Theorem nmcexi 31908

Description: Lemma for nmcopexi 31909 and nmcfnexi 31933. 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 2813	. . . . . . . . 9 ⊢ (𝑚 = (𝑁‘(𝑇‘𝑥)) → (𝑚 ∈ ℝ ↔ (𝑁‘(𝑇‘𝑥)) ∈ ℝ))
4	2, 3	syl5ibrcom 246	. . . . . . . 8 ⊢ (𝑥 ∈ ℋ → (𝑚 = (𝑁‘(𝑇‘𝑥)) → 𝑚 ∈ ℝ))
5	4	imp 405	. . . . . . 7 ⊢ ((𝑥 ∈ ℋ ∧ 𝑚 = (𝑁‘(𝑇‘𝑥))) → 𝑚 ∈ ℝ)
6	5	adantrl 714	. . . . . 6 ⊢ ((𝑥 ∈ ℋ ∧ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))) → 𝑚 ∈ ℝ)
7	6	rexlimiva 3136	. . . . 5 ⊢ (∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥))) → 𝑚 ∈ ℝ)
8	7	abssi 4063	. . . 4 ⊢ {𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))} ⊆ ℝ
9		ax-hv0cl 30885	. . . . . . 7 ⊢ 0ℎ ∈ ℋ
10		norm0 31010	. . . . . . . . 9 ⊢ (normℎ‘0ℎ) = 0
11		0le1 11769	. . . . . . . . 9 ⊢ 0 ≤ 1
12	10, 11	eqbrtri 5170	. . . . . . . 8 ⊢ (normℎ‘0ℎ) ≤ 1
13		nmcex.4	. . . . . . . . 9 ⊢ (𝑁‘(𝑇‘0ℎ)) = 0
14	13	eqcomi 2734	. . . . . . . 8 ⊢ 0 = (𝑁‘(𝑇‘0ℎ))
15	12, 14	pm3.2i 469	. . . . . . 7 ⊢ ((normℎ‘0ℎ) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘0ℎ)))
16		fveq2 6896	. . . . . . . . . 10 ⊢ (𝑥 = 0ℎ → (normℎ‘𝑥) = (normℎ‘0ℎ))
17	16	breq1d 5159	. . . . . . . . 9 ⊢ (𝑥 = 0ℎ → ((normℎ‘𝑥) ≤ 1 ↔ (normℎ‘0ℎ) ≤ 1))
18		2fveq3 6901	. . . . . . . . . 10 ⊢ (𝑥 = 0ℎ → (𝑁‘(𝑇‘𝑥)) = (𝑁‘(𝑇‘0ℎ)))
19	18	eqeq2d 2736	. . . . . . . . 9 ⊢ (𝑥 = 0ℎ → (0 = (𝑁‘(𝑇‘𝑥)) ↔ 0 = (𝑁‘(𝑇‘0ℎ))))
20	17, 19	anbi12d 630	. . . . . . . 8 ⊢ (𝑥 = 0ℎ → (((normℎ‘𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘𝑥))) ↔ ((normℎ‘0ℎ) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘0ℎ)))))
21	20	rspcev 3606	. . . . . . 7 ⊢ ((0ℎ ∈ ℋ ∧ ((normℎ‘0ℎ) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘0ℎ)))) → ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘𝑥))))
22	9, 15, 21	mp2an 690	. . . . . 6 ⊢ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘𝑥)))
23		c0ex 11240	. . . . . . 7 ⊢ 0 ∈ V
24		eqeq1 2729	. . . . . . . . 9 ⊢ (𝑚 = 0 → (𝑚 = (𝑁‘(𝑇‘𝑥)) ↔ 0 = (𝑁‘(𝑇‘𝑥))))
25	24	anbi2d 628	. . . . . . . 8 ⊢ (𝑚 = 0 → (((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥))) ↔ ((normℎ‘𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘𝑥)))))
26	25	rexbidv 3168	. . . . . . 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 3664	. . . . . 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 230	. . . . 5 ⊢ 0 ∈ {𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))}
29	28	ne0ii 4337	. . . 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 13014	. . . . . . . . . 10 ⊢ 2 ∈ ℝ+
32		rpdivcl 13034	. . . . . . . . . 10 ⊢ ((2 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+) → (2 / 𝑦) ∈ ℝ+)
33	31, 32	mpan 688	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ+ → (2 / 𝑦) ∈ ℝ+)
34	33	rpred 13051	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ+ → (2 / 𝑦) ∈ ℝ)
35	34	adantr 479	. . . . . . 7 ⊢ ((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((normℎ‘𝑧) < 𝑦 → (𝑁‘(𝑇‘𝑧)) < 1)) → (2 / 𝑦) ∈ ℝ)
36		rpre 13017	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ ℝ+ → 𝑦 ∈ ℝ)
37	36	adantr 479	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → 𝑦 ∈ ℝ)
38	37	rehalfcld 12492	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (𝑦 / 2) ∈ ℝ)
39	38	recnd 11274	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (𝑦 / 2) ∈ ℂ)
40		simprl 769	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → 𝑥 ∈ ℋ)
41		hvmulcl 30895	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑦 / 2) ∈ ℂ ∧ 𝑥 ∈ ℋ) → ((𝑦 / 2) ·ℎ 𝑥) ∈ ℋ)
42	39, 40, 41	syl2anc 582	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((𝑦 / 2) ·ℎ 𝑥) ∈ ℋ)
43		normcl 31007	. . . . . . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (normℎ‘𝑥) ≤ 1)
46		normcl 31007	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℋ → (normℎ‘𝑥) ∈ ℝ)
47	46	ad2antrl 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (normℎ‘𝑥) ∈ ℝ)
48		1red 11247	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → 1 ∈ ℝ)
49		rphalfcl 13036	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ℝ+ → (𝑦 / 2) ∈ ℝ+)
50	49	adantr 479	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (𝑦 / 2) ∈ ℝ+)
51	47, 48, 50	lemul2d 13095	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((normℎ‘𝑥) ≤ 1 ↔ ((𝑦 / 2) · (normℎ‘𝑥)) ≤ ((𝑦 / 2) · 1)))
52	45, 51	mpbid 231	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((𝑦 / 2) · (normℎ‘𝑥)) ≤ ((𝑦 / 2) · 1))
53		rpcn 13019	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 / 2) ∈ ℝ+ → (𝑦 / 2) ∈ ℂ)
54		norm-iii 31022	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑦 / 2) ∈ ℂ ∧ 𝑥 ∈ ℋ) → (normℎ‘((𝑦 / 2) ·ℎ 𝑥)) = ((abs‘(𝑦 / 2)) · (normℎ‘𝑥)))
55	53, 54	sylan 578	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → (normℎ‘((𝑦 / 2) ·ℎ 𝑥)) = ((abs‘(𝑦 / 2)) · (normℎ‘𝑥)))
56		rpre 13017	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 / 2) ∈ ℝ+ → (𝑦 / 2) ∈ ℝ)
57		rpge0 13022	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 / 2) ∈ ℝ+ → 0 ≤ (𝑦 / 2))
58	56, 57	absidd 15405	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 / 2) ∈ ℝ+ → (abs‘(𝑦 / 2)) = (𝑦 / 2))
59	58	oveq1d 7434	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 / 2) ∈ ℝ+ → ((abs‘(𝑦 / 2)) · (normℎ‘𝑥)) = ((𝑦 / 2) · (normℎ‘𝑥)))
60	59	adantr 479	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → ((abs‘(𝑦 / 2)) · (normℎ‘𝑥)) = ((𝑦 / 2) · (normℎ‘𝑥)))
61	55, 60	eqtr2d 2766	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → ((𝑦 / 2) · (normℎ‘𝑥)) = (normℎ‘((𝑦 / 2) ·ℎ 𝑥)))
62	50, 40, 61	syl2anc 582	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((𝑦 / 2) · (normℎ‘𝑥)) = (normℎ‘((𝑦 / 2) ·ℎ 𝑥)))
63	39	mulridd 11263	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((𝑦 / 2) · 1) = (𝑦 / 2))
64	52, 62, 63	3brtr3d 5180	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (normℎ‘((𝑦 / 2) ·ℎ 𝑥)) ≤ (𝑦 / 2))
65		rphalflt 13038	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℝ+ → (𝑦 / 2) < 𝑦)
66	65	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (𝑦 / 2) < 𝑦)
67	44, 38, 37, 64, 66	lelttrd 11404	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (normℎ‘((𝑦 / 2) ·ℎ 𝑥)) < 𝑦)
68		fveq2 6896	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = ((𝑦 / 2) ·ℎ 𝑥) → (normℎ‘𝑧) = (normℎ‘((𝑦 / 2) ·ℎ 𝑥)))
69	68	breq1d 5159	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = ((𝑦 / 2) ·ℎ 𝑥) → ((normℎ‘𝑧) < 𝑦 ↔ (normℎ‘((𝑦 / 2) ·ℎ 𝑥)) < 𝑦))
70		2fveq3 6901	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = ((𝑦 / 2) ·ℎ 𝑥) → (𝑁‘(𝑇‘𝑧)) = (𝑁‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥))))
71	70	breq1d 5159	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = ((𝑦 / 2) ·ℎ 𝑥) → ((𝑁‘(𝑇‘𝑧)) < 1 ↔ (𝑁‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥))) < 1))
72	69, 71	imbi12d 343	. . . . . . . . . . . . . . . . . 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 3602	. . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (𝑁‘(𝑇‘𝑥)) ∈ ℝ)
77	76, 48, 50	ltmuldiv2d 13099	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (((𝑦 / 2) · (𝑁‘(𝑇‘𝑥))) < 1 ↔ (𝑁‘(𝑇‘𝑥)) < (1 / (𝑦 / 2))))
78	50	rprecred 13062	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (1 / (𝑦 / 2)) ∈ ℝ)
79		ltle 11334	. . . . . . . . . . . . . . . . . 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 582	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((𝑁‘(𝑇‘𝑥)) < (1 / (𝑦 / 2)) → (𝑁‘(𝑇‘𝑥)) ≤ (1 / (𝑦 / 2))))
81	77, 80	sylbid 239	. . . . . . . . . . . . . . . 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 582	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((𝑦 / 2) · (𝑁‘(𝑇‘𝑥))) = (𝑁‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥))))
84	83	breq1d 5159	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (((𝑦 / 2) · (𝑁‘(𝑇‘𝑥))) < 1 ↔ (𝑁‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥))) < 1))
85		rpcn 13019	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℝ+ → 𝑦 ∈ ℂ)
86		rpne0 13025	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℝ+ → 𝑦 ≠ 0)
87		2cn 12320	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℂ
88		2ne0 12349	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ≠ 0
89		recdiv 11953	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → (1 / (𝑦 / 2)) = (2 / 𝑦))
90	87, 88, 89	mpanr12 703	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) → (1 / (𝑦 / 2)) = (2 / 𝑦))
91	85, 86, 90	syl2anc 582	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℝ+ → (1 / (𝑦 / 2)) = (2 / 𝑦))
92	91	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (1 / (𝑦 / 2)) = (2 / 𝑦))
93	92	breq2d 5161	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((𝑁‘(𝑇‘𝑥)) ≤ (1 / (𝑦 / 2)) ↔ (𝑁‘(𝑇‘𝑥)) ≤ (2 / 𝑦)))
94	81, 84, 93	3imtr3d 292	. . . . . . . . . . . . . . 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 405	. . . . . . . . . . . . 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 650	. . . . . . . . . . . 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 466	. . . . . . . . . . 11 ⊢ ((((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((normℎ‘𝑧) < 𝑦 → (𝑁‘(𝑇‘𝑧)) < 1)) ∧ 𝑥 ∈ ℋ) ∧ (normℎ‘𝑥) ≤ 1) → (𝑁‘(𝑇‘𝑥)) ≤ (2 / 𝑦))
99		breq1 5152	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑁‘(𝑇‘𝑥)) → (𝑛 ≤ (2 / 𝑦) ↔ (𝑁‘(𝑇‘𝑥)) ≤ (2 / 𝑦)))
100	98, 99	syl5ibrcom 246	. . . . . . . . . 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 452	. . . . . . . . 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 3144	. . . . . . . 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 1922	. . . . . . 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 2729	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (𝑚 = (𝑁‘(𝑇‘𝑥)) ↔ 𝑛 = (𝑁‘(𝑇‘𝑥))))
105	104	anbi2d 628	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥))) ↔ ((normℎ‘𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇‘𝑥)))))
106	105	rexbidv 3168	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥))) ↔ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑛 = (𝑁‘(𝑇‘𝑥)))))
107	106	ralab 3683	. . . . . . . . 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 5153	. . . . . . . . . . 11 ⊢ (𝑧 = (2 / 𝑦) → (𝑛 ≤ 𝑧 ↔ 𝑛 ≤ (2 / 𝑦)))
109	108	imbi2d 339	. . . . . . . . . 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 1915	. . . . . . . . 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 282	. . . . . . . 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 3606	. . . . . . 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 582	. . . . . 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 3136	. . . . 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 13341	. . . 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 1457	. . 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 2753	. 2 ⊢ (𝑆‘𝑇) = sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))}, ℝ, < )
119		suprcl 12207	. . 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 1457	. 2 ⊢ sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))}, ℝ, < ) ∈ ℝ
121	118, 120	eqeltri 2821	1 ⊢ (𝑆‘𝑇) ∈ ℝ

Syntax hints:   → wi 4   ∧ wa 394  ∀wal 1531   = wceq 1533   ∈ wcel 2098  {cab 2702   ≠ wne 2929  ∀wral 3050  ∃wrex 3059   ⊆ wss 3944  ∅c0 4322   class class class wbr 5149  ‘cfv 6549  (class class class)co 7419  supcsup 9465  ℂcc 11138  ℝcr 11139  0cc0 11140  1c1 11141   · cmul 11145  ℝ^*cxr 11279   < clt 11280   ≤ cle 11281   / cdiv 11903  2c2 12300  ℝ⁺crp 13009  abscabs 15217   ℋchba 30801   ·_ℎ csm 30803  norm_ℎcno 30805  0_ℎc0v 30806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217  ax-pre-sup 11218  ax-hv0cl 30885  ax-hfvmul 30887  ax-hvmul0 30892  ax-hfi 30961  ax-his1 30964  ax-his3 30966  ax-his4 30967

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-sup 9467  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-div 11904  df-nn 12246  df-2 12308  df-3 12309  df-n0 12506  df-z 12592  df-uz 12856  df-rp 13010  df-seq 14003  df-exp 14063  df-cj 15082  df-re 15083  df-im 15084  df-sqrt 15218  df-abs 15219  df-hnorm 30850

This theorem is referenced by:  nmcopexi  31909  nmcfnexi  31933