Theorem nmcexi 32637

Description: Lemma for nmcopexi 32638 and nmcfnexi 32662. 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 2848	. . . . . . . . 9 ⊢ (𝑚 = (𝑁‘(𝑇‘𝑥)) → (𝑚 ∈ ℝ ↔ (𝑁‘(𝑇‘𝑥)) ∈ ℝ))
4	2, 3	syl5ibrcom 250	. . . . . . . 8 ⊢ (𝑥 ∈ ℋ → (𝑚 = (𝑁‘(𝑇‘𝑥)) → 𝑚 ∈ ℝ))
5	4	imp 412	. . . . . . 7 ⊢ ((𝑥 ∈ ℋ ∧ 𝑚 = (𝑁‘(𝑇‘𝑥))) → 𝑚 ∈ ℝ)
6	5	adantrl 729	. . . . . 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 3992	. . . 4 ⊢ {𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))} ⊆ ℝ
9		ax-hv0cl 31614	. . . . . . 7 ⊢ 0ℎ ∈ ℋ
10		norm0 31739	. . . . . . . . 9 ⊢ (normℎ‘0ℎ) = 0
11		0le1 11846	. . . . . . . . 9 ⊢ 0 ≤ 1
12	10, 11	eqbrtri 5125	. . . . . . . 8 ⊢ (normℎ‘0ℎ) ≤ 1
13		nmcex.4	. . . . . . . . 9 ⊢ (𝑁‘(𝑇‘0ℎ)) = 0
14	13	eqcomi 2769	. . . . . . . 8 ⊢ 0 = (𝑁‘(𝑇‘0ℎ))
15	12, 14	pm3.2i 476	. . . . . . 7 ⊢ ((normℎ‘0ℎ) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘0ℎ)))
16		fveq2 6890	. . . . . . . . . 10 ⊢ (𝑥 = 0ℎ → (normℎ‘𝑥) = (normℎ‘0ℎ))
17	16	breq1d 5112	. . . . . . . . 9 ⊢ (𝑥 = 0ℎ → ((normℎ‘𝑥) ≤ 1 ↔ (normℎ‘0ℎ) ≤ 1))
18		2fveq3 6895	. . . . . . . . . 10 ⊢ (𝑥 = 0ℎ → (𝑁‘(𝑇‘𝑥)) = (𝑁‘(𝑇‘0ℎ)))
19	18	eqeq2d 2771	. . . . . . . . 9 ⊢ (𝑥 = 0ℎ → (0 = (𝑁‘(𝑇‘𝑥)) ↔ 0 = (𝑁‘(𝑇‘0ℎ))))
20	17, 19	anbi12d 644	. . . . . . . 8 ⊢ (𝑥 = 0ℎ → (((normℎ‘𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘𝑥))) ↔ ((normℎ‘0ℎ) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘0ℎ)))))
21	20	rspcev 3576	. . . . . . 7 ⊢ ((0ℎ ∈ ℋ ∧ ((normℎ‘0ℎ) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘0ℎ)))) → ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘𝑥))))
22	9, 15, 21	mp2an 705	. . . . . 6 ⊢ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 0 = (𝑁‘(𝑇‘𝑥)))
23		c0ex 11307	. . . . . . 7 ⊢ 0 ∈ V
24		eqeq1 2764	. . . . . . . . 9 ⊢ (𝑚 = 0 → (𝑚 = (𝑁‘(𝑇‘𝑥)) ↔ 0 = (𝑁‘(𝑇‘𝑥))))
25	24	anbi2d 642	. . . . . . . 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 3632	. . . . . 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 234	. . . . 5 ⊢ 0 ∈ {𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))}
29	28	ne0ii 4289	. . . 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 13132	. . . . . . . . . 10 ⊢ 2 ∈ ℝ+
32		rpdivcl 13154	. . . . . . . . . 10 ⊢ ((2 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+) → (2 / 𝑦) ∈ ℝ+)
33	31, 32	mpan 703	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ+ → (2 / 𝑦) ∈ ℝ+)
34	33	rpred 13171	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ+ → (2 / 𝑦) ∈ ℝ)
35	34	adantr 486	. . . . . . 7 ⊢ ((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((normℎ‘𝑧) < 𝑦 → (𝑁‘(𝑇‘𝑧)) < 1)) → (2 / 𝑦) ∈ ℝ)
36		rpre 13136	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ ℝ+ → 𝑦 ∈ ℝ)
37	36	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → 𝑦 ∈ ℝ)
38	37	rehalfcld 12600	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (𝑦 / 2) ∈ ℝ)
39	38	recnd 11344	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (𝑦 / 2) ∈ ℂ)
40		simprl 783	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → 𝑥 ∈ ℋ)
41		hvmulcl 31624	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑦 / 2) ∈ ℂ ∧ 𝑥 ∈ ℋ) → ((𝑦 / 2) ·ℎ 𝑥) ∈ ℋ)
42	39, 40, 41	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((𝑦 / 2) ·ℎ 𝑥) ∈ ℋ)
43		normcl 31736	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 / 2) ·ℎ 𝑥) ∈ ℋ → (normℎ‘((𝑦 / 2) ·ℎ 𝑥)) ∈ ℝ)
44	42, 43	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (normℎ‘((𝑦 / 2) ·ℎ 𝑥)) ∈ ℝ)
45		simprr 785	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (normℎ‘𝑥) ≤ 1)
46		normcl 31736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℋ → (normℎ‘𝑥) ∈ ℝ)
47	46	ad2antrl 741	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (normℎ‘𝑥) ∈ ℝ)
48		1red 11316	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → 1 ∈ ℝ)
49		rphalfcl 13156	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ℝ+ → (𝑦 / 2) ∈ ℝ+)
50	49	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (𝑦 / 2) ∈ ℝ+)
51	47, 48, 50	lemul2d 13215	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((normℎ‘𝑥) ≤ 1 ↔ ((𝑦 / 2) · (normℎ‘𝑥)) ≤ ((𝑦 / 2) · 1)))
52	45, 51	mpbid 235	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((𝑦 / 2) · (normℎ‘𝑥)) ≤ ((𝑦 / 2) · 1))
53		rpcn 13138	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 / 2) ∈ ℝ+ → (𝑦 / 2) ∈ ℂ)
54		norm-iii 31751	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑦 / 2) ∈ ℂ ∧ 𝑥 ∈ ℋ) → (normℎ‘((𝑦 / 2) ·ℎ 𝑥)) = ((abs‘(𝑦 / 2)) · (normℎ‘𝑥)))
55	53, 54	sylan 592	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → (normℎ‘((𝑦 / 2) ·ℎ 𝑥)) = ((abs‘(𝑦 / 2)) · (normℎ‘𝑥)))
56		rpre 13136	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 / 2) ∈ ℝ+ → (𝑦 / 2) ∈ ℝ)
57		rpge0 13141	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 / 2) ∈ ℝ+ → 0 ≤ (𝑦 / 2))
58	56, 57	absidd 15599	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 / 2) ∈ ℝ+ → (abs‘(𝑦 / 2)) = (𝑦 / 2))
59	58	oveq1d 7439	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 / 2) ∈ ℝ+ → ((abs‘(𝑦 / 2)) · (normℎ‘𝑥)) = ((𝑦 / 2) · (normℎ‘𝑥)))
60	59	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → ((abs‘(𝑦 / 2)) · (normℎ‘𝑥)) = ((𝑦 / 2) · (normℎ‘𝑥)))
61	55, 60	eqtr2d 2796	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑦 / 2) ∈ ℝ+ ∧ 𝑥 ∈ ℋ) → ((𝑦 / 2) · (normℎ‘𝑥)) = (normℎ‘((𝑦 / 2) ·ℎ 𝑥)))
62	50, 40, 61	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((𝑦 / 2) · (normℎ‘𝑥)) = (normℎ‘((𝑦 / 2) ·ℎ 𝑥)))
63	39	mulridd 11333	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((𝑦 / 2) · 1) = (𝑦 / 2))
64	52, 62, 63	3brtr3d 5135	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (normℎ‘((𝑦 / 2) ·ℎ 𝑥)) ≤ (𝑦 / 2))
65		rphalflt 13158	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℝ+ → (𝑦 / 2) < 𝑦)
66	65	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (𝑦 / 2) < 𝑦)
67	44, 38, 37, 64, 66	lelttrd 11475	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (normℎ‘((𝑦 / 2) ·ℎ 𝑥)) < 𝑦)
68		fveq2 6890	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = ((𝑦 / 2) ·ℎ 𝑥) → (normℎ‘𝑧) = (normℎ‘((𝑦 / 2) ·ℎ 𝑥)))
69	68	breq1d 5112	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = ((𝑦 / 2) ·ℎ 𝑥) → ((normℎ‘𝑧) < 𝑦 ↔ (normℎ‘((𝑦 / 2) ·ℎ 𝑥)) < 𝑦))
70		2fveq3 6895	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = ((𝑦 / 2) ·ℎ 𝑥) → (𝑁‘(𝑇‘𝑧)) = (𝑁‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥))))
71	70	breq1d 5112	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = ((𝑦 / 2) ·ℎ 𝑥) → ((𝑁‘(𝑇‘𝑧)) < 1 ↔ (𝑁‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥))) < 1))
72	69, 71	imbi12d 347	. . . . . . . . . . . . . . . . . 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 3572	. . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . 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 45	. . . . . . . . . . . . . . 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 741	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (𝑁‘(𝑇‘𝑥)) ∈ ℝ)
77	76, 48, 50	ltmuldiv2d 13219	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (((𝑦 / 2) · (𝑁‘(𝑇‘𝑥))) < 1 ↔ (𝑁‘(𝑇‘𝑥)) < (1 / (𝑦 / 2))))
78	50	rprecred 13182	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (1 / (𝑦 / 2)) ∈ ℝ)
79		ltle 11405	. . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((𝑁‘(𝑇‘𝑥)) < (1 / (𝑦 / 2)) → (𝑁‘(𝑇‘𝑥)) ≤ (1 / (𝑦 / 2))))
81	77, 80	sylbid 243	. . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((𝑦 / 2) · (𝑁‘(𝑇‘𝑥))) = (𝑁‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥))))
84	83	breq1d 5112	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (((𝑦 / 2) · (𝑁‘(𝑇‘𝑥))) < 1 ↔ (𝑁‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥))) < 1))
85		rpcn 13138	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℝ+ → 𝑦 ∈ ℂ)
86		rpne0 13144	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℝ+ → 𝑦 ≠ 0)
87		2cn 12425	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℂ
88		2ne0 12456	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ≠ 0
89		recdiv 12030	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → (1 / (𝑦 / 2)) = (2 / 𝑦))
90	87, 88, 89	mpanr12 718	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) → (1 / (𝑦 / 2)) = (2 / 𝑦))
91	85, 86, 90	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℝ+ → (1 / (𝑦 / 2)) = (2 / 𝑦))
92	91	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → (1 / (𝑦 / 2)) = (2 / 𝑦))
93	92	breq2d 5114	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((𝑁‘(𝑇‘𝑥)) ≤ (1 / (𝑦 / 2)) ↔ (𝑁‘(𝑇‘𝑥)) ≤ (2 / 𝑦)))
94	81, 84, 93	3imtr3d 296	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℝ+ ∧ (𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1)) → ((𝑁‘(𝑇‘((𝑦 / 2) ·ℎ 𝑥))) < 1 → (𝑁‘(𝑇‘𝑥)) ≤ (2 / 𝑦)))
95	75, 94	syld 48	. . . . . . . . . . . . . 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 412	. . . . . . . . . . . . 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 665	. . . . . . . . . . . 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 473	. . . . . . . . . . 11 ⊢ ((((𝑦 ∈ ℝ+ ∧ ∀𝑧 ∈ ℋ ((normℎ‘𝑧) < 𝑦 → (𝑁‘(𝑇‘𝑧)) < 1)) ∧ 𝑥 ∈ ℋ) ∧ (normℎ‘𝑥) ≤ 1) → (𝑁‘(𝑇‘𝑥)) ≤ (2 / 𝑦))
99		breq1 5105	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑁‘(𝑇‘𝑥)) → (𝑛 ≤ (2 / 𝑦) ↔ (𝑁‘(𝑇‘𝑥)) ≤ (2 / 𝑦)))
100	98, 99	syl5ibrcom 250	. . . . . . . . . 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 459	. . . . . . . . 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 1959	. . . . . . 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 2764	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (𝑚 = (𝑁‘(𝑇‘𝑥)) ↔ 𝑛 = (𝑁‘(𝑇‘𝑥))))
105	104	anbi2d 642	. . . . . . . . . . 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 3650	. . . . . . . . 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 5106	. . . . . . . . . . 11 ⊢ (𝑧 = (2 / 𝑦) → (𝑛 ≤ 𝑧 ↔ 𝑛 ≤ (2 / 𝑦)))
109	108	imbi2d 343	. . . . . . . . . 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 1952	. . . . . . . . 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 286	. . . . . . . 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 3576	. . . . . . 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 596	. . . . . 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 13464	. . . 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 1490	. . 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 2783	. 2 ⊢ (𝑆‘𝑇) = sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))}, ℝ, < )
119		suprcl 12284	. . 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 1490	. 2 ⊢ sup({𝑚 ∣ ∃𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 ∧ 𝑚 = (𝑁‘(𝑇‘𝑥)))}, ℝ, < ) ∈ ℝ
121	118, 120	eqeltri 2856	1 ⊢ (𝑆‘𝑇) ∈ ℝ

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1567   = wceq 1569   ∈ wcel 2144  {cab 2738   ≠ wne 2955  ∀wral 3076  ∃wrex 3086   ⊆ wss 3896  ∅c0 4277   class class class wbr 5102  ‘cfv 6543  (class class class)co 7424  supcsup 9439  ℂcc 11205  ℝcr 11206  0cc0 11207  1c1 11208   · cmul 11212  ℝ^*cxr 11349   < clt 11350   ≤ cle 11351   / cdiv 11980  2c2 12404  ℝ⁺crp 13127  abscabs 15410   ℋchba 31530   ·_ℎ csm 31532  norm_ℎcno 31534  0_ℎc0v 31535

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-cnex 11263  ax-resscn 11264  ax-1cn 11265  ax-icn 11266  ax-addcl 11267  ax-addrcl 11268  ax-mulcl 11269  ax-mulrcl 11270  ax-mulcom 11271  ax-addass 11272  ax-mulass 11273  ax-distr 11274  ax-i2m1 11275  ax-1ne0 11276  ax-1rid 11277  ax-rnegex 11278  ax-rrecex 11279  ax-cnre 11280  ax-pre-lttri 11281  ax-pre-lttrn 11282  ax-pre-ltadd 11283  ax-pre-mulgt0 11284  ax-pre-sup 11285  ax-hv0cl 31614  ax-hfvmul 31616  ax-hvmul0 31621  ax-hfi 31690  ax-his1 31693  ax-his3 31695  ax-his4 31696

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7882  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-rdg 8426  df-er 8725  df-en 8982  df-dom 8983  df-sdom 8984  df-sup 9441  df-pnf 11352  df-mnf 11353  df-xr 11354  df-ltxr 11355  df-le 11356  df-sub 11550  df-neg 11551  df-div 11981  df-nn 12343  df-2 12412  df-3 12413  df-n0 12614  df-z 12701  df-uz 12973  df-rp 13128  df-seq 14152  df-exp 14212  df-cj 15275  df-re 15276  df-im 15277  df-sqrt 15411  df-abs 15412  df-hnorm 31579

This theorem is used by:  nmcopexi  32638  nmcfnexi  32662