Theorem nmcexi 32055

Description: Lemma for nmcopexi 32056 and nmcfnexi 32080. 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 ((normℎ‘𝑧) < 𝑦 → (𝑁‘(𝑇‘𝑧)) < 1)
nmcex.2	⊢ (𝑆‘𝑇) = sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))}, ℝ*, < )
nmcex.3	⊢ (𝑥 ∈ ℋ → (𝑁‘(𝑇‘𝑥)) ∈ ℝ)
nmcex.4	⊢ (𝑁‘(𝑇‘0ℎ)) = 0
nmcex.5	⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → ((𝑦 / 2) · (𝑁‘(𝑇‘𝑥))) = (𝑁‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥))))

Assertion

Ref	Expression
nmcexi	⊢ (𝑆‘𝑇) ∈ 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.

Step	Hyp	Ref	Expression
1		nmcex.2	. . 3 ⊢ (𝑆‘𝑇) = sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))}, ℝ*, < )
2		nmcex.3	. . . . . . . . 9 ⊢ (𝑥 ∈ ℋ → (𝑁‘(𝑇‘𝑥)) ∈ ℝ)
3		eleq1 2827	. . . . . . . . 9 ⊢ (𝑚 = (𝑁‘(𝑇‘𝑥)) → (𝑚 ∈ ℝ ↔ (𝑁‘(𝑇‘𝑥)) ∈ ℝ))
4	2, 3	syl5ibrcom 247	. . . . . . . 8 ⊢ (𝑥 ∈ ℋ → (𝑚 = (𝑁‘(𝑇‘𝑥)) → 𝑚 ∈ ℝ))
5	4	imp 406	. . . . . . 7 ⊢ ((𝑥 ∈ ℋ ∧ 𝑚 = (𝑁‘(𝑇‘𝑥))) → 𝑚 ∈ ℝ)
6	5	adantrl 716	. . . . . 6 ⊢ ((𝑥 ∈ ℋ ∧ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))) → 𝑚 ∈ ℝ)
7	6	rexlimiva 3145	. . . . 5 ⊢ (∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥))) → 𝑚 ∈ ℝ)
8	7	abssi 4080	. . . 4 ⊢ {𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))} ⊆ ℝ
9		ax-hv0cl 31032	. . . . . . 7 ⊢ 0ℎ ∈ ℋ
10		norm0 31157	. . . . . . . . 9 ⊢ (normℎ‘0ℎ) = 0
11		0le1 11784	. . . . . . . . 9 ⊢ 0 ≤ 1
12	10, 11	eqbrtri 5169	. . . . . . . 8 ⊢ (normℎ‘0ℎ) ≤ 1
13		nmcex.4	. . . . . . . . 9 ⊢ (𝑁‘(𝑇‘0ℎ)) = 0
14	13	eqcomi 2744	. . . . . . . 8 ⊢ 0 = (𝑁‘(𝑇‘0ℎ))
15	12, 14	pm3.2i 470	. . . . . . 7 ⊢ ((normℎ‘0ℎ) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘0ℎ)))
16		fveq2 6907	. . . . . . . . . 10 ⊢ (𝑥 = 0ℎ → (normℎ‘𝑥) = (normℎ‘0ℎ))
17	16	breq1d 5158	. . . . . . . . 9 ⊢ (𝑥 = 0ℎ → ((normℎ‘𝑥) ≤ 1 ↔ (normℎ‘0ℎ) ≤ 1))
18		2fveq3 6912	. . . . . . . . . 10 ⊢ (𝑥 = 0ℎ → (𝑁‘(𝑇‘𝑥)) = (𝑁‘(𝑇‘0ℎ)))
19	18	eqeq2d 2746	. . . . . . . . 9 ⊢ (𝑥 = 0ℎ → (0 = (𝑁‘(𝑇‘𝑥)) ↔ 0 = (𝑁‘(𝑇‘0ℎ))))
20	17, 19	anbi12d 632	. . . . . . . 8 ⊢ (𝑥 = 0ℎ → (((normℎ‘𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘𝑥))) ↔ ((normℎ‘0ℎ) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘0ℎ)))))
21	20	rspcev 3622	. . . . . . 7 ⊢ ((0ℎ ∈ ℋ ∧ ((normℎ‘0ℎ) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘0ℎ)))) → ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘𝑥))))
22	9, 15, 21	mp2an 692	. . . . . 6 ⊢ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘𝑥)))
23		c0ex 11253	. . . . . . 7 ⊢ 0 ∈ V
24		eqeq1 2739	. . . . . . . . 9 ⊢ (𝑚 = 0 → (𝑚 = (𝑁‘(𝑇‘𝑥)) ↔ 0 = (𝑁‘(𝑇‘𝑥))))
25	24	anbi2d 630	. . . . . . . 8 ⊢ (𝑚 = 0 → (((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥))) ↔ ((normℎ‘𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘𝑥)))))
26	25	rexbidv 3177	. . . . . . 7 ⊢ (𝑚 = 0 → (∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥))) ↔ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘𝑥)))))
27	23, 26	elab 3681	. . . . . 6 ⊢ (0 ∈ {𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))} ↔ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘𝑥))))
28	22, 27	mpbir 231	. . . . 5 ⊢ 0 ∈ {𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))}
29	28	ne0ii 4350	. . . 4 ⊢ {𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))} ≠ ∅
30		nmcex.1	. . . . 5 ⊢ ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℋ ((normℎ‘𝑧) < 𝑦 → (𝑁‘(𝑇‘𝑧)) < 1)
31		2rp 13037	. . . . . . . . . 10 ⊢ 2 ∈ ℝ+
32		rpdivcl 13058	. . . . . . . . . 10 ⊢ ((2 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+) → (2 / 𝑦) ∈ ℝ+)
33	31, 32	mpan 690	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ+ → (2 / 𝑦) ∈ ℝ+)
34	33	rpred 13075	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ+ → (2 / 𝑦) ∈ ℝ)
35	34	adantr 480	. . . . . . 7 ⊢ ((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((normℎ‘𝑧) < 𝑦 → (𝑁‘(𝑇‘𝑧)) < 1)) → (2 / 𝑦) ∈ ℝ)
36		rpre 13041	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ ℝ+ → 𝑦 ∈ ℝ)
37	36	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → 𝑦 ∈ ℝ)
38	37	rehalfcld 12511	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (𝑦 / 2) ∈ ℝ)
39	38	recnd 11287	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (𝑦 / 2) ∈ ℂ)
40		simprl 771	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → 𝑥 ∈ ℋ)
41		hvmulcl 31042	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑦 / 2) ∈ ℂ ∧ 𝑥 ∈ ℋ) → ((𝑦 / 2) ·ℎ 𝑥) ∈ ℋ)
42	39, 40, 41	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((𝑦 / 2) ·ℎ 𝑥) ∈ ℋ)
43		normcl 31154	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 / 2) ·ℎ 𝑥) ∈ ℋ → (normℎ‘((𝑦 / 2) ·ℎ 𝑥)) ∈ ℝ)
44	42, 43	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (normℎ‘((𝑦 / 2) ·ℎ 𝑥)) ∈ ℝ)
45		simprr 773	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (normℎ‘𝑥) ≤ 1)
46		normcl 31154	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℋ → (normℎ‘𝑥) ∈ ℝ)
47	46	ad2antrl 728	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (normℎ‘𝑥) ∈ ℝ)
48		1red 11260	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → 1 ∈ ℝ)
49		rphalfcl 13060	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ℝ+ → (𝑦 / 2) ∈ ℝ+)
50	49	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (𝑦 / 2) ∈ ℝ+)
51	47, 48, 50	lemul2d 13119	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((normℎ‘𝑥) ≤ 1 ↔ ((𝑦 / 2) · (normℎ‘𝑥)) ≤ ((𝑦 / 2) · 1)))
52	45, 51	mpbid 232	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((𝑦 / 2) · (normℎ‘𝑥)) ≤ ((𝑦 / 2) · 1))
53		rpcn 13043	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 / 2) ∈ ℝ+ → (𝑦 / 2) ∈ ℂ)
54		norm-iii 31169	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑦 / 2) ∈ ℂ ∧ 𝑥 ∈ ℋ) → (normℎ‘((𝑦 / 2) ·ℎ 𝑥)) = ((abs‘(𝑦 / 2)) · (normℎ‘𝑥)))
55	53, 54	sylan 580	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → (normℎ‘((𝑦 / 2) ·ℎ 𝑥)) = ((abs‘(𝑦 / 2)) · (normℎ‘𝑥)))
56		rpre 13041	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 / 2) ∈ ℝ+ → (𝑦 / 2) ∈ ℝ)
57		rpge0 13046	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 / 2) ∈ ℝ+ → 0 ≤ (𝑦 / 2))
58	56, 57	absidd 15458	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 / 2) ∈ ℝ+ → (abs‘(𝑦 / 2)) = (𝑦 / 2))
59	58	oveq1d 7446	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 / 2) ∈ ℝ+ → ((abs‘(𝑦 / 2)) · (normℎ‘𝑥)) = ((𝑦 / 2) · (normℎ‘𝑥)))
60	59	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → ((abs‘(𝑦 / 2)) · (normℎ‘𝑥)) = ((𝑦 / 2) · (normℎ‘𝑥)))
61	55, 60	eqtr2d 2776	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → ((𝑦 / 2) · (normℎ‘𝑥)) = (normℎ‘((𝑦 / 2) ·ℎ 𝑥)))
62	50, 40, 61	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((𝑦 / 2) · (normℎ‘𝑥)) = (normℎ‘((𝑦 / 2) ·ℎ 𝑥)))
63	39	mulridd 11276	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((𝑦 / 2) · 1) = (𝑦 / 2))
64	52, 62, 63	3brtr3d 5179	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (normℎ‘((𝑦 / 2) ·ℎ 𝑥)) ≤ (𝑦 / 2))
65		rphalflt 13062	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℝ+ → (𝑦 / 2) < 𝑦)
66	65	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (𝑦 / 2) < 𝑦)
67	44, 38, 37, 64, 66	lelttrd 11417	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (normℎ‘((𝑦 / 2) ·ℎ 𝑥)) < 𝑦)
68		fveq2 6907	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = ((𝑦 / 2) ·ℎ 𝑥) → (normℎ‘𝑧) = (normℎ‘((𝑦 / 2) ·ℎ 𝑥)))
69	68	breq1d 5158	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = ((𝑦 / 2) ·ℎ 𝑥) → ((normℎ‘𝑧) < 𝑦 ↔ (normℎ‘((𝑦 / 2) ·ℎ 𝑥)) < 𝑦))
70		2fveq3 6912	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = ((𝑦 / 2) ·ℎ 𝑥) → (𝑁‘(𝑇‘𝑧)) = (𝑁‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥))))
71	70	breq1d 5158	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = ((𝑦 / 2) ·ℎ 𝑥) → ((𝑁‘(𝑇‘𝑧)) < 1 ↔ (𝑁‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥))) < 1))
72	69, 71	imbi12d 344	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = ((𝑦 / 2) ·ℎ 𝑥) → (((normℎ‘𝑧) < 𝑦 → (𝑁‘(𝑇‘𝑧)) < 1) ↔ ((normℎ‘((𝑦 / 2) ·ℎ 𝑥)) < 𝑦 → (𝑁‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥))) < 1)))
73	72	rspcv 3618	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 / 2) ·ℎ 𝑥) ∈ ℋ → (∀𝑧 ∈ ℋ ((normℎ‘𝑧) < 𝑦 → (𝑁‘(𝑇‘𝑧)) < 1) → ((normℎ‘((𝑦 / 2) ·ℎ 𝑥)) < 𝑦 → (𝑁‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥))) < 1)))
74	42, 73	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (∀𝑧 ∈ ℋ ((normℎ‘𝑧) < 𝑦 → (𝑁‘(𝑇‘𝑧)) < 1) → ((normℎ‘((𝑦 / 2) ·ℎ 𝑥)) < 𝑦 → (𝑁‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥))) < 1)))
75	67, 74	mpid 44	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (∀𝑧 ∈ ℋ ((normℎ‘𝑧) < 𝑦 → (𝑁‘(𝑇‘𝑧)) < 1) → (𝑁‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥))) < 1))
76	2	ad2antrl 728	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (𝑁‘(𝑇‘𝑥)) ∈ ℝ)
77	76, 48, 50	ltmuldiv2d 13123	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (((𝑦 / 2) · (𝑁‘(𝑇‘𝑥))) < 1 ↔ (𝑁‘(𝑇‘𝑥)) < (1 / (𝑦 / 2))))
78	50	rprecred 13086	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (1 / (𝑦 / 2)) ∈ ℝ)
79		ltle 11347	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁‘(𝑇‘𝑥)) ∈ ℝ ∧ (1 / (𝑦 / 2)) ∈ ℝ) → ((𝑁‘(𝑇‘𝑥)) < (1 / (𝑦 / 2)) → (𝑁‘(𝑇‘𝑥)) ≤ (1 / (𝑦 / 2))))
80	76, 78, 79	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((𝑁‘(𝑇‘𝑥)) < (1 / (𝑦 / 2)) → (𝑁‘(𝑇‘𝑥)) ≤ (1 / (𝑦 / 2))))
81	77, 80	sylbid 240	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (((𝑦 / 2) · (𝑁‘(𝑇‘𝑥))) < 1 → (𝑁‘(𝑇‘𝑥)) ≤ (1 / (𝑦 / 2))))
82		nmcex.5	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → ((𝑦 / 2) · (𝑁‘(𝑇‘𝑥))) = (𝑁‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥))))
83	50, 40, 82	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((𝑦 / 2) · (𝑁‘(𝑇‘𝑥))) = (𝑁‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥))))
84	83	breq1d 5158	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (((𝑦 / 2) · (𝑁‘(𝑇‘𝑥))) < 1 ↔ (𝑁‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥))) < 1))
85		rpcn 13043	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℝ+ → 𝑦 ∈ ℂ)
86		rpne0 13049	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℝ+ → 𝑦 ≠ 0)
87		2cn 12339	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℂ
88		2ne0 12368	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ≠ 0
89		recdiv 11971	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → (1 / (𝑦 / 2)) = (2 / 𝑦))
90	87, 88, 89	mpanr12 705	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) → (1 / (𝑦 / 2)) = (2 / 𝑦))
91	85, 86, 90	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℝ+ → (1 / (𝑦 / 2)) = (2 / 𝑦))
92	91	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (1 / (𝑦 / 2)) = (2 / 𝑦))
93	92	breq2d 5160	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((𝑁‘(𝑇‘𝑥)) ≤ (1 / (𝑦 / 2)) ↔ (𝑁‘(𝑇‘𝑥)) ≤ (2 / 𝑦)))
94	81, 84, 93	3imtr3d 293	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((𝑁‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥))) < 1 → (𝑁‘(𝑇‘𝑥)) ≤ (2 / 𝑦)))
95	75, 94	syld 47	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (∀𝑧 ∈ ℋ ((normℎ‘𝑧) < 𝑦 → (𝑁‘(𝑇‘𝑧)) < 1) → (𝑁‘(𝑇‘𝑥)) ≤ (2 / 𝑦)))
96	95	imp 406	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) ∧ ∀𝑧 ∈ ℋ ((normℎ‘𝑧) < 𝑦 → (𝑁‘(𝑇‘𝑧)) < 1)) → (𝑁‘(𝑇‘𝑥)) ≤ (2 / 𝑦))
97	96	an32s 652	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((normℎ‘𝑧) < 𝑦 → (𝑁‘(𝑇‘𝑧)) < 1)) ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (𝑁‘(𝑇‘𝑥)) ≤ (2 / 𝑦))
98	97	anassrs 467	. . . . . . . . . . 11 ⊢ ((((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((normℎ‘𝑧) < 𝑦 → (𝑁‘(𝑇‘𝑧)) < 1)) ∧ 𝑥 ∈ ℋ) ∧ (normℎ‘𝑥) ≤ 1) → (𝑁‘(𝑇‘𝑥)) ≤ (2 / 𝑦))
99		breq1 5151	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑁‘(𝑇‘𝑥)) → (𝑛 ≤ (2 / 𝑦) ↔ (𝑁‘(𝑇‘𝑥)) ≤ (2 / 𝑦)))
100	98, 99	syl5ibrcom 247	. . . . . . . . . 10 ⊢ ((((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((normℎ‘𝑧) < 𝑦 → (𝑁‘(𝑇‘𝑧)) < 1)) ∧ 𝑥 ∈ ℋ) ∧ (normℎ‘𝑥) ≤ 1) → (𝑛 = (𝑁‘(𝑇‘𝑥)) → 𝑛 ≤ (2 / 𝑦)))
101	100	expimpd 453	. . . . . . . . 9 ⊢ (((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((normℎ‘𝑧) < 𝑦 → (𝑁‘(𝑇‘𝑧)) < 1)) ∧ 𝑥 ∈ ℋ) → (((normℎ‘𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇‘𝑥))) → 𝑛 ≤ (2 / 𝑦)))
102	101	rexlimdva 3153	. . . . . . . 8 ⊢ ((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((normℎ‘𝑧) < 𝑦 → (𝑁‘(𝑇‘𝑧)) < 1)) → (∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇‘𝑥))) → 𝑛 ≤ (2 / 𝑦)))
103	102	alrimiv 1925	. . . . . . 7 ⊢ ((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((normℎ‘𝑧) < 𝑦 → (𝑁‘(𝑇‘𝑧)) < 1)) → ∀𝑛(∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇‘𝑥))) → 𝑛 ≤ (2 / 𝑦)))
104		eqeq1 2739	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (𝑚 = (𝑁‘(𝑇‘𝑥)) ↔ 𝑛 = (𝑁‘(𝑇‘𝑥))))
105	104	anbi2d 630	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥))) ↔ ((normℎ‘𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇‘𝑥)))))
106	105	rexbidv 3177	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥))) ↔ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇‘𝑥)))))
107	106	ralab 3700	. . . . . . . . 9 ⊢ (∀𝑛 ∈ {𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))}𝑛 ≤ 𝑧 ↔ ∀𝑛(∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇‘𝑥))) → 𝑛 ≤ 𝑧))
108		breq2 5152	. . . . . . . . . . 11 ⊢ (𝑧 = (2 / 𝑦) → (𝑛 ≤ 𝑧 ↔ 𝑛 ≤ (2 / 𝑦)))
109	108	imbi2d 340	. . . . . . . . . 10 ⊢ (𝑧 = (2 / 𝑦) → ((∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇‘𝑥))) → 𝑛 ≤ 𝑧) ↔ (∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇‘𝑥))) → 𝑛 ≤ (2 / 𝑦))))
110	109	albidv 1918	. . . . . . . . 9 ⊢ (𝑧 = (2 / 𝑦) → (∀𝑛(∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇‘𝑥))) → 𝑛 ≤ 𝑧) ↔ ∀𝑛(∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇‘𝑥))) → 𝑛 ≤ (2 / 𝑦))))
111	107, 110	bitrid 283	. . . . . . . 8 ⊢ (𝑧 = (2 / 𝑦) → (∀𝑛 ∈ {𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))}𝑛 ≤ 𝑧 ↔ ∀𝑛(∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇‘𝑥))) → 𝑛 ≤ (2 / 𝑦))))
112	111	rspcev 3622	. . . . . . 7 ⊢ (((2 / 𝑦) ∈ ℝ ∧ ∀𝑛(∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇‘𝑥))) → 𝑛 ≤ (2 / 𝑦))) → ∃𝑧 ∈ ℝ ∀𝑛 ∈ {𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))}𝑛 ≤ 𝑧)
113	35, 103, 112	syl2anc 584	. . . . . 6 ⊢ ((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((normℎ‘𝑧) < 𝑦 → (𝑁‘(𝑇‘𝑧)) < 1)) → ∃𝑧 ∈ ℝ ∀𝑛 ∈ {𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))}𝑛 ≤ 𝑧)
114	113	rexlimiva 3145	. . . . 5 ⊢ (∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℋ ((normℎ‘𝑧) < 𝑦 → (𝑁‘(𝑇‘𝑧)) < 1) → ∃𝑧 ∈ ℝ ∀𝑛 ∈ {𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))}𝑛 ≤ 𝑧)
115	30, 114	ax-mp 5	. . . 4 ⊢ ∃𝑧 ∈ ℝ ∀𝑛 ∈ {𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))}𝑛 ≤ 𝑧
116		supxrre 13366	. . . 4 ⊢ (({𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))} ⊆ ℝ ∧ {𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))} ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑛 ∈ {𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))}𝑛 ≤ 𝑧) → sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))}, ℝ*, < ) = sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))}, ℝ, < ))
117	8, 29, 115, 116	mp3an 1460	. . 3 ⊢ sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))}, ℝ*, < ) = sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))}, ℝ, < )
118	1, 117	eqtri 2763	. 2 ⊢ (𝑆‘𝑇) = sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))}, ℝ, < )
119		suprcl 12226	. . 3 ⊢ (({𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))} ⊆ ℝ ∧ {𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))} ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑛 ∈ {𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))}𝑛 ≤ 𝑧) → sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))}, ℝ, < ) ∈ ℝ)
120	8, 29, 115, 119	mp3an 1460	. 2 ⊢ sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))}, ℝ, < ) ∈ ℝ
121	118, 120	eqeltri 2835	1 ⊢ (𝑆‘𝑇) ∈ ℝ

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1535   = wceq 1537   ∈ wcel 2106  {cab 2712   ≠ wne 2938  ∀wral 3059  ∃wrex 3068   ⊆ wss 3963  ∅c0 4339   class class class wbr 5148  ‘cfv 6563  (class class class)co 7431  supcsup 9478  ℂcc 11151  ℝcr 11152  0cc0 11153  1c1 11154   · cmul 11158  ℝ^*cxr 11292   < clt 11293   ≤ cle 11294   / cdiv 11918  2c2 12319  ℝ⁺crp 13032  abscabs 15270   ℋchba 30948   ·_ℎ csm 30950  norm_ℎcno 30952  0_ℎc0v 30953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231  ax-hv0cl 31032  ax-hfvmul 31034  ax-hvmul0 31039  ax-hfi 31108  ax-his1 31111  ax-his3 31113  ax-his4 31114

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-sup 9480  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-rp 13033  df-seq 14040  df-exp 14100  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-hnorm 30997

This theorem is referenced by:  nmcopexi  32056  nmcfnexi  32080