Theorem strlem1 5691

Description: Lemma for strong state theorem: if closed subspace A is not contained in B, there is a unit vector u in their difference.

Hypotheses

Ref	Expression
strlem1.1	⊢ A ∈ Cℋ
strlem1.2	⊢ B ∈ Cℋ

Assertion

Ref	Expression
strlem1	⊢ (¬ A ⊆ B → ∃u ∈ (A ∖ B)(norm ‘u) = 1)

Distinct variable group(s):   u,A   u,B

Proof of Theorem strlem1

Step	Hyp	Ref	Expression
1		ssdif0 1748	. . . 4 ⊢ (A ⊆ B ↔ (A ∖ B) = ∅)
2	1	negbii 162	. . 3 ⊢ (¬ A ⊆ B ↔ ¬ (A ∖ B) = ∅)
3		n0 1714	. . 3 ⊢ (¬ (A ∖ B) = ∅ ↔ ∃x x ∈ (A ∖ B))
4	2, 3	bitr 151	. 2 ⊢ (¬ A ⊆ B ↔ ∃x x ∈ (A ∖ B))
5		fveq2 2832	. . . . . 6 ⊢ (u = ((1 / (norm ‘x)) ·s x) → (norm ‘u) = (norm ‘((1 / (norm ‘x)) ·s x)))
6	5	cleq1d 1109	. . . . 5 ⊢ (u = ((1 / (norm ‘x)) ·s x) → ((norm ‘u) = 1 ↔ (norm ‘((1 / (norm ‘x)) ·s x)) = 1))
7	6	rcla4ev 1403	. . . 4 ⊢ ((((1 / (norm ‘x)) ·s x) ∈ (A ∖ B) ∧ (norm ‘((1 / (norm ‘x)) ·s x)) = 1) → ∃u ∈ (A ∖ B)(norm ‘u) = 1)
8		ax1re 4064	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
9		redivclt 4276	. . . . . . . . . . 11 ⊢ (((1 ∈ ℝ ∧ (norm ‘x) ∈ ℝ) ∧ (norm ‘x) ≠ 0) → (1 / (norm ‘x)) ∈ ℝ)
10	8, 9	mpan11 529	. . . . . . . . . 10 ⊢ (((norm ‘x) ∈ ℝ ∧ (norm ‘x) ≠ 0) → (1 / (norm ‘x)) ∈ ℝ)
11		eldifi 1591	. . . . . . . . . . 11 ⊢ (x ∈ (A ∖ B) → x ∈ A)
12		strlem1.1	. . . . . . . . . . . 12 ⊢ A ∈ Cℋ
13	12	chel 5137	. . . . . . . . . . 11 ⊢ (x ∈ A → x ∈ ℋ )
14		normclt 5076	. . . . . . . . . . 11 ⊢ (x ∈ ℋ → (norm ‘x) ∈ ℝ)
15	11, 13, 14	3syl 21	. . . . . . . . . 10 ⊢ (x ∈ (A ∖ B) → (norm ‘x) ∈ ℝ)
16		strlem1.2	. . . . . . . . . . . . . . . 16 ⊢ B ∈ Cℋ
17		ch0 5133	. . . . . . . . . . . . . . . 16 ⊢ (B ∈ Cℋ → 0v ∈ B)
18	16, 17	ax-mp 6	. . . . . . . . . . . . . . 15 ⊢ 0v ∈ B
19		eldifn 1592	. . . . . . . . . . . . . . 15 ⊢ (0v ∈ (A ∖ B) → ¬ 0v ∈ B)
20	18, 19	mt2 96	. . . . . . . . . . . . . 14 ⊢ ¬ 0v ∈ (A ∖ B)
21		eleq1 1149	. . . . . . . . . . . . . 14 ⊢ (x = 0v → (x ∈ (A ∖ B) ↔ 0v ∈ (A ∖ B)))
22	20, 21	mtbiri 539	. . . . . . . . . . . . 13 ⊢ (x = 0v → ¬ x ∈ (A ∖ B))
23	22	con2i 89	. . . . . . . . . . . 12 ⊢ (x ∈ (A ∖ B) → ¬ x = 0v)
24		norm-it 5080	. . . . . . . . . . . . 13 ⊢ (x ∈ ℋ → ((norm ‘x) = 0 ↔ x = 0v))
25	11, 13, 24	3syl 21	. . . . . . . . . . . 12 ⊢ (x ∈ (A ∖ B) → ((norm ‘x) = 0 ↔ x = 0v))
26	23, 25	mtbird 537	. . . . . . . . . . 11 ⊢ (x ∈ (A ∖ B) → ¬ (norm ‘x) = 0)
27		df-ne 1192	. . . . . . . . . . 11 ⊢ ((norm ‘x) ≠ 0 ↔ ¬ (norm ‘x) = 0)
28	26, 27	sylibr 175	. . . . . . . . . 10 ⊢ (x ∈ (A ∖ B) → (norm ‘x) ≠ 0)
29	10, 15, 28	sylanc 361	. . . . . . . . 9 ⊢ (x ∈ (A ∖ B) → (1 / (norm ‘x)) ∈ ℝ)
30	29	recnd 4099	. . . . . . . 8 ⊢ (x ∈ (A ∖ B) → (1 / (norm ‘x)) ∈ ℂ)
31	12	chshi 5132	. . . . . . . . . 10 ⊢ A ∈ Sℋ
32		shmulclt 5124	. . . . . . . . . 10 ⊢ (A ∈ Sℋ → (((1 / (norm ‘x)) ∈ ℂ ∧ x ∈ A) → ((1 / (norm ‘x)) ·s x) ∈ A))
33	31, 32	ax-mp 6	. . . . . . . . 9 ⊢ (((1 / (norm ‘x)) ∈ ℂ ∧ x ∈ A) → ((1 / (norm ‘x)) ·s x) ∈ A)
34	33	exp 291	. . . . . . . 8 ⊢ ((1 / (norm ‘x)) ∈ ℂ → (x ∈ A → ((1 / (norm ‘x)) ·s x) ∈ A))
35	30, 34	syl 12	. . . . . . 7 ⊢ (x ∈ (A ∖ B) → (x ∈ A → ((1 / (norm ‘x)) ·s x) ∈ A))
36	15	recnd 4099	. . . . . . . . . 10 ⊢ (x ∈ (A ∖ B) → (norm ‘x) ∈ ℂ)
37	16	chshi 5132	. . . . . . . . . . . 12 ⊢ B ∈ Sℋ
38		shmulclt 5124	. . . . . . . . . . . 12 ⊢ (B ∈ Sℋ → (((norm ‘x) ∈ ℂ ∧ ((1 / (norm ‘x)) ·s x) ∈ B) → ((norm ‘x) ·s ((1 / (norm ‘x)) ·s x)) ∈ B))
39	37, 38	ax-mp 6	. . . . . . . . . . 11 ⊢ (((norm ‘x) ∈ ℂ ∧ ((1 / (norm ‘x)) ·s x) ∈ B) → ((norm ‘x) ·s ((1 / (norm ‘x)) ·s x)) ∈ B)
40	39	exp 291	. . . . . . . . . 10 ⊢ ((norm ‘x) ∈ ℂ → (((1 / (norm ‘x)) ·s x) ∈ B → ((norm ‘x) ·s ((1 / (norm ‘x)) ·s x)) ∈ B))
41	36, 40	syl 12	. . . . . . . . 9 ⊢ (x ∈ (A ∖ B) → (((1 / (norm ‘x)) ·s x) ∈ B → ((norm ‘x) ·s ((1 / (norm ‘x)) ·s x)) ∈ B))
42		recidt 4235	. . . . . . . . . . . . 13 ⊢ (((norm ‘x) ∈ ℂ ∧ (norm ‘x) ≠ 0) → ((norm ‘x) · (1 / (norm ‘x))) = 1)
43	42, 36, 28	sylanc 361	. . . . . . . . . . . 12 ⊢ (x ∈ (A ∖ B) → ((norm ‘x) · (1 / (norm ‘x))) = 1)
44	43	opreq1d 3012	. . . . . . . . . . 11 ⊢ (x ∈ (A ∖ B) → (((norm ‘x) · (1 / (norm ‘x))) ·s x) = (1 ·s x))
45		ax-hvmulass 4992	. . . . . . . . . . . 12 ⊢ (((norm ‘x) ∈ ℂ ∧ (1 / (norm ‘x)) ∈ ℂ ∧ x ∈ ℋ ) → (((norm ‘x) · (1 / (norm ‘x))) ·s x) = ((norm ‘x) ·s ((1 / (norm ‘x)) ·s x)))
46	11, 13	syl 12	. . . . . . . . . . . 12 ⊢ (x ∈ (A ∖ B) → x ∈ ℋ )
47	45, 36, 30, 46	syl3anc 629	. . . . . . . . . . 11 ⊢ (x ∈ (A ∖ B) → (((norm ‘x) · (1 / (norm ‘x))) ·s x) = ((norm ‘x) ·s ((1 / (norm ‘x)) ·s x)))
48		ax-hvmulid 4991	. . . . . . . . . . . 12 ⊢ (x ∈ ℋ → (1 ·s x) = x)
49	11, 13, 48	3syl 21	. . . . . . . . . . 11 ⊢ (x ∈ (A ∖ B) → (1 ·s x) = x)
50	44, 47, 49	3eqtr3d 1133	. . . . . . . . . 10 ⊢ (x ∈ (A ∖ B) → ((norm ‘x) ·s ((1 / (norm ‘x)) ·s x)) = x)
51	50	eleq1d 1155	. . . . . . . . 9 ⊢ (x ∈ (A ∖ B) → (((norm ‘x) ·s ((1 / (norm ‘x)) ·s x)) ∈ B ↔ x ∈ B))
52	41, 51	sylibd 177	. . . . . . . 8 ⊢ (x ∈ (A ∖ B) → (((1 / (norm ‘x)) ·s x) ∈ B → x ∈ B))
53	52	con3d 87	. . . . . . 7 ⊢ (x ∈ (A ∖ B) → (¬ x ∈ B → ¬ ((1 / (norm ‘x)) ·s x) ∈ B))
54	35, 53	anim12d 431	. . . . . 6 ⊢ (x ∈ (A ∖ B) → ((x ∈ A ∧ ¬ x ∈ B) → (((1 / (norm ‘x)) ·s x) ∈ A ∧ ¬ ((1 / (norm ‘x)) ·s x) ∈ B)))
55		eldif 1496	. . . . . 6 ⊢ (x ∈ (A ∖ B) ↔ (x ∈ A ∧ ¬ x ∈ B))
56		eldif 1496	. . . . . 6 ⊢ (((1 / (norm ‘x)) ·s x) ∈ (A ∖ B) ↔ (((1 / (norm ‘x)) ·s x) ∈ A ∧ ¬ ((1 / (norm ‘x)) ·s x) ∈ B))
57	54, 55, 56	3imtr4g 426	. . . . 5 ⊢ (x ∈ (A ∖ B) → (x ∈ (A ∖ B) → ((1 / (norm ‘x)) ·s x) ∈ (A ∖ B)))
58	57	pm2.43i 58	. . . 4 ⊢ (x ∈ (A ∖ B) → ((1 / (norm ‘x)) ·s x) ∈ (A ∖ B))
59		norm-iiit 5088	. . . . . 6 ⊢ (((1 / (norm ‘x)) ∈ ℂ ∧ x ∈ ℋ ) → (norm ‘((1 / (norm ‘x)) ·s x)) = ((abs ‘(1 / (norm ‘x))) · (norm ‘x)))
60	59, 30, 46	sylanc 361	. . . . 5 ⊢ (x ∈ (A ∖ B) → (norm ‘((1 / (norm ‘x)) ·s x)) = ((abs ‘(1 / (norm ‘x))) · (norm ‘x)))
61		absidt 4862	. . . . . . 7 ⊢ ((1 / (norm ‘x)) ∈ ℝ → (0 ≤ (1 / (norm ‘x)) → (abs ‘(1 / (norm ‘x))) = (1 / (norm ‘x))))
62		divge0t 4403	. . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ (norm ‘x) ∈ ℝ) → ((0 ≤ 1 ∧ 0 < (norm ‘x)) → 0 ≤ (1 / (norm ‘x))))
63	15, 8	jctil 240	. . . . . . . 8 ⊢ (x ∈ (A ∖ B) → (1 ∈ ℝ ∧ (norm ‘x) ∈ ℝ))
64		normgt0t 5078	. . . . . . . . . . 11 ⊢ (x ∈ ℋ → (¬ x = 0v ↔ 0 < (norm ‘x)))
65	11, 13, 64	3syl 21	. . . . . . . . . 10 ⊢ (x ∈ (A ∖ B) → (¬ x = 0v ↔ 0 < (norm ‘x)))
66	23, 65	mpbid 170	. . . . . . . . 9 ⊢ (x ∈ (A ∖ B) → 0 < (norm ‘x))
67		ax0re 4063	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
68		lt01 4377	. . . . . . . . . 10 ⊢ 0 < 1
69	67, 8, 68	ltlei 4303	. . . . . . . . 9 ⊢ 0 ≤ 1
70	66, 69	jctil 240	. . . . . . . 8 ⊢ (x ∈ (A ∖ B) → (0 ≤ 1 ∧ 0 < (norm ‘x)))
71	62, 63, 70	sylc 62	. . . . . . 7 ⊢ (x ∈ (A ∖ B) → 0 ≤ (1 / (norm ‘x)))
72	61, 29, 71	sylc 62	. . . . . 6 ⊢ (x ∈ (A ∖ B) → (abs ‘(1 / (norm ‘x))) = (1 / (norm ‘x)))
73	72	opreq1d 3012	. . . . 5 ⊢ (x ∈ (A ∖ B) → ((abs ‘(1 / (norm ‘x))) · (norm ‘x)) = ((1 / (norm ‘x)) · (norm ‘x)))
74		axmulcom 4071	. . . . . . 7 ⊢ (((1 / (norm ‘x)) ∈ ℂ ∧ (norm ‘x) ∈ ℂ) → ((1 / (norm ‘x)) · (norm ‘x)) = ((norm ‘x) · (1 / (norm ‘x))))
75	74, 30, 36	sylanc 361	. . . . . 6 ⊢ (x ∈ (A ∖ B) → ((1 / (norm ‘x)) · (norm ‘x)) = ((norm ‘x) · (1 / (norm ‘x))))
76	75, 43	eqtrd 1128	. . . . 5 ⊢ (x ∈ (A ∖ B) → ((1 / (norm ‘x)) · (norm ‘x)) = 1)
77	60, 73, 76	3eqtrd 1132	. . . 4 ⊢ (x ∈ (A ∖ B) → (norm ‘((1 / (norm ‘x)) ·s x)) = 1)
78	7, 58, 77	sylanc 361	. . 3 ⊢ (x ∈ (A ∖ B) → ∃u ∈ (A ∖ B)(norm ‘u) = 1)
79	78	19.23aiv 952	. 2 ⊢ (∃x x ∈ (A ∖ B) → ∃u ∈ (A ∖ B)(norm ‘u) = 1)
80	4, 79	sylbi 174	1 ⊢ (¬ A ⊆ B → ∃u ∈ (A ∖ B)(norm ‘u) = 1)

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092   ≠ wne 1190  ∃wrex 1202   ∖ cdif 1484   ⊆ wss 1487  ∅c0 1707   class class class wbr 2054   ‘cfv 2422  (class class class)co 3001  ℂcc 4026  ℝcr 4027  0cc0 4028  1c1 4029   · cmulc 4032   < clt 4033   / cdiv 4091   ≤ cle 4092  abscabs 4789   ℋ chil 4958   ·_s csm 4960  0_vc0v 4961  normcno 4964   S_ℋ csh 4967   C_ℋ cch 4968

This theorem is referenced by:  str 5698

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079  ax-hilex 4983  ax-hvzercl 4987  ax-hvmulcl 4989  ax-hvmulid 4991  ax-hvmulass 4992  ax-hvmulzer 4995  ax-hicl 5043  ax-his1 5045  ax-his3 5047  ax-his4 5048

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-sup 2154  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1st 3087  df-2nd 3088  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-mp 3883  df-ltp 3884  df-plpr 3958  df-mpr 3959  df-enr 3960  df-nr 3961  df-plr 3962  df-mr 3963  df-ltr 3964  df-0r 3965  df-1r 3966  df-m1r 3967  df-c 4034  df-0 4035  df-1 4036  df-i 4037  df-r 4038  df-plus 4039  df-mul 4040  df-lt 4041  df-sub 4133  df-neg 4135  df-div 4216  df-le 4277  df-n 4423  df-2 4462  df-n0 4535  df-z 4564  df-seq 4661  df-exp 4676  df-sqr 4728  df-re 4790  df-im 4791  df-cj 4792  df-abs 4793  df-hnorm 5074  df-sh 5114  df-ch 5127

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