strlem1 - Hilbert Space Explorer

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Theorem strlem1 31503

Description: Lemma for strong state theorem: if closed subspace 𝐴 is not contained in 𝐵, there is a unit vector 𝑢 in their difference. (Contributed by NM, 25-Oct-1999.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
strlem1.1	⊢ 𝐴 ∈ C_ℋ
strlem1.2	⊢ 𝐵 ∈ C_ℋ

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

Distinct variable groups: 𝑢,𝐴 𝑢,𝐵

Proof of Theorem strlem1 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		neq0 4346	. . 3 ⊢ (¬ (𝐴 ∖ 𝐵) = ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵))
2		ssdif0 4364	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∖ 𝐵) = ∅)
3	1, 2	xchnxbir 333	. 2 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ ∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵))
4		eldifi 4127	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ 𝐴)
5		strlem1.1	. . . . . . . . . . . 12 ⊢ 𝐴 ∈ C_ℋ
6	5	cheli 30485	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℋ)
7		normcl 30378	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℋ → (norm_ℎ‘𝑥) ∈ ℝ)
8	4, 6, 7	3syl 18	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (norm_ℎ‘𝑥) ∈ ℝ)
9		strlem1.2	. . . . . . . . . . . . . . . 16 ⊢ 𝐵 ∈ C_ℋ
10		ch0 30481	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ C_ℋ → 0_ℎ ∈ 𝐵)
11	9, 10	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 0_ℎ ∈ 𝐵
12		eldifn 4128	. . . . . . . . . . . . . . 15 ⊢ (0_ℎ ∈ (𝐴 ∖ 𝐵) → ¬ 0_ℎ ∈ 𝐵)
13	11, 12	mt2 199	. . . . . . . . . . . . . 14 ⊢ ¬ 0_ℎ ∈ (𝐴 ∖ 𝐵)
14		eleq1 2822	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 0_ℎ → (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ 0_ℎ ∈ (𝐴 ∖ 𝐵)))
15	13, 14	mtbiri 327	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0_ℎ → ¬ 𝑥 ∈ (𝐴 ∖ 𝐵))
16	15	con2i 139	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ¬ 𝑥 = 0_ℎ)
17		norm-i 30382	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℋ → ((norm_ℎ‘𝑥) = 0 ↔ 𝑥 = 0_ℎ))
18	4, 6, 17	3syl 18	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ((norm_ℎ‘𝑥) = 0 ↔ 𝑥 = 0_ℎ))
19	16, 18	mtbird 325	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ¬ (norm_ℎ‘𝑥) = 0)
20	19	neqned 2948	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (norm_ℎ‘𝑥) ≠ 0)
21	8, 20	rereccld 12041	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (1 / (norm_ℎ‘𝑥)) ∈ ℝ)
22	21	recnd 11242	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (1 / (norm_ℎ‘𝑥)) ∈ ℂ)
23	5	chshii 30480	. . . . . . . . . 10 ⊢ 𝐴 ∈ S_ℋ
24		shmulcl 30471	. . . . . . . . . 10 ⊢ ((𝐴 ∈ S_ℋ ∧ (1 / (norm_ℎ‘𝑥)) ∈ ℂ ∧ 𝑥 ∈ 𝐴) → ((1 / (norm_ℎ‘𝑥)) ·_ℎ 𝑥) ∈ 𝐴)
25	23, 24	mp3an1 1449	. . . . . . . . 9 ⊢ (((1 / (norm_ℎ‘𝑥)) ∈ ℂ ∧ 𝑥 ∈ 𝐴) → ((1 / (norm_ℎ‘𝑥)) ·_ℎ 𝑥) ∈ 𝐴)
26	25	ex 414	. . . . . . . 8 ⊢ ((1 / (norm_ℎ‘𝑥)) ∈ ℂ → (𝑥 ∈ 𝐴 → ((1 / (norm_ℎ‘𝑥)) ·_ℎ 𝑥) ∈ 𝐴))
27	22, 26	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (𝑥 ∈ 𝐴 → ((1 / (norm_ℎ‘𝑥)) ·_ℎ 𝑥) ∈ 𝐴))
28	8	recnd 11242	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (norm_ℎ‘𝑥) ∈ ℂ)
29	9	chshii 30480	. . . . . . . . . . . 12 ⊢ 𝐵 ∈ S_ℋ
30		shmulcl 30471	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ S_ℋ ∧ (norm_ℎ‘𝑥) ∈ ℂ ∧ ((1 / (norm_ℎ‘𝑥)) ·_ℎ 𝑥) ∈ 𝐵) → ((norm_ℎ‘𝑥) ·_ℎ ((1 / (norm_ℎ‘𝑥)) ·_ℎ 𝑥)) ∈ 𝐵)
31	29, 30	mp3an1 1449	. . . . . . . . . . 11 ⊢ (((norm_ℎ‘𝑥) ∈ ℂ ∧ ((1 / (norm_ℎ‘𝑥)) ·_ℎ 𝑥) ∈ 𝐵) → ((norm_ℎ‘𝑥) ·_ℎ ((1 / (norm_ℎ‘𝑥)) ·_ℎ 𝑥)) ∈ 𝐵)
32	31	ex 414	. . . . . . . . . 10 ⊢ ((norm_ℎ‘𝑥) ∈ ℂ → (((1 / (norm_ℎ‘𝑥)) ·_ℎ 𝑥) ∈ 𝐵 → ((norm_ℎ‘𝑥) ·_ℎ ((1 / (norm_ℎ‘𝑥)) ·_ℎ 𝑥)) ∈ 𝐵))
33	28, 32	syl 17	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (((1 / (norm_ℎ‘𝑥)) ·_ℎ 𝑥) ∈ 𝐵 → ((norm_ℎ‘𝑥) ·_ℎ ((1 / (norm_ℎ‘𝑥)) ·_ℎ 𝑥)) ∈ 𝐵))
34	28, 20	recidd 11985	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ((norm_ℎ‘𝑥) · (1 / (norm_ℎ‘𝑥))) = 1)
35	34	oveq1d 7424	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (((norm_ℎ‘𝑥) · (1 / (norm_ℎ‘𝑥))) ·_ℎ 𝑥) = (1 ·_ℎ 𝑥))
36	4, 6	syl 17	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ ℋ)
37		ax-hvmulass 30260	. . . . . . . . . . . 12 ⊢ (((norm_ℎ‘𝑥) ∈ ℂ ∧ (1 / (norm_ℎ‘𝑥)) ∈ ℂ ∧ 𝑥 ∈ ℋ) → (((norm_ℎ‘𝑥) · (1 / (norm_ℎ‘𝑥))) ·_ℎ 𝑥) = ((norm_ℎ‘𝑥) ·_ℎ ((1 / (norm_ℎ‘𝑥)) ·_ℎ 𝑥)))
38	28, 22, 36, 37	syl3anc 1372	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (((norm_ℎ‘𝑥) · (1 / (norm_ℎ‘𝑥))) ·_ℎ 𝑥) = ((norm_ℎ‘𝑥) ·_ℎ ((1 / (norm_ℎ‘𝑥)) ·_ℎ 𝑥)))
39		ax-hvmulid 30259	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℋ → (1 ·_ℎ 𝑥) = 𝑥)
40	4, 6, 39	3syl 18	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (1 ·_ℎ 𝑥) = 𝑥)
41	35, 38, 40	3eqtr3d 2781	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ((norm_ℎ‘𝑥) ·_ℎ ((1 / (norm_ℎ‘𝑥)) ·_ℎ 𝑥)) = 𝑥)
42	41	eleq1d 2819	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (((norm_ℎ‘𝑥) ·_ℎ ((1 / (norm_ℎ‘𝑥)) ·_ℎ 𝑥)) ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
43	33, 42	sylibd 238	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (((1 / (norm_ℎ‘𝑥)) ·_ℎ 𝑥) ∈ 𝐵 → 𝑥 ∈ 𝐵))
44	43	con3d 152	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (¬ 𝑥 ∈ 𝐵 → ¬ ((1 / (norm_ℎ‘𝑥)) ·_ℎ 𝑥) ∈ 𝐵))
45	27, 44	anim12d 610	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → (((1 / (norm_ℎ‘𝑥)) ·_ℎ 𝑥) ∈ 𝐴 ∧ ¬ ((1 / (norm_ℎ‘𝑥)) ·_ℎ 𝑥) ∈ 𝐵)))
46		eldif 3959	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
47		eldif 3959	. . . . . 6 ⊢ (((1 / (norm_ℎ‘𝑥)) ·_ℎ 𝑥) ∈ (𝐴 ∖ 𝐵) ↔ (((1 / (norm_ℎ‘𝑥)) ·_ℎ 𝑥) ∈ 𝐴 ∧ ¬ ((1 / (norm_ℎ‘𝑥)) ·_ℎ 𝑥) ∈ 𝐵))
48	45, 46, 47	3imtr4g 296	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (𝑥 ∈ (𝐴 ∖ 𝐵) → ((1 / (norm_ℎ‘𝑥)) ·_ℎ 𝑥) ∈ (𝐴 ∖ 𝐵)))
49	48	pm2.43i 52	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ((1 / (norm_ℎ‘𝑥)) ·_ℎ 𝑥) ∈ (𝐴 ∖ 𝐵))
50		norm-iii 30393	. . . . . 6 ⊢ (((1 / (norm_ℎ‘𝑥)) ∈ ℂ ∧ 𝑥 ∈ ℋ) → (norm_ℎ‘((1 / (norm_ℎ‘𝑥)) ·_ℎ 𝑥)) = ((abs‘(1 / (norm_ℎ‘𝑥))) · (norm_ℎ‘𝑥)))
51	22, 36, 50	syl2anc 585	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (norm_ℎ‘((1 / (norm_ℎ‘𝑥)) ·_ℎ 𝑥)) = ((abs‘(1 / (norm_ℎ‘𝑥))) · (norm_ℎ‘𝑥)))
52	15	necon2ai 2971	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ≠ 0_ℎ)
53		normgt0 30380	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℋ → (𝑥 ≠ 0_ℎ ↔ 0 < (norm_ℎ‘𝑥)))
54	4, 6, 53	3syl 18	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (𝑥 ≠ 0_ℎ ↔ 0 < (norm_ℎ‘𝑥)))
55	52, 54	mpbid 231	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → 0 < (norm_ℎ‘𝑥))
56		1re 11214	. . . . . . . . 9 ⊢ 1 ∈ ℝ
57		0le1 11737	. . . . . . . . 9 ⊢ 0 ≤ 1
58		divge0 12083	. . . . . . . . 9 ⊢ (((1 ∈ ℝ ∧ 0 ≤ 1) ∧ ((norm_ℎ‘𝑥) ∈ ℝ ∧ 0 < (norm_ℎ‘𝑥))) → 0 ≤ (1 / (norm_ℎ‘𝑥)))
59	56, 57, 58	mpanl12 701	. . . . . . . 8 ⊢ (((norm_ℎ‘𝑥) ∈ ℝ ∧ 0 < (norm_ℎ‘𝑥)) → 0 ≤ (1 / (norm_ℎ‘𝑥)))
60	8, 55, 59	syl2anc 585	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → 0 ≤ (1 / (norm_ℎ‘𝑥)))
61	21, 60	absidd 15369	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (abs‘(1 / (norm_ℎ‘𝑥))) = (1 / (norm_ℎ‘𝑥)))
62	61	oveq1d 7424	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ((abs‘(1 / (norm_ℎ‘𝑥))) · (norm_ℎ‘𝑥)) = ((1 / (norm_ℎ‘𝑥)) · (norm_ℎ‘𝑥)))
63	28, 20	recid2d 11986	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ((1 / (norm_ℎ‘𝑥)) · (norm_ℎ‘𝑥)) = 1)
64	51, 62, 63	3eqtrd 2777	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (norm_ℎ‘((1 / (norm_ℎ‘𝑥)) ·_ℎ 𝑥)) = 1)
65		fveqeq2 6901	. . . . 5 ⊢ (𝑢 = ((1 / (norm_ℎ‘𝑥)) ·_ℎ 𝑥) → ((norm_ℎ‘𝑢) = 1 ↔ (norm_ℎ‘((1 / (norm_ℎ‘𝑥)) ·_ℎ 𝑥)) = 1))
66	65	rspcev 3613	. . . 4 ⊢ ((((1 / (norm_ℎ‘𝑥)) ·_ℎ 𝑥) ∈ (𝐴 ∖ 𝐵) ∧ (norm_ℎ‘((1 / (norm_ℎ‘𝑥)) ·_ℎ 𝑥)) = 1) → ∃𝑢 ∈ (𝐴 ∖ 𝐵)(norm_ℎ‘𝑢) = 1)
67	49, 64, 66	syl2anc 585	. . 3 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ∃𝑢 ∈ (𝐴 ∖ 𝐵)(norm_ℎ‘𝑢) = 1)
68	67	exlimiv 1934	. 2 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵) → ∃𝑢 ∈ (𝐴 ∖ 𝐵)(norm_ℎ‘𝑢) = 1)
69	3, 68	sylbi 216	1 ⊢ (¬ 𝐴 ⊆ 𝐵 → ∃𝑢 ∈ (𝐴 ∖ 𝐵)(norm_ℎ‘𝑢) = 1)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ≠ wne 2941 ∃wrex 3071 ∖ cdif 3946 ⊆ wss 3949 ∅c0 4323 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 ℂcc 11108 ℝcr 11109 0cc0 11110 1c1 11111 · cmul 11115 < clt 11248 ≤ cle 11249 / cdiv 11871 abscabs 15181 ℋchba 30172 ·_ℎ csm 30174 norm_ℎcno 30176 0_ℎc0v 30177 S_ℋ csh 30181 C_ℋ cch 30182

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-hilex 30252 ax-hfvadd 30253 ax-hv0cl 30256 ax-hfvmul 30258 ax-hvmulid 30259 ax-hvmulass 30260 ax-hvmul0 30263 ax-hfi 30332 ax-his1 30335 ax-his3 30337 ax-his4 30338

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-sup 9437 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-rp 12975 df-seq 13967 df-exp 14028 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-hnorm 30221 df-sh 30460 df-ch 30474

This theorem is referenced by: stri 31510 hstri 31518

W3C validator