Theorem strlem1 32269

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 4352	. . 3 ⊢ (¬ (𝐴 ∖ 𝐵) = ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵))
2		ssdif0 4366	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∖ 𝐵) = ∅)
3	1, 2	xchnxbir 333	. 2 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ ∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵))
4		eldifi 4131	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ 𝐴)
5		strlem1.1	. . . . . . . . . . . 12 ⊢ 𝐴 ∈ Cℋ
6	5	cheli 31251	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℋ)
7		normcl 31144	. . . . . . . . . . 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 31247	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ Cℋ → 0ℎ ∈ 𝐵)
11	9, 10	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 0ℎ ∈ 𝐵
12		eldifn 4132	. . . . . . . . . . . . . . 15 ⊢ (0ℎ ∈ (𝐴 ∖ 𝐵) → ¬ 0ℎ ∈ 𝐵)
13	11, 12	mt2 200	. . . . . . . . . . . . . 14 ⊢ ¬ 0ℎ ∈ (𝐴 ∖ 𝐵)
14		eleq1 2829	. . . . . . . . . . . . . 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 31148	. . . . . . . . . . . . 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 2947	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (normℎ‘𝑥) ≠ 0)
21	8, 20	rereccld 12094	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (1 / (normℎ‘𝑥)) ∈ ℝ)
22	21	recnd 11289	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (1 / (normℎ‘𝑥)) ∈ ℂ)
23	5	chshii 31246	. . . . . . . . . 10 ⊢ 𝐴 ∈ Sℋ
24		shmulcl 31237	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Sℋ ∧ (1 / (normℎ‘𝑥)) ∈ ℂ ∧ 𝑥 ∈ 𝐴) → ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) ∈ 𝐴)
25	23, 24	mp3an1 1450	. . . . . . . . 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 11289	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (normℎ‘𝑥) ∈ ℂ)
29	9	chshii 31246	. . . . . . . . . . . 12 ⊢ 𝐵 ∈ Sℋ
30		shmulcl 31237	. . . . . . . . . . . 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 1450	. . . . . . . . . . 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 12038	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ((normℎ‘𝑥) · (1 / (normℎ‘𝑥))) = 1)
35	34	oveq1d 7446	. . . . . . . . . . 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 31026	. . . . . . . . . . . 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 1373	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (((normℎ‘𝑥) · (1 / (normℎ‘𝑥))) ·ℎ 𝑥) = ((normℎ‘𝑥) ·ℎ ((1 / (normℎ‘𝑥)) ·ℎ 𝑥)))
39		ax-hvmulid 31025	. . . . . . . . . . . 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 2785	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ((normℎ‘𝑥) ·ℎ ((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) = 𝑥)
42	41	eleq1d 2826	. . . . . . . . 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 3961	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
47		eldif 3961	. . . . . 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 31159	. . . . . 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 2970	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ≠ 0ℎ)
53		normgt0 31146	. . . . . . . . . 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 11261	. . . . . . . . 9 ⊢ 1 ∈ ℝ
57		0le1 11786	. . . . . . . . 9 ⊢ 0 ≤ 1
58		divge0 12137	. . . . . . . . 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 15461	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (abs‘(1 / (normℎ‘𝑥))) = (1 / (normℎ‘𝑥)))
62	61	oveq1d 7446	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ((abs‘(1 / (normℎ‘𝑥))) · (normℎ‘𝑥)) = ((1 / (normℎ‘𝑥)) · (normℎ‘𝑥)))
63	28, 20	recid2d 12039	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ((1 / (normℎ‘𝑥)) · (normℎ‘𝑥)) = 1)
64	51, 62, 63	3eqtrd 2781	. . . 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 3622	. . . 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 1930	. 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 1540  ∃wex 1779   ∈ wcel 2108   ≠ wne 2940  ∃wrex 3070   ∖ cdif 3948   ⊆ wss 3951  ∅c0 4333   class class class wbr 5143  ‘cfv 6561  (class class class)co 7431  ℂcc 11153  ℝcr 11154  0cc0 11155  1c1 11156   · cmul 11160   < clt 11295   ≤ cle 11296   / cdiv 11920  abscabs 15273   ℋchba 30938   ·_ℎ csm 30940  norm_ℎcno 30942  0_ℎc0v 30943   S_ℋ csh 30947   C_ℋ cch 30948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-hilex 31018  ax-hfvadd 31019  ax-hv0cl 31022  ax-hfvmul 31024  ax-hvmulid 31025  ax-hvmulass 31026  ax-hvmul0 31029  ax-hfi 31098  ax-his1 31101  ax-his3 31103  ax-his4 31104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-seq 14043  df-exp 14103  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-hnorm 30987  df-sh 31226  df-ch 31240

This theorem is referenced by:  stri  32276  hstri  32284