Theorem nmcexi 32229

Description: Lemma for nmcopexi 32230 and nmcfnexi 32254. 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 2850	. . . . . . . . 9 ⊢ (𝑚 = (𝑁‘(𝑇‘𝑥)) → (𝑚 ∈ ℝ ↔ (𝑁‘(𝑇‘𝑥)) ∈ ℝ))
4	2, 3	syl5ibrcom 249	. . . . . . . 8 ⊢ (𝑥 ∈ ℋ → (𝑚 = (𝑁‘(𝑇‘𝑥)) → 𝑚 ∈ ℝ))
5	4	imp 410	. . . . . . 7 ⊢ ((𝑥 ∈ ℋ ∧ 𝑚 = (𝑁‘(𝑇‘𝑥))) → 𝑚 ∈ ℝ)
6	5	adantrl 726	. . . . . 6 ⊢ ((𝑥 ∈ ℋ ∧ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))) → 𝑚 ∈ ℝ)
7	6	rexlimiva 3155	. . . . 5 ⊢ (∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥))) → 𝑚 ∈ ℝ)
8	7	abssi 4021	. . . 4 ⊢ {𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))} ⊆ ℝ
9		ax-hv0cl 31206	. . . . . . 7 ⊢ 0ℎ ∈ ℋ
10		norm0 31331	. . . . . . . . 9 ⊢ (normℎ‘0ℎ) = 0
11		0le1 11710	. . . . . . . . 9 ⊢ 0 ≤ 1
12	10, 11	eqbrtri 5121	. . . . . . . 8 ⊢ (normℎ‘0ℎ) ≤ 1
13		nmcex.4	. . . . . . . . 9 ⊢ (𝑁‘(𝑇‘0ℎ)) = 0
14	13	eqcomi 2771	. . . . . . . 8 ⊢ 0 = (𝑁‘(𝑇‘0ℎ))
15	12, 14	pm3.2i 474	. . . . . . 7 ⊢ ((normℎ‘0ℎ) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘0ℎ)))
16		fveq2 6867	. . . . . . . . . 10 ⊢ (𝑥 = 0ℎ → (normℎ‘𝑥) = (normℎ‘0ℎ))
17	16	breq1d 5110	. . . . . . . . 9 ⊢ (𝑥 = 0ℎ → ((normℎ‘𝑥) ≤ 1 ↔ (normℎ‘0ℎ) ≤ 1))
18		2fveq3 6872	. . . . . . . . . 10 ⊢ (𝑥 = 0ℎ → (𝑁‘(𝑇‘𝑥)) = (𝑁‘(𝑇‘0ℎ)))
19	18	eqeq2d 2773	. . . . . . . . 9 ⊢ (𝑥 = 0ℎ → (0 = (𝑁‘(𝑇‘𝑥)) ↔ 0 = (𝑁‘(𝑇‘0ℎ))))
20	17, 19	anbi12d 641	. . . . . . . 8 ⊢ (𝑥 = 0ℎ → (((normℎ‘𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘𝑥))) ↔ ((normℎ‘0ℎ) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘0ℎ)))))
21	20	rspcev 3581	. . . . . . 7 ⊢ ((0ℎ ∈ ℋ ∧ ((normℎ‘0ℎ) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘0ℎ)))) → ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘𝑥))))
22	9, 15, 21	mp2an 702	. . . . . 6 ⊢ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘𝑥)))
23		c0ex 11173	. . . . . . 7 ⊢ 0 ∈ V
24		eqeq1 2766	. . . . . . . . 9 ⊢ (𝑚 = 0 → (𝑚 = (𝑁‘(𝑇‘𝑥)) ↔ 0 = (𝑁‘(𝑇‘𝑥))))
25	24	anbi2d 639	. . . . . . . 8 ⊢ (𝑚 = 0 → (((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥))) ↔ ((normℎ‘𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘𝑥)))))
26	25	rexbidv 3186	. . . . . . 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 3638	. . . . . 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 233	. . . . 5 ⊢ 0 ∈ {𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))}
29	28	ne0ii 4296	. . . 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 12998	. . . . . . . . . 10 ⊢ 2 ∈ ℝ+
32		rpdivcl 13020	. . . . . . . . . 10 ⊢ ((2 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+) → (2 / 𝑦) ∈ ℝ+)
33	31, 32	mpan 700	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ+ → (2 / 𝑦) ∈ ℝ+)
34	33	rpred 13037	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ+ → (2 / 𝑦) ∈ ℝ)
35	34	adantr 484	. . . . . . 7 ⊢ ((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((normℎ‘𝑧) < 𝑦 → (𝑁‘(𝑇‘𝑧)) < 1)) → (2 / 𝑦) ∈ ℝ)
36		rpre 13002	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ ℝ+ → 𝑦 ∈ ℝ)
37	36	adantr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → 𝑦 ∈ ℝ)
38	37	rehalfcld 12468	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (𝑦 / 2) ∈ ℝ)
39	38	recnd 11210	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (𝑦 / 2) ∈ ℂ)
40		simprl 780	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → 𝑥 ∈ ℋ)
41		hvmulcl 31216	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑦 / 2) ∈ ℂ ∧ 𝑥 ∈ ℋ) → ((𝑦 / 2) ·ℎ 𝑥) ∈ ℋ)
42	39, 40, 41	syl2anc 593	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((𝑦 / 2) ·ℎ 𝑥) ∈ ℋ)
43		normcl 31328	. . . . . . . . . . . . . . . . . 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 782	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (normℎ‘𝑥) ≤ 1)
46		normcl 31328	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℋ → (normℎ‘𝑥) ∈ ℝ)
47	46	ad2antrl 738	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (normℎ‘𝑥) ∈ ℝ)
48		1red 11182	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → 1 ∈ ℝ)
49		rphalfcl 13022	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ℝ+ → (𝑦 / 2) ∈ ℝ+)
50	49	adantr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (𝑦 / 2) ∈ ℝ+)
51	47, 48, 50	lemul2d 13081	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((normℎ‘𝑥) ≤ 1 ↔ ((𝑦 / 2) · (normℎ‘𝑥)) ≤ ((𝑦 / 2) · 1)))
52	45, 51	mpbid 234	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((𝑦 / 2) · (normℎ‘𝑥)) ≤ ((𝑦 / 2) · 1))
53		rpcn 13004	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 / 2) ∈ ℝ+ → (𝑦 / 2) ∈ ℂ)
54		norm-iii 31343	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑦 / 2) ∈ ℂ ∧ 𝑥 ∈ ℋ) → (normℎ‘((𝑦 / 2) ·ℎ 𝑥)) = ((abs‘(𝑦 / 2)) · (normℎ‘𝑥)))
55	53, 54	sylan 589	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → (normℎ‘((𝑦 / 2) ·ℎ 𝑥)) = ((abs‘(𝑦 / 2)) · (normℎ‘𝑥)))
56		rpre 13002	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 / 2) ∈ ℝ+ → (𝑦 / 2) ∈ ℝ)
57		rpge0 13007	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 / 2) ∈ ℝ+ → 0 ≤ (𝑦 / 2))
58	56, 57	absidd 15450	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 / 2) ∈ ℝ+ → (abs‘(𝑦 / 2)) = (𝑦 / 2))
59	58	oveq1d 7411	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 / 2) ∈ ℝ+ → ((abs‘(𝑦 / 2)) · (normℎ‘𝑥)) = ((𝑦 / 2) · (normℎ‘𝑥)))
60	59	adantr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → ((abs‘(𝑦 / 2)) · (normℎ‘𝑥)) = ((𝑦 / 2) · (normℎ‘𝑥)))
61	55, 60	eqtr2d 2798	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → ((𝑦 / 2) · (normℎ‘𝑥)) = (normℎ‘((𝑦 / 2) ·ℎ 𝑥)))
62	50, 40, 61	syl2anc 593	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((𝑦 / 2) · (normℎ‘𝑥)) = (normℎ‘((𝑦 / 2) ·ℎ 𝑥)))
63	39	mulridd 11199	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((𝑦 / 2) · 1) = (𝑦 / 2))
64	52, 62, 63	3brtr3d 5131	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (normℎ‘((𝑦 / 2) ·ℎ 𝑥)) ≤ (𝑦 / 2))
65		rphalflt 13024	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℝ+ → (𝑦 / 2) < 𝑦)
66	65	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (𝑦 / 2) < 𝑦)
67	44, 38, 37, 64, 66	lelttrd 11341	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (normℎ‘((𝑦 / 2) ·ℎ 𝑥)) < 𝑦)
68		fveq2 6867	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = ((𝑦 / 2) ·ℎ 𝑥) → (normℎ‘𝑧) = (normℎ‘((𝑦 / 2) ·ℎ 𝑥)))
69	68	breq1d 5110	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = ((𝑦 / 2) ·ℎ 𝑥) → ((normℎ‘𝑧) < 𝑦 ↔ (normℎ‘((𝑦 / 2) ·ℎ 𝑥)) < 𝑦))
70		2fveq3 6872	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = ((𝑦 / 2) ·ℎ 𝑥) → (𝑁‘(𝑇‘𝑧)) = (𝑁‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥))))
71	70	breq1d 5110	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = ((𝑦 / 2) ·ℎ 𝑥) → ((𝑁‘(𝑇‘𝑧)) < 1 ↔ (𝑁‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥))) < 1))
72	69, 71	imbi12d 346	. . . . . . . . . . . . . . . . . 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 3577	. . . . . . . . . . . . . . . . 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 738	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (𝑁‘(𝑇‘𝑥)) ∈ ℝ)
77	76, 48, 50	ltmuldiv2d 13085	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (((𝑦 / 2) · (𝑁‘(𝑇‘𝑥))) < 1 ↔ (𝑁‘(𝑇‘𝑥)) < (1 / (𝑦 / 2))))
78	50	rprecred 13048	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (1 / (𝑦 / 2)) ∈ ℝ)
79		ltle 11271	. . . . . . . . . . . . . . . . . 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 593	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((𝑁‘(𝑇‘𝑥)) < (1 / (𝑦 / 2)) → (𝑁‘(𝑇‘𝑥)) ≤ (1 / (𝑦 / 2))))
81	77, 80	sylbid 242	. . . . . . . . . . . . . . . 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 593	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((𝑦 / 2) · (𝑁‘(𝑇‘𝑥))) = (𝑁‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥))))
84	83	breq1d 5110	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (((𝑦 / 2) · (𝑁‘(𝑇‘𝑥))) < 1 ↔ (𝑁‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥))) < 1))
85		rpcn 13004	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℝ+ → 𝑦 ∈ ℂ)
86		rpne0 13010	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℝ+ → 𝑦 ≠ 0)
87		2cn 12293	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℂ
88		2ne0 12324	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ≠ 0
89		recdiv 11897	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → (1 / (𝑦 / 2)) = (2 / 𝑦))
90	87, 88, 89	mpanr12 715	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) → (1 / (𝑦 / 2)) = (2 / 𝑦))
91	85, 86, 90	syl2anc 593	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℝ+ → (1 / (𝑦 / 2)) = (2 / 𝑦))
92	91	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (1 / (𝑦 / 2)) = (2 / 𝑦))
93	92	breq2d 5112	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((𝑁‘(𝑇‘𝑥)) ≤ (1 / (𝑦 / 2)) ↔ (𝑁‘(𝑇‘𝑥)) ≤ (2 / 𝑦)))
94	81, 84, 93	3imtr3d 295	. . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . 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 662	. . . . . . . . . . . 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 471	. . . . . . . . . . 11 ⊢ ((((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((normℎ‘𝑧) < 𝑦 → (𝑁‘(𝑇‘𝑧)) < 1)) ∧ 𝑥 ∈ ℋ) ∧ (normℎ‘𝑥) ≤ 1) → (𝑁‘(𝑇‘𝑥)) ≤ (2 / 𝑦))
99		breq1 5103	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑁‘(𝑇‘𝑥)) → (𝑛 ≤ (2 / 𝑦) ↔ (𝑁‘(𝑇‘𝑥)) ≤ (2 / 𝑦)))
100	98, 99	syl5ibrcom 249	. . . . . . . . . 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 457	. . . . . . . . 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 3163	. . . . . . . 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 1947	. . . . . . 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 2766	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (𝑚 = (𝑁‘(𝑇‘𝑥)) ↔ 𝑛 = (𝑁‘(𝑇‘𝑥))))
105	104	anbi2d 639	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥))) ↔ ((normℎ‘𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇‘𝑥)))))
106	105	rexbidv 3186	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥))) ↔ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇‘𝑥)))))
107	106	ralab 3656	. . . . . . . . 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 5104	. . . . . . . . . . 11 ⊢ (𝑧 = (2 / 𝑦) → (𝑛 ≤ 𝑧 ↔ 𝑛 ≤ (2 / 𝑦)))
109	108	imbi2d 342	. . . . . . . . . 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 1940	. . . . . . . . 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 285	. . . . . . . 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 3581	. . . . . . 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 593	. . . . . 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 3155	. . . . 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 13330	. . . 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 1482	. . 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 2785	. 2 ⊢ (𝑆‘𝑇) = sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))}, ℝ, < )
119		suprcl 12152	. . 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 1482	. 2 ⊢ sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))}, ℝ, < ) ∈ ℝ
121	118, 120	eqeltri 2858	1 ⊢ (𝑆‘𝑇) ∈ ℝ

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1558   = wceq 1560   ∈ wcel 2142  {cab 2740   ≠ wne 2957  ∀wral 3076  ∃wrex 3086   ⊆ wss 3904  ∅c0 4285   class class class wbr 5100  ‘cfv 6521  (class class class)co 7396  supcsup 9386  ℂcc 11071  ℝcr 11072  0cc0 11073  1c1 11074   · cmul 11078  ℝ^*cxr 11215   < clt 11216   ≤ cle 11217   / cdiv 11844  2c2 12272  ℝ⁺crp 12993  abscabs 15261   ℋchba 31122   ·_ℎ csm 31124  norm_ℎcno 31126  0_ℎc0v 31127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-cnex 11129  ax-resscn 11130  ax-1cn 11131  ax-icn 11132  ax-addcl 11133  ax-addrcl 11134  ax-mulcl 11135  ax-mulrcl 11136  ax-mulcom 11137  ax-addass 11138  ax-mulass 11139  ax-distr 11140  ax-i2m1 11141  ax-1ne0 11142  ax-1rid 11143  ax-rnegex 11144  ax-rrecex 11145  ax-cnre 11146  ax-pre-lttri 11147  ax-pre-lttrn 11148  ax-pre-ltadd 11149  ax-pre-mulgt0 11150  ax-pre-sup 11151  ax-hv0cl 31206  ax-hfvmul 31208  ax-hvmul0 31213  ax-hfi 31282  ax-his1 31285  ax-his3 31287  ax-his4 31288

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-er 8678  df-en 8928  df-dom 8929  df-sdom 8930  df-sup 9388  df-pnf 11218  df-mnf 11219  df-xr 11220  df-ltxr 11221  df-le 11222  df-sub 11416  df-neg 11417  df-div 11845  df-nn 12211  df-2 12280  df-3 12281  df-n0 12482  df-z 12569  df-uz 12840  df-rp 12994  df-seq 14015  df-exp 14075  df-cj 15126  df-re 15127  df-im 15128  df-sqrt 15262  df-abs 15263  df-hnorm 31171

This theorem is referenced by:  nmcopexi  32230  nmcfnexi  32254