Theorem strlem1 32278

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 ∈ Cℋ
strlem1.2	⊢ 𝐵 ∈ Cℋ

Assertion

Ref	Expression
strlem1	⊢ (¬ 𝐴 ⊆ 𝐵 → ∃𝑢 ∈ (𝐴 ∖ 𝐵)(normℎ‘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 4357	. . 3 ⊢ (¬ (𝐴 ∖ 𝐵) = ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵))
2		ssdif0 4371	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∖ 𝐵) = ∅)
3	1, 2	xchnxbir 333	. 2 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ ∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵))
4		eldifi 4140	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ 𝐴)
5		strlem1.1	. . . . . . . . . . . 12 ⊢ 𝐴 ∈ Cℋ
6	5	cheli 31260	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℋ)
7		normcl 31153	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℋ → (normℎ‘𝑥) ∈ ℝ)
8	4, 6, 7	3syl 18	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (normℎ‘𝑥) ∈ ℝ)
9		strlem1.2	. . . . . . . . . . . . . . . 16 ⊢ 𝐵 ∈ Cℋ
10		ch0 31256	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ Cℋ → 0ℎ ∈ 𝐵)
11	9, 10	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 0ℎ ∈ 𝐵
12		eldifn 4141	. . . . . . . . . . . . . . 15 ⊢ (0ℎ ∈ (𝐴 ∖ 𝐵) → ¬ 0ℎ ∈ 𝐵)
13	11, 12	mt2 200	. . . . . . . . . . . . . 14 ⊢ ¬ 0ℎ ∈ (𝐴 ∖ 𝐵)
14		eleq1 2826	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 0ℎ → (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ 0ℎ ∈ (𝐴 ∖ 𝐵)))
15	13, 14	mtbiri 327	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0ℎ → ¬ 𝑥 ∈ (𝐴 ∖ 𝐵))
16	15	con2i 139	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ¬ 𝑥 = 0ℎ)
17		norm-i 31157	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℋ → ((normℎ‘𝑥) = 0 ↔ 𝑥 = 0ℎ))
18	4, 6, 17	3syl 18	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ((normℎ‘𝑥) = 0 ↔ 𝑥 = 0ℎ))
19	16, 18	mtbird 325	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ¬ (normℎ‘𝑥) = 0)
20	19	neqned 2944	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (normℎ‘𝑥) ≠ 0)
21	8, 20	rereccld 12091	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (1 / (normℎ‘𝑥)) ∈ ℝ)
22	21	recnd 11286	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (1 / (normℎ‘𝑥)) ∈ ℂ)
23	5	chshii 31255	. . . . . . . . . 10 ⊢ 𝐴 ∈ Sℋ
24		shmulcl 31246	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Sℋ ∧ (1 / (normℎ‘𝑥)) ∈ ℂ ∧ 𝑥 ∈ 𝐴) → ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) ∈ 𝐴)
25	23, 24	mp3an1 1447	. . . . . . . . 9 ⊢ (((1 / (normℎ‘𝑥)) ∈ ℂ ∧ 𝑥 ∈ 𝐴) → ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) ∈ 𝐴)
26	25	ex 412	. . . . . . . 8 ⊢ ((1 / (normℎ‘𝑥)) ∈ ℂ → (𝑥 ∈ 𝐴 → ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) ∈ 𝐴))
27	22, 26	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (𝑥 ∈ 𝐴 → ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) ∈ 𝐴))
28	8	recnd 11286	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (normℎ‘𝑥) ∈ ℂ)
29	9	chshii 31255	. . . . . . . . . . . 12 ⊢ 𝐵 ∈ Sℋ
30		shmulcl 31246	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ Sℋ ∧ (normℎ‘𝑥) ∈ ℂ ∧ ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) ∈ 𝐵) → ((normℎ‘𝑥) ·ℎ ((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) ∈ 𝐵)
31	29, 30	mp3an1 1447	. . . . . . . . . . 11 ⊢ (((normℎ‘𝑥) ∈ ℂ ∧ ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) ∈ 𝐵) → ((normℎ‘𝑥) ·ℎ ((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) ∈ 𝐵)
32	31	ex 412	. . . . . . . . . 10 ⊢ ((normℎ‘𝑥) ∈ ℂ → (((1 / (normℎ‘𝑥)) ·ℎ 𝑥) ∈ 𝐵 → ((normℎ‘𝑥) ·ℎ ((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) ∈ 𝐵))
33	28, 32	syl 17	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (((1 / (normℎ‘𝑥)) ·ℎ 𝑥) ∈ 𝐵 → ((normℎ‘𝑥) ·ℎ ((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) ∈ 𝐵))
34	28, 20	recidd 12035	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ((normℎ‘𝑥) · (1 / (normℎ‘𝑥))) = 1)
35	34	oveq1d 7445	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (((normℎ‘𝑥) · (1 / (normℎ‘𝑥))) ·ℎ 𝑥) = (1 ·ℎ 𝑥))
36	4, 6	syl 17	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ ℋ)
37		ax-hvmulass 31035	. . . . . . . . . . . 12 ⊢ (((normℎ‘𝑥) ∈ ℂ ∧ (1 / (normℎ‘𝑥)) ∈ ℂ ∧ 𝑥 ∈ ℋ) → (((normℎ‘𝑥) · (1 / (normℎ‘𝑥))) ·ℎ 𝑥) = ((normℎ‘𝑥) ·ℎ ((1 / (normℎ‘𝑥)) ·ℎ 𝑥)))
38	28, 22, 36, 37	syl3anc 1370	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (((normℎ‘𝑥) · (1 / (normℎ‘𝑥))) ·ℎ 𝑥) = ((normℎ‘𝑥) ·ℎ ((1 / (normℎ‘𝑥)) ·ℎ 𝑥)))
39		ax-hvmulid 31034	. . . . . . . . . . . 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 2782	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ((normℎ‘𝑥) ·ℎ ((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) = 𝑥)
42	41	eleq1d 2823	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (((normℎ‘𝑥) ·ℎ ((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
43	33, 42	sylibd 239	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (((1 / (normℎ‘𝑥)) ·ℎ 𝑥) ∈ 𝐵 → 𝑥 ∈ 𝐵))
44	43	con3d 152	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (¬ 𝑥 ∈ 𝐵 → ¬ ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) ∈ 𝐵))
45	27, 44	anim12d 609	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → (((1 / (normℎ‘𝑥)) ·ℎ 𝑥) ∈ 𝐴 ∧ ¬ ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) ∈ 𝐵)))
46		eldif 3972	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
47		eldif 3972	. . . . . 6 ⊢ (((1 / (normℎ‘𝑥)) ·ℎ 𝑥) ∈ (𝐴 ∖ 𝐵) ↔ (((1 / (normℎ‘𝑥)) ·ℎ 𝑥) ∈ 𝐴 ∧ ¬ ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) ∈ 𝐵))
48	45, 46, 47	3imtr4g 296	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (𝑥 ∈ (𝐴 ∖ 𝐵) → ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) ∈ (𝐴 ∖ 𝐵)))
49	48	pm2.43i 52	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) ∈ (𝐴 ∖ 𝐵))
50		norm-iii 31168	. . . . . 6 ⊢ (((1 / (normℎ‘𝑥)) ∈ ℂ ∧ 𝑥 ∈ ℋ) → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) = ((abs‘(1 / (normℎ‘𝑥))) · (normℎ‘𝑥)))
51	22, 36, 50	syl2anc 584	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) = ((abs‘(1 / (normℎ‘𝑥))) · (normℎ‘𝑥)))
52	15	necon2ai 2967	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ≠ 0ℎ)
53		normgt0 31155	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℋ → (𝑥 ≠ 0ℎ ↔ 0 < (normℎ‘𝑥)))
54	4, 6, 53	3syl 18	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (𝑥 ≠ 0ℎ ↔ 0 < (normℎ‘𝑥)))
55	52, 54	mpbid 232	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → 0 < (normℎ‘𝑥))
56		1re 11258	. . . . . . . . 9 ⊢ 1 ∈ ℝ
57		0le1 11783	. . . . . . . . 9 ⊢ 0 ≤ 1
58		divge0 12134	. . . . . . . . 9 ⊢ (((1 ∈ ℝ ∧ 0 ≤ 1) ∧ ((normℎ‘𝑥) ∈ ℝ ∧ 0 < (normℎ‘𝑥))) → 0 ≤ (1 / (normℎ‘𝑥)))
59	56, 57, 58	mpanl12 702	. . . . . . . 8 ⊢ (((normℎ‘𝑥) ∈ ℝ ∧ 0 < (normℎ‘𝑥)) → 0 ≤ (1 / (normℎ‘𝑥)))
60	8, 55, 59	syl2anc 584	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → 0 ≤ (1 / (normℎ‘𝑥)))
61	21, 60	absidd 15457	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (abs‘(1 / (normℎ‘𝑥))) = (1 / (normℎ‘𝑥)))
62	61	oveq1d 7445	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ((abs‘(1 / (normℎ‘𝑥))) · (normℎ‘𝑥)) = ((1 / (normℎ‘𝑥)) · (normℎ‘𝑥)))
63	28, 20	recid2d 12036	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ((1 / (normℎ‘𝑥)) · (normℎ‘𝑥)) = 1)
64	51, 62, 63	3eqtrd 2778	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) = 1)
65		fveqeq2 6915	. . . . 5 ⊢ (𝑢 = ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) → ((normℎ‘𝑢) = 1 ↔ (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) = 1))
66	65	rspcev 3621	. . . 4 ⊢ ((((1 / (normℎ‘𝑥)) ·ℎ 𝑥) ∈ (𝐴 ∖ 𝐵) ∧ (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) = 1) → ∃𝑢 ∈ (𝐴 ∖ 𝐵)(normℎ‘𝑢) = 1)
67	49, 64, 66	syl2anc 584	. . 3 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ∃𝑢 ∈ (𝐴 ∖ 𝐵)(normℎ‘𝑢) = 1)
68	67	exlimiv 1927	. 2 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵) → ∃𝑢 ∈ (𝐴 ∖ 𝐵)(normℎ‘𝑢) = 1)
69	3, 68	sylbi 217	1 ⊢ (¬ 𝐴 ⊆ 𝐵 → ∃𝑢 ∈ (𝐴 ∖ 𝐵)(normℎ‘𝑢) = 1)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1536  ∃wex 1775   ∈ wcel 2105   ≠ wne 2937  ∃wrex 3067   ∖ cdif 3959   ⊆ wss 3962  ∅c0 4338   class class class wbr 5147  ‘cfv 6562  (class class class)co 7430  ℂcc 11150  ℝcr 11151  0cc0 11152  1c1 11153   · cmul 11157   < clt 11292   ≤ cle 11293   / cdiv 11917  abscabs 15269   ℋchba 30947   ·_ℎ csm 30949  norm_ℎcno 30951  0_ℎc0v 30952   S_ℋ csh 30956   C_ℋ cch 30957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230  ax-hilex 31027  ax-hfvadd 31028  ax-hv0cl 31031  ax-hfvmul 31033  ax-hvmulid 31034  ax-hvmulass 31035  ax-hvmul0 31038  ax-hfi 31107  ax-his1 31110  ax-his3 31112  ax-his4 31113

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-sup 9479  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-z 12611  df-uz 12876  df-rp 13032  df-seq 14039  df-exp 14099  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-hnorm 30996  df-sh 31235  df-ch 31249

This theorem is referenced by:  stri  32285  hstri  32293