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Theorem nmcexi 29013
Description: Lemma for nmcopexi 29014 and nmcfnexi 29038. 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 2718 . . . . . . . . 9 (𝑚 = (𝑁‘(𝑇𝑥)) → (𝑚 ∈ ℝ ↔ (𝑁‘(𝑇𝑥)) ∈ ℝ))
42, 3syl5ibrcom 237 . . . . . . . 8 (𝑥 ∈ ℋ → (𝑚 = (𝑁‘(𝑇𝑥)) → 𝑚 ∈ ℝ))
54imp 444 . . . . . . 7 ((𝑥 ∈ ℋ ∧ 𝑚 = (𝑁‘(𝑇𝑥))) → 𝑚 ∈ ℝ)
65adantrl 752 . . . . . 6 ((𝑥 ∈ ℋ ∧ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))) → 𝑚 ∈ ℝ)
76rexlimiva 3057 . . . . 5 (∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥))) → 𝑚 ∈ ℝ)
87abssi 3710 . . . 4 {𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))} ⊆ ℝ
9 ax-hv0cl 27988 . . . . . . 7 0 ∈ ℋ
10 norm0 28113 . . . . . . . . 9 (norm‘0) = 0
11 0le1 10589 . . . . . . . . 9 0 ≤ 1
1210, 11eqbrtri 4706 . . . . . . . 8 (norm‘0) ≤ 1
13 nmcex.4 . . . . . . . . 9 (𝑁‘(𝑇‘0)) = 0
1413eqcomi 2660 . . . . . . . 8 0 = (𝑁‘(𝑇‘0))
1512, 14pm3.2i 470 . . . . . . 7 ((norm‘0) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘0)))
16 fveq2 6229 . . . . . . . . . 10 (𝑥 = 0 → (norm𝑥) = (norm‘0))
1716breq1d 4695 . . . . . . . . 9 (𝑥 = 0 → ((norm𝑥) ≤ 1 ↔ (norm‘0) ≤ 1))
18 fveq2 6229 . . . . . . . . . . 11 (𝑥 = 0 → (𝑇𝑥) = (𝑇‘0))
1918fveq2d 6233 . . . . . . . . . 10 (𝑥 = 0 → (𝑁‘(𝑇𝑥)) = (𝑁‘(𝑇‘0)))
2019eqeq2d 2661 . . . . . . . . 9 (𝑥 = 0 → (0 = (𝑁‘(𝑇𝑥)) ↔ 0 = (𝑁‘(𝑇‘0))))
2117, 20anbi12d 747 . . . . . . . 8 (𝑥 = 0 → (((norm𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇𝑥))) ↔ ((norm‘0) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘0)))))
2221rspcev 3340 . . . . . . 7 ((0 ∈ ℋ ∧ ((norm‘0) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘0)))) → ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇𝑥))))
239, 15, 22mp2an 708 . . . . . 6 𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇𝑥)))
24 c0ex 10072 . . . . . . 7 0 ∈ V
25 eqeq1 2655 . . . . . . . . 9 (𝑚 = 0 → (𝑚 = (𝑁‘(𝑇𝑥)) ↔ 0 = (𝑁‘(𝑇𝑥))))
2625anbi2d 740 . . . . . . . 8 (𝑚 = 0 → (((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥))) ↔ ((norm𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇𝑥)))))
2726rexbidv 3081 . . . . . . 7 (𝑚 = 0 → (∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥))) ↔ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇𝑥)))))
2824, 27elab 3382 . . . . . 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 3956 . . . 4 {𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))} ≠ ∅
31 nmcex.1 . . . . 5 𝑦 ∈ ℝ+𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1)
32 2rp 11875 . . . . . . . . . 10 2 ∈ ℝ+
33 rpdivcl 11894 . . . . . . . . . 10 ((2 ∈ ℝ+𝑦 ∈ ℝ+) → (2 / 𝑦) ∈ ℝ+)
3432, 33mpan 706 . . . . . . . . 9 (𝑦 ∈ ℝ+ → (2 / 𝑦) ∈ ℝ+)
3534rpred 11910 . . . . . . . 8 (𝑦 ∈ ℝ+ → (2 / 𝑦) ∈ ℝ)
3635adantr 480 . . . . . . 7 ((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1)) → (2 / 𝑦) ∈ ℝ)
37 rpre 11877 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ ℝ+𝑦 ∈ ℝ)
3837adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → 𝑦 ∈ ℝ)
3938rehalfcld 11317 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (𝑦 / 2) ∈ ℝ)
4039recnd 10106 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (𝑦 / 2) ∈ ℂ)
41 simprl 809 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → 𝑥 ∈ ℋ)
42 hvmulcl 27998 . . . . . . . . . . . . . . . . . . 19 (((𝑦 / 2) ∈ ℂ ∧ 𝑥 ∈ ℋ) → ((𝑦 / 2) · 𝑥) ∈ ℋ)
4340, 41, 42syl2anc 694 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → ((𝑦 / 2) · 𝑥) ∈ ℋ)
44 normcl 28110 . . . . . . . . . . . . . . . . . 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 811 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (norm𝑥) ≤ 1)
47 normcl 28110 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ ℋ → (norm𝑥) ∈ ℝ)
4847ad2antrl 764 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (norm𝑥) ∈ ℝ)
49 1red 10093 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → 1 ∈ ℝ)
50 rphalfcl 11896 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ ℝ+ → (𝑦 / 2) ∈ ℝ+)
5150adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (𝑦 / 2) ∈ ℝ+)
5248, 49, 51lemul2d 11954 . . . . . . . . . . . . . . . . . . 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 11879 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 / 2) ∈ ℝ+ → (𝑦 / 2) ∈ ℂ)
55 norm-iii 28125 . . . . . . . . . . . . . . . . . . . . 21 (((𝑦 / 2) ∈ ℂ ∧ 𝑥 ∈ ℋ) → (norm‘((𝑦 / 2) · 𝑥)) = ((abs‘(𝑦 / 2)) · (norm𝑥)))
5654, 55sylan 487 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 / 2) ∈ ℝ+𝑥 ∈ ℋ) → (norm‘((𝑦 / 2) · 𝑥)) = ((abs‘(𝑦 / 2)) · (norm𝑥)))
57 rpre 11877 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 / 2) ∈ ℝ+ → (𝑦 / 2) ∈ ℝ)
58 rpge0 11883 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 / 2) ∈ ℝ+ → 0 ≤ (𝑦 / 2))
5957, 58absidd 14205 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 / 2) ∈ ℝ+ → (abs‘(𝑦 / 2)) = (𝑦 / 2))
6059oveq1d 6705 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 / 2) ∈ ℝ+ → ((abs‘(𝑦 / 2)) · (norm𝑥)) = ((𝑦 / 2) · (norm𝑥)))
6160adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 / 2) ∈ ℝ+𝑥 ∈ ℋ) → ((abs‘(𝑦 / 2)) · (norm𝑥)) = ((𝑦 / 2) · (norm𝑥)))
6256, 61eqtr2d 2686 . . . . . . . . . . . . . . . . . . 19 (((𝑦 / 2) ∈ ℝ+𝑥 ∈ ℋ) → ((𝑦 / 2) · (norm𝑥)) = (norm‘((𝑦 / 2) · 𝑥)))
6351, 41, 62syl2anc 694 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → ((𝑦 / 2) · (norm𝑥)) = (norm‘((𝑦 / 2) · 𝑥)))
6440mulid1d 10095 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → ((𝑦 / 2) · 1) = (𝑦 / 2))
6553, 63, 643brtr3d 4716 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (norm‘((𝑦 / 2) · 𝑥)) ≤ (𝑦 / 2))
66 rphalflt 11898 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ℝ+ → (𝑦 / 2) < 𝑦)
6766adantr 480 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (𝑦 / 2) < 𝑦)
6845, 39, 38, 65, 67lelttrd 10233 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (norm‘((𝑦 / 2) · 𝑥)) < 𝑦)
69 fveq2 6229 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = ((𝑦 / 2) · 𝑥) → (norm𝑧) = (norm‘((𝑦 / 2) · 𝑥)))
7069breq1d 4695 . . . . . . . . . . . . . . . . . . 19 (𝑧 = ((𝑦 / 2) · 𝑥) → ((norm𝑧) < 𝑦 ↔ (norm‘((𝑦 / 2) · 𝑥)) < 𝑦))
71 fveq2 6229 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = ((𝑦 / 2) · 𝑥) → (𝑇𝑧) = (𝑇‘((𝑦 / 2) · 𝑥)))
7271fveq2d 6233 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = ((𝑦 / 2) · 𝑥) → (𝑁‘(𝑇𝑧)) = (𝑁‘(𝑇‘((𝑦 / 2) · 𝑥))))
7372breq1d 4695 . . . . . . . . . . . . . . . . . . 19 (𝑧 = ((𝑦 / 2) · 𝑥) → ((𝑁‘(𝑇𝑧)) < 1 ↔ (𝑁‘(𝑇‘((𝑦 / 2) · 𝑥))) < 1))
7470, 73imbi12d 333 . . . . . . . . . . . . . . . . . 18 (𝑧 = ((𝑦 / 2) · 𝑥) → (((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1) ↔ ((norm‘((𝑦 / 2) · 𝑥)) < 𝑦 → (𝑁‘(𝑇‘((𝑦 / 2) · 𝑥))) < 1)))
7574rspcv 3336 . . . . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (𝑁‘(𝑇𝑥)) ∈ ℝ)
7978, 49, 51ltmuldiv2d 11958 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (((𝑦 / 2) · (𝑁‘(𝑇𝑥))) < 1 ↔ (𝑁‘(𝑇𝑥)) < (1 / (𝑦 / 2))))
8051rprecred 11921 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (1 / (𝑦 / 2)) ∈ ℝ)
81 ltle 10164 . . . . . . . . . . . . . . . . . 18 (((𝑁‘(𝑇𝑥)) ∈ ℝ ∧ (1 / (𝑦 / 2)) ∈ ℝ) → ((𝑁‘(𝑇𝑥)) < (1 / (𝑦 / 2)) → (𝑁‘(𝑇𝑥)) ≤ (1 / (𝑦 / 2))))
8278, 80, 81syl2anc 694 . . . . . . . . . . . . . . . . 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 694 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → ((𝑦 / 2) · (𝑁‘(𝑇𝑥))) = (𝑁‘(𝑇‘((𝑦 / 2) · 𝑥))))
8685breq1d 4695 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (((𝑦 / 2) · (𝑁‘(𝑇𝑥))) < 1 ↔ (𝑁‘(𝑇‘((𝑦 / 2) · 𝑥))) < 1))
87 rpcn 11879 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ℝ+𝑦 ∈ ℂ)
88 rpne0 11886 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ℝ+𝑦 ≠ 0)
89 2cn 11129 . . . . . . . . . . . . . . . . . . . 20 2 ∈ ℂ
90 2ne0 11151 . . . . . . . . . . . . . . . . . . . 20 2 ≠ 0
91 recdiv 10769 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → (1 / (𝑦 / 2)) = (2 / 𝑦))
9289, 90, 91mpanr12 721 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) → (1 / (𝑦 / 2)) = (2 / 𝑦))
9387, 88, 92syl2anc 694 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ℝ+ → (1 / (𝑦 / 2)) = (2 / 𝑦))
9493adantr 480 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (1 / (𝑦 / 2)) = (2 / 𝑦))
9594breq2d 4697 . . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . 13 (((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) ∧ ∀𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1)) → (𝑁‘(𝑇𝑥)) ≤ (2 / 𝑦))
9998an32s 863 . . . . . . . . . . . 12 (((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1)) ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (𝑁‘(𝑇𝑥)) ≤ (2 / 𝑦))
10099anassrs 681 . . . . . . . . . . 11 ((((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1)) ∧ 𝑥 ∈ ℋ) ∧ (norm𝑥) ≤ 1) → (𝑁‘(𝑇𝑥)) ≤ (2 / 𝑦))
101 breq1 4688 . . . . . . . . . . 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 3060 . . . . . . . 8 ((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1)) → (∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇𝑥))) → 𝑛 ≤ (2 / 𝑦)))
105104alrimiv 1895 . . . . . . 7 ((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1)) → ∀𝑛(∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇𝑥))) → 𝑛 ≤ (2 / 𝑦)))
106 eqeq1 2655 . . . . . . . . . . . 12 (𝑚 = 𝑛 → (𝑚 = (𝑁‘(𝑇𝑥)) ↔ 𝑛 = (𝑁‘(𝑇𝑥))))
107106anbi2d 740 . . . . . . . . . . 11 (𝑚 = 𝑛 → (((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥))) ↔ ((norm𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇𝑥)))))
108107rexbidv 3081 . . . . . . . . . 10 (𝑚 = 𝑛 → (∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥))) ↔ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇𝑥)))))
109108ralab 3400 . . . . . . . . 9 (∀𝑛 ∈ {𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}𝑛𝑧 ↔ ∀𝑛(∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇𝑥))) → 𝑛𝑧))
110 breq2 4689 . . . . . . . . . . 11 (𝑧 = (2 / 𝑦) → (𝑛𝑧𝑛 ≤ (2 / 𝑦)))
111110imbi2d 329 . . . . . . . . . 10 (𝑧 = (2 / 𝑦) → ((∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇𝑥))) → 𝑛𝑧) ↔ (∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇𝑥))) → 𝑛 ≤ (2 / 𝑦))))
112111albidv 1889 . . . . . . . . 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 3340 . . . . . . 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 694 . . . . . 6 ((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1)) → ∃𝑧 ∈ ℝ ∀𝑛 ∈ {𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}𝑛𝑧)
116115rexlimiva 3057 . . . . 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 12195 . . . 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 1464 . . 3 sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}, ℝ*, < ) = sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}, ℝ, < )
1201, 119eqtri 2673 . 2 (𝑆𝑇) = sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}, ℝ, < )
121 suprcl 11021 . . 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 1464 . 2 sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}, ℝ, < ) ∈ ℝ
123120, 122eqeltri 2726 1 (𝑆𝑇) ∈ ℝ
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1521   = wceq 1523  wcel 2030  {cab 2637  wne 2823  wral 2941  wrex 2942  wss 3607  c0 3948   class class class wbr 4685  cfv 5926  (class class class)co 6690  supcsup 8387  cc 9972  cr 9973  0cc0 9974  1c1 9975   · cmul 9979  *cxr 10111   < clt 10112  cle 10113   / cdiv 10722  2c2 11108  +crp 11870  abscabs 14018  chil 27904   · csm 27906  normcno 27908  0c0v 27909
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-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-abs 14020  df-hnorm 27953
This theorem is referenced by:  nmcopexi  29014  nmcfnexi  29038
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