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Theorem nmcexi 29490
Description: Lemma for nmcopexi 29491 and nmcfnexi 29515. 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 2872 . . . . . . . . 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 3246 . . . . 5 (∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥))) → 𝑚 ∈ ℝ)
87abssi 3973 . . . 4 {𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))} ⊆ ℝ
9 ax-hv0cl 28467 . . . . . . 7 0 ∈ ℋ
10 norm0 28592 . . . . . . . . 9 (norm‘0) = 0
11 0le1 11017 . . . . . . . . 9 0 ≤ 1
1210, 11eqbrtri 4989 . . . . . . . 8 (norm‘0) ≤ 1
13 nmcex.4 . . . . . . . . 9 (𝑁‘(𝑇‘0)) = 0
1413eqcomi 2806 . . . . . . . 8 0 = (𝑁‘(𝑇‘0))
1512, 14pm3.2i 471 . . . . . . 7 ((norm‘0) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘0)))
16 fveq2 6545 . . . . . . . . . 10 (𝑥 = 0 → (norm𝑥) = (norm‘0))
1716breq1d 4978 . . . . . . . . 9 (𝑥 = 0 → ((norm𝑥) ≤ 1 ↔ (norm‘0) ≤ 1))
18 2fveq3 6550 . . . . . . . . . 10 (𝑥 = 0 → (𝑁‘(𝑇𝑥)) = (𝑁‘(𝑇‘0)))
1918eqeq2d 2807 . . . . . . . . 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 3561 . . . . . . 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 10488 . . . . . . 7 0 ∈ V
24 eqeq1 2801 . . . . . . . . 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 3262 . . . . . . 7 (𝑚 = 0 → (∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥))) ↔ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇𝑥)))))
2723, 26elab 3608 . . . . . 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 4229 . . . 4 {𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))} ≠ ∅
30 nmcex.1 . . . . 5 𝑦 ∈ ℝ+𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1)
31 2rp 12248 . . . . . . . . . 10 2 ∈ ℝ+
32 rpdivcl 12268 . . . . . . . . . 10 ((2 ∈ ℝ+𝑦 ∈ ℝ+) → (2 / 𝑦) ∈ ℝ+)
3331, 32mpan 686 . . . . . . . . 9 (𝑦 ∈ ℝ+ → (2 / 𝑦) ∈ ℝ+)
3433rpred 12285 . . . . . . . 8 (𝑦 ∈ ℝ+ → (2 / 𝑦) ∈ ℝ)
3534adantr 481 . . . . . . 7 ((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1)) → (2 / 𝑦) ∈ ℝ)
36 rpre 12251 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ ℝ+𝑦 ∈ ℝ)
3736adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → 𝑦 ∈ ℝ)
3837rehalfcld 11738 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (𝑦 / 2) ∈ ℝ)
3938recnd 10522 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (𝑦 / 2) ∈ ℂ)
40 simprl 767 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → 𝑥 ∈ ℋ)
41 hvmulcl 28477 . . . . . . . . . . . . . . . . . . 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 28589 . . . . . . . . . . . . . . . . . 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 28589 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ ℋ → (norm𝑥) ∈ ℝ)
4746ad2antrl 724 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (norm𝑥) ∈ ℝ)
48 1red 10495 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → 1 ∈ ℝ)
49 rphalfcl 12270 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ ℝ+ → (𝑦 / 2) ∈ ℝ+)
5049adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (𝑦 / 2) ∈ ℝ+)
5147, 48, 50lemul2d 12329 . . . . . . . . . . . . . . . . . . 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 12253 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 / 2) ∈ ℝ+ → (𝑦 / 2) ∈ ℂ)
54 norm-iii 28604 . . . . . . . . . . . . . . . . . . . . 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 12251 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 / 2) ∈ ℝ+ → (𝑦 / 2) ∈ ℝ)
57 rpge0 12256 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 / 2) ∈ ℝ+ → 0 ≤ (𝑦 / 2))
5856, 57absidd 14620 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 / 2) ∈ ℝ+ → (abs‘(𝑦 / 2)) = (𝑦 / 2))
5958oveq1d 7038 . . . . . . . . . . . . . . . . . . . . 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 2834 . . . . . . . . . . . . . . . . . . 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 10511 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → ((𝑦 / 2) · 1) = (𝑦 / 2))
6452, 62, 633brtr3d 4999 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (norm‘((𝑦 / 2) · 𝑥)) ≤ (𝑦 / 2))
65 rphalflt 12272 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ℝ+ → (𝑦 / 2) < 𝑦)
6665adantr 481 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (𝑦 / 2) < 𝑦)
6744, 38, 37, 64, 66lelttrd 10651 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (norm‘((𝑦 / 2) · 𝑥)) < 𝑦)
68 fveq2 6545 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = ((𝑦 / 2) · 𝑥) → (norm𝑧) = (norm‘((𝑦 / 2) · 𝑥)))
6968breq1d 4978 . . . . . . . . . . . . . . . . . . 19 (𝑧 = ((𝑦 / 2) · 𝑥) → ((norm𝑧) < 𝑦 ↔ (norm‘((𝑦 / 2) · 𝑥)) < 𝑦))
70 2fveq3 6550 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = ((𝑦 / 2) · 𝑥) → (𝑁‘(𝑇𝑧)) = (𝑁‘(𝑇‘((𝑦 / 2) · 𝑥))))
7170breq1d 4978 . . . . . . . . . . . . . . . . . . 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 3557 . . . . . . . . . . . . . . . . 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 12333 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (((𝑦 / 2) · (𝑁‘(𝑇𝑥))) < 1 ↔ (𝑁‘(𝑇𝑥)) < (1 / (𝑦 / 2))))
7850rprecred 12296 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (1 / (𝑦 / 2)) ∈ ℝ)
79 ltle 10582 . . . . . . . . . . . . . . . . . 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 4978 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (norm𝑥) ≤ 1)) → (((𝑦 / 2) · (𝑁‘(𝑇𝑥))) < 1 ↔ (𝑁‘(𝑇‘((𝑦 / 2) · 𝑥))) < 1))
85 rpcn 12253 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ℝ+𝑦 ∈ ℂ)
86 rpne0 12259 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ℝ+𝑦 ≠ 0)
87 2cn 11566 . . . . . . . . . . . . . . . . . . . 20 2 ∈ ℂ
88 2ne0 11595 . . . . . . . . . . . . . . . . . . . 20 2 ≠ 0
89 recdiv 11200 . . . . . . . . . . . . . . . . . . . 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 4980 . . . . . . . . . . . . . . . 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 4971 . . . . . . . . . . 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 3249 . . . . . . . 8 ((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1)) → (∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇𝑥))) → 𝑛 ≤ (2 / 𝑦)))
103102alrimiv 1909 . . . . . . 7 ((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((norm𝑧) < 𝑦 → (𝑁‘(𝑇𝑧)) < 1)) → ∀𝑛(∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇𝑥))) → 𝑛 ≤ (2 / 𝑦)))
104 eqeq1 2801 . . . . . . . . . . . 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 3262 . . . . . . . . . 10 (𝑚 = 𝑛 → (∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥))) ↔ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇𝑥)))))
107106ralab 3625 . . . . . . . . 9 (∀𝑛 ∈ {𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}𝑛𝑧 ↔ ∀𝑛(∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇𝑥))) → 𝑛𝑧))
108 breq2 4972 . . . . . . . . . . 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 1902 . . . . . . . . 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 3561 . . . . . . 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 3246 . . . . 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 12574 . . . 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 1453 . . 3 sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}, ℝ*, < ) = sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}, ℝ, < )
1181, 117eqtri 2821 . 2 (𝑆𝑇) = sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}, ℝ, < )
119 suprcl 11455 . . 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 1453 . 2 sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇𝑥)))}, ℝ, < ) ∈ ℝ
121118, 120eqeltri 2881 1 (𝑆𝑇) ∈ ℝ
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1523   = wceq 1525  wcel 2083  {cab 2777  wne 2986  wral 3107  wrex 3108  wss 3865  c0 4217   class class class wbr 4968  cfv 6232  (class class class)co 7023  supcsup 8757  cc 10388  cr 10389  0cc0 10390  1c1 10391   · cmul 10395  *cxr 10527   < clt 10528  cle 10529   / cdiv 11151  2c2 11546  +crp 12243  abscabs 14431  chba 28383   · csm 28385  normcno 28387  0c0v 28388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-cnex 10446  ax-resscn 10447  ax-1cn 10448  ax-icn 10449  ax-addcl 10450  ax-addrcl 10451  ax-mulcl 10452  ax-mulrcl 10453  ax-mulcom 10454  ax-addass 10455  ax-mulass 10456  ax-distr 10457  ax-i2m1 10458  ax-1ne0 10459  ax-1rid 10460  ax-rnegex 10461  ax-rrecex 10462  ax-cnre 10463  ax-pre-lttri 10464  ax-pre-lttrn 10465  ax-pre-ltadd 10466  ax-pre-mulgt0 10467  ax-pre-sup 10468  ax-hv0cl 28467  ax-hfvmul 28469  ax-hvmul0 28474  ax-hfi 28543  ax-his1 28546  ax-his3 28548  ax-his4 28549
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-om 7444  df-2nd 7553  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-er 8146  df-en 8365  df-dom 8366  df-sdom 8367  df-sup 8759  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-sub 10725  df-neg 10726  df-div 11152  df-nn 11493  df-2 11554  df-3 11555  df-n0 11752  df-z 11836  df-uz 12098  df-rp 12244  df-seq 13224  df-exp 13284  df-cj 14296  df-re 14297  df-im 14298  df-sqrt 14432  df-abs 14433  df-hnorm 28432
This theorem is referenced by:  nmcopexi  29491  nmcfnexi  29515
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