Theorem strlem1 32336

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 4293	. . 3 ⊢ (¬ (𝐴 ∖ 𝐵) = ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵))
2		ssdif0 4307	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∖ 𝐵) = ∅)
3	1, 2	xchnxbir 333	. 2 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ ∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵))
4		eldifi 4072	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ 𝐴)
5		strlem1.1	. . . . . . . . . . . 12 ⊢ 𝐴 ∈ Cℋ
6	5	cheli 31318	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℋ)
7		normcl 31211	. . . . . . . . . . 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 31314	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ Cℋ → 0ℎ ∈ 𝐵)
11	9, 10	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 0ℎ ∈ 𝐵
12		eldifn 4073	. . . . . . . . . . . . . . 15 ⊢ (0ℎ ∈ (𝐴 ∖ 𝐵) → ¬ 0ℎ ∈ 𝐵)
13	11, 12	mt2 200	. . . . . . . . . . . . . 14 ⊢ ¬ 0ℎ ∈ (𝐴 ∖ 𝐵)
14		eleq1 2825	. . . . . . . . . . . . . 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 31215	. . . . . . . . . . . . 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 2940	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (normℎ‘𝑥) ≠ 0)
21	8, 20	rereccld 11973	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (1 / (normℎ‘𝑥)) ∈ ℝ)
22	21	recnd 11164	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (1 / (normℎ‘𝑥)) ∈ ℂ)
23	5	chshii 31313	. . . . . . . . . 10 ⊢ 𝐴 ∈ Sℋ
24		shmulcl 31304	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Sℋ ∧ (1 / (normℎ‘𝑥)) ∈ ℂ ∧ 𝑥 ∈ 𝐴) → ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) ∈ 𝐴)
25	23, 24	mp3an1 1451	. . . . . . . . 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 11164	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (normℎ‘𝑥) ∈ ℂ)
29	9	chshii 31313	. . . . . . . . . . . 12 ⊢ 𝐵 ∈ Sℋ
30		shmulcl 31304	. . . . . . . . . . . 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 1451	. . . . . . . . . . 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 11917	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ((normℎ‘𝑥) · (1 / (normℎ‘𝑥))) = 1)
35	34	oveq1d 7375	. . . . . . . . . . 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 31093	. . . . . . . . . . . 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 1374	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (((normℎ‘𝑥) · (1 / (normℎ‘𝑥))) ·ℎ 𝑥) = ((normℎ‘𝑥) ·ℎ ((1 / (normℎ‘𝑥)) ·ℎ 𝑥)))
39		ax-hvmulid 31092	. . . . . . . . . . . 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 2780	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ((normℎ‘𝑥) ·ℎ ((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) = 𝑥)
42	41	eleq1d 2822	. . . . . . . . 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 610	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → (((1 / (normℎ‘𝑥)) ·ℎ 𝑥) ∈ 𝐴 ∧ ¬ ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) ∈ 𝐵)))
46		eldif 3900	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
47		eldif 3900	. . . . . 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 31226	. . . . . 6 ⊢ (((1 / (normℎ‘𝑥)) ∈ ℂ ∧ 𝑥 ∈ ℋ) → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) = ((abs‘(1 / (normℎ‘𝑥))) · (normℎ‘𝑥)))
51	22, 36, 50	syl2anc 585	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) = ((abs‘(1 / (normℎ‘𝑥))) · (normℎ‘𝑥)))
52	15	necon2ai 2962	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ≠ 0ℎ)
53		normgt0 31213	. . . . . . . . . 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 11135	. . . . . . . . 9 ⊢ 1 ∈ ℝ
57		0le1 11664	. . . . . . . . 9 ⊢ 0 ≤ 1
58		divge0 12016	. . . . . . . . 9 ⊢ (((1 ∈ ℝ ∧ 0 ≤ 1) ∧ ((normℎ‘𝑥) ∈ ℝ ∧ 0 < (normℎ‘𝑥))) → 0 ≤ (1 / (normℎ‘𝑥)))
59	56, 57, 58	mpanl12 703	. . . . . . . 8 ⊢ (((normℎ‘𝑥) ∈ ℝ ∧ 0 < (normℎ‘𝑥)) → 0 ≤ (1 / (normℎ‘𝑥)))
60	8, 55, 59	syl2anc 585	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → 0 ≤ (1 / (normℎ‘𝑥)))
61	21, 60	absidd 15376	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (abs‘(1 / (normℎ‘𝑥))) = (1 / (normℎ‘𝑥)))
62	61	oveq1d 7375	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ((abs‘(1 / (normℎ‘𝑥))) · (normℎ‘𝑥)) = ((1 / (normℎ‘𝑥)) · (normℎ‘𝑥)))
63	28, 20	recid2d 11918	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ((1 / (normℎ‘𝑥)) · (normℎ‘𝑥)) = 1)
64	51, 62, 63	3eqtrd 2776	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) = 1)
65		fveqeq2 6843	. . . . 5 ⊢ (𝑢 = ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) → ((normℎ‘𝑢) = 1 ↔ (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) = 1))
66	65	rspcev 3565	. . . 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 585	. . 3 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ∃𝑢 ∈ (𝐴 ∖ 𝐵)(normℎ‘𝑢) = 1)
68	67	exlimiv 1932	. 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 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3062   ∖ cdif 3887   ⊆ wss 3890  ∅c0 4274   class class class wbr 5086  ‘cfv 6492  (class class class)co 7360  ℂcc 11027  ℝcr 11028  0cc0 11029  1c1 11030   · cmul 11034   < clt 11170   ≤ cle 11171   / cdiv 11798  abscabs 15187   ℋchba 31005   ·_ℎ csm 31007  norm_ℎcno 31009  0_ℎc0v 31010   S_ℋ csh 31014   C_ℋ cch 31015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107  ax-hilex 31085  ax-hfvadd 31086  ax-hv0cl 31089  ax-hfvmul 31091  ax-hvmulid 31092  ax-hvmulass 31093  ax-hvmul0 31096  ax-hfi 31165  ax-his1 31168  ax-his3 31170  ax-his4 31171

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-sup 9348  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12166  df-2 12235  df-3 12236  df-n0 12429  df-z 12516  df-uz 12780  df-rp 12934  df-seq 13955  df-exp 14015  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-hnorm 31054  df-sh 31293  df-ch 31307

This theorem is referenced by:  stri  32343  hstri  32351