Theorem nmcopexlem5 9950

Description: Lemma for nmcopex 9952.

Hypotheses

Ref	Expression
nmcopex.1	⊢ T ∈ LinOp
nmcopex.2	⊢ T ∈ ConOp
nmcopexlem4.3	⊢ A = {k ∈ ℕ∣(1 / k) < y}
nmcopexlem4.4	⊢ M = sup(A, ℝ, ◡ < )

Assertion

Ref	Expression
nmcopexlem5	⊢ (((y ∈ ℝ ⋀ 0 < y) ⋀ (n ∈ ℕ ⋀ M ≤ n) ⋀ (x ∈ ℋ ⋀ (normh ‘x) ≤ 1)) → (normh ‘((1 / n) ·h x)) < y)

Distinct variable groups:   x,k,M   y,k,T,x   x,n,y,T

Proof of Theorem nmcopexlem5

Step	Hyp	Ref	Expression
1		hvmulclt 8878	. . . . . 6 ⊢ (((1 / n) ∈ ℂ ⋀ x ∈ ℋ ) → ((1 / n) ·h x) ∈ ℋ )
2		rerecclt 5805	. . . . . . . 8 ⊢ ((n ∈ ℝ ⋀ n ≠ 0) → (1 / n) ∈ ℝ)
3		nnret 5931	. . . . . . . 8 ⊢ (n ∈ ℕ → n ∈ ℝ)
4		nnne0t 5951	. . . . . . . 8 ⊢ (n ∈ ℕ → n ≠ 0)
5	2, 3, 4	sylanc 473	. . . . . . 7 ⊢ (n ∈ ℕ → (1 / n) ∈ ℝ)
6	5	recnd 5327	. . . . . 6 ⊢ (n ∈ ℕ → (1 / n) ∈ ℂ)
7	1, 6	sylan 450	. . . . 5 ⊢ ((n ∈ ℕ ⋀ x ∈ ℋ ) → ((1 / n) ·h x) ∈ ℋ )
8		normclt 8986	. . . . 5 ⊢ (((1 / n) ·h x) ∈ ℋ → (normh ‘((1 / n) ·h x)) ∈ ℝ)
9	7, 8	syl 10	. . . 4 ⊢ ((n ∈ ℕ ⋀ x ∈ ℋ ) → (normh ‘((1 / n) ·h x)) ∈ ℝ)
10	9	ad2ant2r 411	. . 3 ⊢ (((n ∈ ℕ ⋀ M ≤ n) ⋀ (x ∈ ℋ ⋀ (normh ‘x) ≤ 1)) → (normh ‘((1 / n) ·h x)) ∈ ℝ)
11	10	3adant1 799	. 2 ⊢ (((y ∈ ℝ ⋀ 0 < y) ⋀ (n ∈ ℕ ⋀ M ≤ n) ⋀ (x ∈ ℋ ⋀ (normh ‘x) ≤ 1)) → (normh ‘((1 / n) ·h x)) ∈ ℝ)
12		nmcopex.1	. . . . . 6 ⊢ T ∈ LinOp
13		nmcopex.2	. . . . . 6 ⊢ T ∈ ConOp
14		nmcopexlem4.3	. . . . . 6 ⊢ A = {k ∈ ℕ∣(1 / k) < y}
15		nmcopexlem4.4	. . . . . 6 ⊢ M = sup(A, ℝ, ◡ < )
16	12, 13, 14, 15	nmcopexlem4 9949	. . . . 5 ⊢ ((y ∈ ℝ ⋀ 0 < y) → (M ∈ ℕ ⋀ (1 / M) < y))
17	16	pm3.26d 321	. . . 4 ⊢ ((y ∈ ℝ ⋀ 0 < y) → M ∈ ℕ)
18		rerecclt 5805	. . . . 5 ⊢ ((M ∈ ℝ ⋀ M ≠ 0) → (1 / M) ∈ ℝ)
19		nnret 5931	. . . . 5 ⊢ (M ∈ ℕ → M ∈ ℝ)
20		nnne0t 5951	. . . . 5 ⊢ (M ∈ ℕ → M ≠ 0)
21	18, 19, 20	sylanc 473	. . . 4 ⊢ (M ∈ ℕ → (1 / M) ∈ ℝ)
22	17, 21	syl 10	. . 3 ⊢ ((y ∈ ℝ ⋀ 0 < y) → (1 / M) ∈ ℝ)
23	22	3ad2ant1 802	. 2 ⊢ (((y ∈ ℝ ⋀ 0 < y) ⋀ (n ∈ ℕ ⋀ M ≤ n) ⋀ (x ∈ ℋ ⋀ (normh ‘x) ≤ 1)) → (1 / M) ∈ ℝ)
24		pm3.26 319	. . 3 ⊢ ((y ∈ ℝ ⋀ 0 < y) → y ∈ ℝ)
25	24	3ad2ant1 802	. 2 ⊢ (((y ∈ ℝ ⋀ 0 < y) ⋀ (n ∈ ℕ ⋀ M ≤ n) ⋀ (x ∈ ℋ ⋀ (normh ‘x) ≤ 1)) → y ∈ ℝ)
26	10	3adant1 799	. . . 4 ⊢ ((M ∈ ℕ ⋀ (n ∈ ℕ ⋀ M ≤ n) ⋀ (x ∈ ℋ ⋀ (normh ‘x) ≤ 1)) → (normh ‘((1 / n) ·h x)) ∈ ℝ)
27	5	ad2antrr 406	. . . . 5 ⊢ (((n ∈ ℕ ⋀ M ≤ n) ⋀ (x ∈ ℋ ⋀ (normh ‘x) ≤ 1)) → (1 / n) ∈ ℝ)
28	27	3adant1 799	. . . 4 ⊢ ((M ∈ ℕ ⋀ (n ∈ ℕ ⋀ M ≤ n) ⋀ (x ∈ ℋ ⋀ (normh ‘x) ≤ 1)) → (1 / n) ∈ ℝ)
29	21	3ad2ant1 802	. . . 4 ⊢ ((M ∈ ℕ ⋀ (n ∈ ℕ ⋀ M ≤ n) ⋀ (x ∈ ℋ ⋀ (normh ‘x) ≤ 1)) → (1 / M) ∈ ℝ)
30		lemul2itOLD 5842	. . . . . 6 ⊢ ((((normh ‘x) ∈ ℝ ⋀ 1 ∈ ℝ ⋀ (1 / n) ∈ ℝ) ⋀ (0 ≤ (1 / n) ⋀ (normh ‘x) ≤ 1)) → ((1 / n) · (normh ‘x)) ≤ ((1 / n) · 1))
31		normclt 8986	. . . . . . . . 9 ⊢ (x ∈ ℋ → (normh ‘x) ∈ ℝ)
32	31	ad2antrl 408	. . . . . . . 8 ⊢ ((M ∈ ℕ ⋀ (x ∈ ℋ ⋀ (normh ‘x) ≤ 1)) → (normh ‘x) ∈ ℝ)
33	32	3adant2 800	. . . . . . 7 ⊢ ((M ∈ ℕ ⋀ (n ∈ ℕ ⋀ M ≤ n) ⋀ (x ∈ ℋ ⋀ (normh ‘x) ≤ 1)) → (normh ‘x) ∈ ℝ)
34		1re 5447	. . . . . . . 8 ⊢ 1 ∈ ℝ
35	34	a1i 8	. . . . . . 7 ⊢ ((M ∈ ℕ ⋀ (n ∈ ℕ ⋀ M ≤ n) ⋀ (x ∈ ℋ ⋀ (normh ‘x) ≤ 1)) → 1 ∈ ℝ)
36	33, 35, 28	3jca 821	. . . . . 6 ⊢ ((M ∈ ℕ ⋀ (n ∈ ℕ ⋀ M ≤ n) ⋀ (x ∈ ℋ ⋀ (normh ‘x) ≤ 1)) → ((normh ‘x) ∈ ℝ ⋀ 1 ∈ ℝ ⋀ (1 / n) ∈ ℝ))
37		0re 5452	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
38		lt01 5692	. . . . . . . . . . . 12 ⊢ 0 < 1
39	37, 34, 38	ltlei 5593	. . . . . . . . . . 11 ⊢ 0 ≤ 1
40		divge0t 5858	. . . . . . . . . . 11 ⊢ (((1 ∈ ℝ ⋀ 0 ≤ 1) ⋀ (n ∈ ℝ ⋀ 0 < n)) → 0 ≤ (1 / n))
41	34, 39, 40	mpanl12 710	. . . . . . . . . 10 ⊢ ((n ∈ ℝ ⋀ 0 < n) → 0 ≤ (1 / n))
42		nngt0t 5948	. . . . . . . . . 10 ⊢ (n ∈ ℕ → 0 < n)
43	41, 3, 42	sylanc 473	. . . . . . . . 9 ⊢ (n ∈ ℕ → 0 ≤ (1 / n))
44	43	ad2antrl 408	. . . . . . . 8 ⊢ ((M ∈ ℕ ⋀ (n ∈ ℕ ⋀ M ≤ n)) → 0 ≤ (1 / n))
45	44	3adant3 801	. . . . . . 7 ⊢ ((M ∈ ℕ ⋀ (n ∈ ℕ ⋀ M ≤ n) ⋀ (x ∈ ℋ ⋀ (normh ‘x) ≤ 1)) → 0 ≤ (1 / n))
46		pm3.27 323	. . . . . . . 8 ⊢ ((x ∈ ℋ ⋀ (normh ‘x) ≤ 1) → (normh ‘x) ≤ 1)
47	46	3ad2ant3 804	. . . . . . 7 ⊢ ((M ∈ ℕ ⋀ (n ∈ ℕ ⋀ M ≤ n) ⋀ (x ∈ ℋ ⋀ (normh ‘x) ≤ 1)) → (normh ‘x) ≤ 1)
48	45, 47	jca 288	. . . . . 6 ⊢ ((M ∈ ℕ ⋀ (n ∈ ℕ ⋀ M ≤ n) ⋀ (x ∈ ℋ ⋀ (normh ‘x) ≤ 1)) → (0 ≤ (1 / n) ⋀ (normh ‘x) ≤ 1))
49	30, 36, 48	sylanc 473	. . . . 5 ⊢ ((M ∈ ℕ ⋀ (n ∈ ℕ ⋀ M ≤ n) ⋀ (x ∈ ℋ ⋀ (normh ‘x) ≤ 1)) → ((1 / n) · (normh ‘x)) ≤ ((1 / n) · 1))
50		norm-iiit 9002	. . . . . . . . 9 ⊢ (((1 / n) ∈ ℂ ⋀ x ∈ ℋ ) → (normh ‘((1 / n) ·h x)) = ((abs ‘(1 / n)) · (normh ‘x)))
51	50, 6	sylan 450	. . . . . . . 8 ⊢ ((n ∈ ℕ ⋀ x ∈ ℋ ) → (normh ‘((1 / n) ·h x)) = ((abs ‘(1 / n)) · (normh ‘x)))
52		absidt 6862	. . . . . . . . . . 11 ⊢ (((1 / n) ∈ ℝ ⋀ 0 ≤ (1 / n)) → (abs ‘(1 / n)) = (1 / n))
53	52, 5, 43	sylanc 473	. . . . . . . . . 10 ⊢ (n ∈ ℕ → (abs ‘(1 / n)) = (1 / n))
54	53	adantr 391	. . . . . . . . 9 ⊢ ((n ∈ ℕ ⋀ x ∈ ℋ ) → (abs ‘(1 / n)) = (1 / n))
55	54	opreq1d 3981	. . . . . . . 8 ⊢ ((n ∈ ℕ ⋀ x ∈ ℋ ) → ((abs ‘(1 / n)) · (normh ‘x)) = ((1 / n) · (normh ‘x)))
56	51, 55	eqtr2d 1511	. . . . . . 7 ⊢ ((n ∈ ℕ ⋀ x ∈ ℋ ) → ((1 / n) · (normh ‘x)) = (normh ‘((1 / n) ·h x)))
57	56	ad2ant2r 411	. . . . . 6 ⊢ (((n ∈ ℕ ⋀ M ≤ n) ⋀ (x ∈ ℋ ⋀ (normh ‘x) ≤ 1)) → ((1 / n) · (normh ‘x)) = (normh ‘((1 / n) ·h x)))
58	57	3adant1 799	. . . . 5 ⊢ ((M ∈ ℕ ⋀ (n ∈ ℕ ⋀ M ≤ n) ⋀ (x ∈ ℋ ⋀ (normh ‘x) ≤ 1)) → ((1 / n) · (normh ‘x)) = (normh ‘((1 / n) ·h x)))
59		ax1id 5294	. . . . . . . 8 ⊢ ((1 / n) ∈ ℂ → ((1 / n) · 1) = (1 / n))
60	6, 59	syl 10	. . . . . . 7 ⊢ (n ∈ ℕ → ((1 / n) · 1) = (1 / n))
61	60	ad2antrl 408	. . . . . 6 ⊢ ((M ∈ ℕ ⋀ (n ∈ ℕ ⋀ M ≤ n)) → ((1 / n) · 1) = (1 / n))
62	61	3adant3 801	. . . . 5 ⊢ ((M ∈ ℕ ⋀ (n ∈ ℕ ⋀ M ≤ n) ⋀ (x ∈ ℋ ⋀ (normh ‘x) ≤ 1)) → ((1 / n) · 1) = (1 / n))
63	49, 58, 62	3brtr3d 2649	. . . 4 ⊢ ((M ∈ ℕ ⋀ (n ∈ ℕ ⋀ M ≤ n) ⋀ (x ∈ ℋ ⋀ (normh ‘x) ≤ 1)) → (normh ‘((1 / n) ·h x)) ≤ (1 / n))
64		lerect 5887	. . . . . . . 8 ⊢ (((M ∈ ℝ ⋀ 0 < M) ⋀ (n ∈ ℝ ⋀ 0 < n)) → (M ≤ n ↔ (1 / n) ≤ (1 / M)))
65		nngt0t 5948	. . . . . . . . 9 ⊢ (M ∈ ℕ → 0 < M)
66	19, 65	jca 288	. . . . . . . 8 ⊢ (M ∈ ℕ → (M ∈ ℝ ⋀ 0 < M))
67	3, 42	jca 288	. . . . . . . 8 ⊢ (n ∈ ℕ → (n ∈ ℝ ⋀ 0 < n))
68	64, 66, 67	syl2an 456	. . . . . . 7 ⊢ ((M ∈ ℕ ⋀ n ∈ ℕ) → (M ≤ n ↔ (1 / n) ≤ (1 / M)))
69	68	biimpa 418	. . . . . 6 ⊢ (((M ∈ ℕ ⋀ n ∈ ℕ) ⋀ M ≤ n) → (1 / n) ≤ (1 / M))
70	69	anasss 442	. . . . 5 ⊢ ((M ∈ ℕ ⋀ (n ∈ ℕ ⋀ M ≤ n)) → (1 / n) ≤ (1 / M))
71	70	3adant3 801	. . . 4 ⊢ ((M ∈ ℕ ⋀ (n ∈ ℕ ⋀ M ≤ n) ⋀ (x ∈ ℋ ⋀ (normh ‘x) ≤ 1)) → (1 / n) ≤ (1 / M))
72	26, 28, 29, 63, 71	letrd 5538	. . 3 ⊢ ((M ∈ ℕ ⋀ (n ∈ ℕ ⋀ M ≤ n) ⋀ (x ∈ ℋ ⋀ (normh ‘x) ≤ 1)) → (normh ‘((1 / n) ·h x)) ≤ (1 / M))
73	72, 17	syl3an1 861	. 2 ⊢ (((y ∈ ℝ ⋀ 0 < y) ⋀ (n ∈ ℕ ⋀ M ≤ n) ⋀ (x ∈ ℋ ⋀ (normh ‘x) ≤ 1)) → (normh ‘((1 / n) ·h x)) ≤ (1 / M))
74	16	pm3.27d 325	. . 3 ⊢ ((y ∈ ℝ ⋀ 0 < y) → (1 / M) < y)
75	74	3ad2ant1 802	. 2 ⊢ (((y ∈ ℝ ⋀ 0 < y) ⋀ (n ∈ ℕ ⋀ M ≤ n) ⋀ (x ∈ ℋ ⋀ (normh ‘x) ≤ 1)) → (1 / M) < y)
76	11, 23, 25, 73, 75	lelttrd 5539	1 ⊢ (((y ∈ ℝ ⋀ 0 < y) ⋀ (n ∈ ℕ ⋀ M ≤ n) ⋀ (x ∈ ℋ ⋀ (normh ‘x) ≤ 1)) → (normh ‘((1 / n) ·h x)) < y)

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223   ⋀ w3a 777   = wceq 958   ∈ wcel 960   ≠ wne 1588  {crab 1651   class class class wbr 2624  ^◡ccnv 3175   ‘cfv 3188  (class class class)co 3969  supcsup 4582  ℂcc 5244  ℝcr 5245  0cc0 5246  1c1 5247   · cmul 5251   / cdiv 5306   ≤ cle 5307  ℕcn 5308   < clt 5498  abscabs 6751   ℋ chil 8783   ·_h csm 8785  norm_hcno 8789  ConOpcco 8810  LinOpclo 8811

This theorem is referenced by:  nmcopexlem6 9951

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634  ax-hv0cl 8868  ax-hfvmul 8870  ax-hvmul0 8875  ax-hfi 8941  ax-his1 8944  ax-his3 8946  ax-his4 8947

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-en 4374  df-dom 4375  df-sdom 4376  df-sup 4583  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-ltr 5182  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-lt 5259  df-sub 5368  df-neg 5370  df-pnf 5499  df-mnf 5500  df-xr 5501  df-ltxr 5502  df-le 5503  df-div 5715  df-n 5927  df-2 5972  df-n0 6102  df-z 6138  df-seq1 6309  df-uz 6419  df-exp 6570  df-sqr 6671  df-re 6752  df-im 6753  df-cj 6754  df-abs 6755  df-hnorm 8832

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