Theorem nmcexi 32058

Description: Lemma for nmcopexi 32059 and nmcfnexi 32083. 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 2832	. . . . . . . . 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 715	. . . . . 6 ⊢ ((𝑥 ∈ ℋ ∧ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))) → 𝑚 ∈ ℝ)
7	6	rexlimiva 3153	. . . . 5 ⊢ (∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥))) → 𝑚 ∈ ℝ)
8	7	abssi 4093	. . . 4 ⊢ {𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))} ⊆ ℝ
9		ax-hv0cl 31035	. . . . . . 7 ⊢ 0ℎ ∈ ℋ
10		norm0 31160	. . . . . . . . 9 ⊢ (normℎ‘0ℎ) = 0
11		0le1 11813	. . . . . . . . 9 ⊢ 0 ≤ 1
12	10, 11	eqbrtri 5187	. . . . . . . 8 ⊢ (normℎ‘0ℎ) ≤ 1
13		nmcex.4	. . . . . . . . 9 ⊢ (𝑁‘(𝑇‘0ℎ)) = 0
14	13	eqcomi 2749	. . . . . . . 8 ⊢ 0 = (𝑁‘(𝑇‘0ℎ))
15	12, 14	pm3.2i 470	. . . . . . 7 ⊢ ((normℎ‘0ℎ) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘0ℎ)))
16		fveq2 6920	. . . . . . . . . 10 ⊢ (𝑥 = 0ℎ → (normℎ‘𝑥) = (normℎ‘0ℎ))
17	16	breq1d 5176	. . . . . . . . 9 ⊢ (𝑥 = 0ℎ → ((normℎ‘𝑥) ≤ 1 ↔ (normℎ‘0ℎ) ≤ 1))
18		2fveq3 6925	. . . . . . . . . 10 ⊢ (𝑥 = 0ℎ → (𝑁‘(𝑇‘𝑥)) = (𝑁‘(𝑇‘0ℎ)))
19	18	eqeq2d 2751	. . . . . . . . 9 ⊢ (𝑥 = 0ℎ → (0 = (𝑁‘(𝑇‘𝑥)) ↔ 0 = (𝑁‘(𝑇‘0ℎ))))
20	17, 19	anbi12d 631	. . . . . . . 8 ⊢ (𝑥 = 0ℎ → (((normℎ‘𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘𝑥))) ↔ ((normℎ‘0ℎ) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘0ℎ)))))
21	20	rspcev 3635	. . . . . . 7 ⊢ ((0ℎ ∈ ℋ ∧ ((normℎ‘0ℎ) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘0ℎ)))) → ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘𝑥))))
22	9, 15, 21	mp2an 691	. . . . . 6 ⊢ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘𝑥)))
23		c0ex 11284	. . . . . . 7 ⊢ 0 ∈ V
24		eqeq1 2744	. . . . . . . . 9 ⊢ (𝑚 = 0 → (𝑚 = (𝑁‘(𝑇‘𝑥)) ↔ 0 = (𝑁‘(𝑇‘𝑥))))
25	24	anbi2d 629	. . . . . . . 8 ⊢ (𝑚 = 0 → (((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥))) ↔ ((normℎ‘𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘𝑥)))))
26	25	rexbidv 3185	. . . . . . 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 3694	. . . . . 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 4367	. . . 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 13062	. . . . . . . . . 10 ⊢ 2 ∈ ℝ+
32		rpdivcl 13082	. . . . . . . . . 10 ⊢ ((2 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+) → (2 / 𝑦) ∈ ℝ+)
33	31, 32	mpan 689	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ+ → (2 / 𝑦) ∈ ℝ+)
34	33	rpred 13099	. . . . . . . 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 13065	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ ℝ+ → 𝑦 ∈ ℝ)
37	36	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → 𝑦 ∈ ℝ)
38	37	rehalfcld 12540	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (𝑦 / 2) ∈ ℝ)
39	38	recnd 11318	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (𝑦 / 2) ∈ ℂ)
40		simprl 770	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → 𝑥 ∈ ℋ)
41		hvmulcl 31045	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑦 / 2) ∈ ℂ ∧ 𝑥 ∈ ℋ) → ((𝑦 / 2) ·ℎ 𝑥) ∈ ℋ)
42	39, 40, 41	syl2anc 583	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((𝑦 / 2) ·ℎ 𝑥) ∈ ℋ)
43		normcl 31157	. . . . . . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (normℎ‘𝑥) ≤ 1)
46		normcl 31157	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℋ → (normℎ‘𝑥) ∈ ℝ)
47	46	ad2antrl 727	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (normℎ‘𝑥) ∈ ℝ)
48		1red 11291	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → 1 ∈ ℝ)
49		rphalfcl 13084	. . . . . . . . . . . . . . . . . . . . 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 13143	. . . . . . . . . . . . . . . . . . 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 13067	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 / 2) ∈ ℝ+ → (𝑦 / 2) ∈ ℂ)
54		norm-iii 31172	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑦 / 2) ∈ ℂ ∧ 𝑥 ∈ ℋ) → (normℎ‘((𝑦 / 2) ·ℎ 𝑥)) = ((abs‘(𝑦 / 2)) · (normℎ‘𝑥)))
55	53, 54	sylan 579	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → (normℎ‘((𝑦 / 2) ·ℎ 𝑥)) = ((abs‘(𝑦 / 2)) · (normℎ‘𝑥)))
56		rpre 13065	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 / 2) ∈ ℝ+ → (𝑦 / 2) ∈ ℝ)
57		rpge0 13070	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 / 2) ∈ ℝ+ → 0 ≤ (𝑦 / 2))
58	56, 57	absidd 15471	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 / 2) ∈ ℝ+ → (abs‘(𝑦 / 2)) = (𝑦 / 2))
59	58	oveq1d 7463	. . . . . . . . . . . . . . . . . . . . 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 2781	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → ((𝑦 / 2) · (normℎ‘𝑥)) = (normℎ‘((𝑦 / 2) ·ℎ 𝑥)))
62	50, 40, 61	syl2anc 583	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((𝑦 / 2) · (normℎ‘𝑥)) = (normℎ‘((𝑦 / 2) ·ℎ 𝑥)))
63	39	mulridd 11307	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((𝑦 / 2) · 1) = (𝑦 / 2))
64	52, 62, 63	3brtr3d 5197	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (normℎ‘((𝑦 / 2) ·ℎ 𝑥)) ≤ (𝑦 / 2))
65		rphalflt 13086	. . . . . . . . . . . . . . . . . 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 11448	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (normℎ‘((𝑦 / 2) ·ℎ 𝑥)) < 𝑦)
68		fveq2 6920	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = ((𝑦 / 2) ·ℎ 𝑥) → (normℎ‘𝑧) = (normℎ‘((𝑦 / 2) ·ℎ 𝑥)))
69	68	breq1d 5176	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = ((𝑦 / 2) ·ℎ 𝑥) → ((normℎ‘𝑧) < 𝑦 ↔ (normℎ‘((𝑦 / 2) ·ℎ 𝑥)) < 𝑦))
70		2fveq3 6925	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = ((𝑦 / 2) ·ℎ 𝑥) → (𝑁‘(𝑇‘𝑧)) = (𝑁‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥))))
71	70	breq1d 5176	. . . . . . . . . . . . . . . . . . 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 3631	. . . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (𝑁‘(𝑇‘𝑥)) ∈ ℝ)
77	76, 48, 50	ltmuldiv2d 13147	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (((𝑦 / 2) · (𝑁‘(𝑇‘𝑥))) < 1 ↔ (𝑁‘(𝑇‘𝑥)) < (1 / (𝑦 / 2))))
78	50	rprecred 13110	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (1 / (𝑦 / 2)) ∈ ℝ)
79		ltle 11378	. . . . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((𝑦 / 2) · (𝑁‘(𝑇‘𝑥))) = (𝑁‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥))))
84	83	breq1d 5176	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (((𝑦 / 2) · (𝑁‘(𝑇‘𝑥))) < 1 ↔ (𝑁‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥))) < 1))
85		rpcn 13067	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℝ+ → 𝑦 ∈ ℂ)
86		rpne0 13073	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℝ+ → 𝑦 ≠ 0)
87		2cn 12368	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℂ
88		2ne0 12397	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ≠ 0
89		recdiv 12000	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → (1 / (𝑦 / 2)) = (2 / 𝑦))
90	87, 88, 89	mpanr12 704	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) → (1 / (𝑦 / 2)) = (2 / 𝑦))
91	85, 86, 90	syl2anc 583	. . . . . . . . . . . . . . . . . 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 5178	. . . . . . . . . . . . . . . 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 651	. . . . . . . . . . . 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 5169	. . . . . . . . . . 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 3161	. . . . . . . 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 1926	. . . . . . 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 2744	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (𝑚 = (𝑁‘(𝑇‘𝑥)) ↔ 𝑛 = (𝑁‘(𝑇‘𝑥))))
105	104	anbi2d 629	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥))) ↔ ((normℎ‘𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇‘𝑥)))))
106	105	rexbidv 3185	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥))) ↔ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇‘𝑥)))))
107	106	ralab 3713	. . . . . . . . 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 5170	. . . . . . . . . . 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 1919	. . . . . . . . 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 3635	. . . . . . 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 583	. . . . . 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 3153	. . . . 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 13389	. . . 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 1461	. . 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 2768	. 2 ⊢ (𝑆‘𝑇) = sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))}, ℝ, < )
119		suprcl 12255	. . 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 1461	. 2 ⊢ sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))}, ℝ, < ) ∈ ℝ
121	118, 120	eqeltri 2840	1 ⊢ (𝑆‘𝑇) ∈ ℝ

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1535   = wceq 1537   ∈ wcel 2108  {cab 2717   ≠ wne 2946  ∀wral 3067  ∃wrex 3076   ⊆ wss 3976  ∅c0 4352   class class class wbr 5166  ‘cfv 6573  (class class class)co 7448  supcsup 9509  ℂcc 11182  ℝcr 11183  0cc0 11184  1c1 11185   · cmul 11189  ℝ^*cxr 11323   < clt 11324   ≤ cle 11325   / cdiv 11947  2c2 12348  ℝ⁺crp 13057  abscabs 15283   ℋchba 30951   ·_ℎ csm 30953  norm_ℎcno 30955  0_ℎc0v 30956

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262  ax-hv0cl 31035  ax-hfvmul 31037  ax-hvmul0 31042  ax-hfi 31111  ax-his1 31114  ax-his3 31116  ax-his4 31117

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-sup 9511  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-z 12640  df-uz 12904  df-rp 13058  df-seq 14053  df-exp 14113  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-hnorm 31000

This theorem is referenced by:  nmcopexi  32059  nmcfnexi  32083