑥)) → 𝑛 ≤ (2 / 𝑦)))
103102expimpd 628 . . . . . . . . 9 (((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1)) ∧ 𝑥 ∈ ℋ) → (((norm𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇𝑥))) → 𝑛 ≤ (2 / 𝑦)))
104103rexlimdva 3024 . . . . . . . 8 ((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1)) → (∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇𝑥))) → 𝑛 ≤ (2 / 𝑦)))
105104alrimiv 1852 . . . . . . 7 ((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1)) → ∀𝑛(∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇𝑥))) → 𝑛 ≤ (2 / 𝑦)))
106 eqeq1 2625 . . . . . . . . . . . 12 (𝑚 = 𝑛 → (𝑚 = (𝑁‘(𝑇𝑥)) ↔ 𝑛 = (𝑁‘(𝑇𝑥))))
107106anbi2d 739 . . . . . . . . . . 11 (𝑚 = 𝑛 → (((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥))) ↔ ((norm𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇𝑥)))))
108107rexbidv 3045 . . . . . . . . . 10 (𝑚 = 𝑛 → (∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥))) ↔ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇𝑥)))))
109108ralab 3349 . . . . . . . . 9 (∀𝑛 ∈ {𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}𝑛𝑧 ↔ ∀𝑛(∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇𝑥))) → 𝑛𝑧))
110 breq2 4617 . . . . . . . . . . 11 (𝑧 = (2 / 𝑦) → (𝑛𝑧𝑛 ≤ (2 / 𝑦)))
111110imbi2d 330 . . . . . . . . . 10 (𝑧 = (2 / 𝑦) → ((∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇𝑥))) → 𝑛𝑧) ↔ (∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇𝑥))) → 𝑛 ≤ (2 / 𝑦))))
112111albidv 1846 . . . . . . . . 9 (𝑧 = (2 / 𝑦) → (∀𝑛(∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇𝑥))) → 𝑛𝑧) ↔ ∀𝑛(∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇𝑥))) → 𝑛 ≤ (2 / 𝑦))))
113109, 112syl5bb 272 . . . . . . . 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))))
114113rspcev 3295 . . . . . . 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)
11536, 105, 114syl2anc 692 . . . . . 6 ((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1)) → ∃𝑧 ∈ ℝ ∀𝑛 ∈ {𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}𝑛𝑧)
116115rexlimiva 3021 . . . . 5 (∃𝑦 ∈ ℝ+𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1) → ∃𝑧 ∈ ℝ ∀𝑛 ∈ {𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}𝑛𝑧)
11731, 116ax-mp 5 . . . 4 𝑧 ∈ ℝ ∀𝑛 ∈ {𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}𝑛𝑧
118 supxrre 12100 . . . 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, < ))
1198, 30, 117, 118mp3an 1421 . . 3 sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}, ℝ*, < ) = sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}, ℝ, < )
1201, 119eqtri 2643 . 2 (𝑆𝑇) = sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}, ℝ, < )
121 suprcl 10927 . . 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)
1228, 30, 117, 121mp3an 1421 . 2 sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}, ℝ, < ) ∈ ℝ
123120, 122eqeltri 2694 1 (𝑆𝑇) ∈ ℝ
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1478   = wceq 1480  wcel 1987  {cab 2607  wne 2790  wral 2907  wrex 2908  wss 3555  c0 3891   class class class wbr 4613  cfv 5847  (class class class)co 6604  supcsup 8290  cc 9878  cr 9879  0cc0 9880  1c1 9881   · cmul 9885  *cxr 10017   < clt 10018  cle 10019   / cdiv 10628  2c2 11014  +crp 11776  abscabs 13908  chil 27622   · csm 27624  normcno 27626  0c0v 27627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958  ax-hv0cl 27706  ax-hfvmul 27708  ax-hvmul0 27713  ax-hfi 27782  ax-his1 27785  ax-his3 27787  ax-his4 27788
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-sup 8292  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-z 11322  df-uz 11632  df-rp 11777  df-seq 12742  df-exp 12801  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-hnorm 27671
This theorem is referenced by:  nmcopexi  28732  nmcfnexi  28756
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