Theorem strlem1 32212

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 4305	. . 3 ⊢ (¬ (𝐴 ∖ 𝐵) = ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵))
2		ssdif0 4319	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∖ 𝐵) = ∅)
3	1, 2	xchnxbir 333	. 2 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ ∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵))
4		eldifi 4084	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ 𝐴)
5		strlem1.1	. . . . . . . . . . . 12 ⊢ 𝐴 ∈ Cℋ
6	5	cheli 31194	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℋ)
7		normcl 31087	. . . . . . . . . . 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 31190	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ Cℋ → 0ℎ ∈ 𝐵)
11	9, 10	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 0ℎ ∈ 𝐵
12		eldifn 4085	. . . . . . . . . . . . . . 15 ⊢ (0ℎ ∈ (𝐴 ∖ 𝐵) → ¬ 0ℎ ∈ 𝐵)
13	11, 12	mt2 200	. . . . . . . . . . . . . 14 ⊢ ¬ 0ℎ ∈ (𝐴 ∖ 𝐵)
14		eleq1 2816	. . . . . . . . . . . . . 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 31091	. . . . . . . . . . . . 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 2932	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (normℎ‘𝑥) ≠ 0)
21	8, 20	rereccld 11969	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (1 / (normℎ‘𝑥)) ∈ ℝ)
22	21	recnd 11162	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (1 / (normℎ‘𝑥)) ∈ ℂ)
23	5	chshii 31189	. . . . . . . . . 10 ⊢ 𝐴 ∈ Sℋ
24		shmulcl 31180	. . . . . . . . . 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 11162	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (normℎ‘𝑥) ∈ ℂ)
29	9	chshii 31189	. . . . . . . . . . . 12 ⊢ 𝐵 ∈ Sℋ
30		shmulcl 31180	. . . . . . . . . . . 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 11913	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ((normℎ‘𝑥) · (1 / (normℎ‘𝑥))) = 1)
35	34	oveq1d 7368	. . . . . . . . . . 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 30969	. . . . . . . . . . . 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 30968	. . . . . . . . . . . 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 2772	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ((normℎ‘𝑥) ·ℎ ((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) = 𝑥)
42	41	eleq1d 2813	. . . . . . . . 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 3915	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
47		eldif 3915	. . . . . 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 31102	. . . . . 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 2954	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ≠ 0ℎ)
53		normgt0 31089	. . . . . . . . . 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 11134	. . . . . . . . 9 ⊢ 1 ∈ ℝ
57		0le1 11661	. . . . . . . . 9 ⊢ 0 ≤ 1
58		divge0 12012	. . . . . . . . 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 15348	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (abs‘(1 / (normℎ‘𝑥))) = (1 / (normℎ‘𝑥)))
62	61	oveq1d 7368	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ((abs‘(1 / (normℎ‘𝑥))) · (normℎ‘𝑥)) = ((1 / (normℎ‘𝑥)) · (normℎ‘𝑥)))
63	28, 20	recid2d 11914	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ((1 / (normℎ‘𝑥)) · (normℎ‘𝑥)) = 1)
64	51, 62, 63	3eqtrd 2768	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) = 1)
65		fveqeq2 6835	. . . . 5 ⊢ (𝑢 = ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) → ((normℎ‘𝑢) = 1 ↔ (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) = 1))
66	65	rspcev 3579	. . . 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 2109   ≠ wne 2925  ∃wrex 3053   ∖ cdif 3902   ⊆ wss 3905  ∅c0 4286   class class class wbr 5095  ‘cfv 6486  (class class class)co 7353  ℂcc 11026  ℝcr 11027  0cc0 11028  1c1 11029   · cmul 11033   < clt 11168   ≤ cle 11169   / cdiv 11795  abscabs 15159   ℋchba 30881   ·_ℎ csm 30883  norm_ℎcno 30885  0_ℎc0v 30886   S_ℋ csh 30890   C_ℋ cch 30891

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105  ax-pre-sup 11106  ax-hilex 30961  ax-hfvadd 30962  ax-hv0cl 30965  ax-hfvmul 30967  ax-hvmulid 30968  ax-hvmulass 30969  ax-hvmul0 30972  ax-hfi 31041  ax-his1 31044  ax-his3 31046  ax-his4 31047

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8632  df-en 8880  df-dom 8881  df-sdom 8882  df-sup 9351  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-div 11796  df-nn 12147  df-2 12209  df-3 12210  df-n0 12403  df-z 12490  df-uz 12754  df-rp 12912  df-seq 13927  df-exp 13987  df-cj 15024  df-re 15025  df-im 15026  df-sqrt 15160  df-abs 15161  df-hnorm 30930  df-sh 31169  df-ch 31183

This theorem is referenced by:  stri  32219  hstri  32227