Theorem strlem1 32152

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 4311	. . 3 ⊢ (¬ (𝐴 ∖ 𝐵) = ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵))
2		ssdif0 4325	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∖ 𝐵) = ∅)
3	1, 2	xchnxbir 333	. 2 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ ∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵))
4		eldifi 4090	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝑥 ∈ 𝐴)
5		strlem1.1	. . . . . . . . . . . 12 ⊢ 𝐴 ∈ Cℋ
6	5	cheli 31134	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℋ)
7		normcl 31027	. . . . . . . . . . 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 31130	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ Cℋ → 0ℎ ∈ 𝐵)
11	9, 10	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 0ℎ ∈ 𝐵
12		eldifn 4091	. . . . . . . . . . . . . . 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 31031	. . . . . . . . . . . . 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 11985	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (1 / (normℎ‘𝑥)) ∈ ℝ)
22	21	recnd 11178	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (1 / (normℎ‘𝑥)) ∈ ℂ)
23	5	chshii 31129	. . . . . . . . . 10 ⊢ 𝐴 ∈ Sℋ
24		shmulcl 31120	. . . . . . . . . 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 11178	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (normℎ‘𝑥) ∈ ℂ)
29	9	chshii 31129	. . . . . . . . . . . 12 ⊢ 𝐵 ∈ Sℋ
30		shmulcl 31120	. . . . . . . . . . . 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 11929	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ((normℎ‘𝑥) · (1 / (normℎ‘𝑥))) = 1)
35	34	oveq1d 7384	. . . . . . . . . . 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 30909	. . . . . . . . . . . 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 30908	. . . . . . . . . . . 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 3921	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
47		eldif 3921	. . . . . 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 31042	. . . . . 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 31029	. . . . . . . . . 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 11150	. . . . . . . . 9 ⊢ 1 ∈ ℝ
57		0le1 11677	. . . . . . . . 9 ⊢ 0 ≤ 1
58		divge0 12028	. . . . . . . . 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 15365	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → (abs‘(1 / (normℎ‘𝑥))) = (1 / (normℎ‘𝑥)))
62	61	oveq1d 7384	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) → ((abs‘(1 / (normℎ‘𝑥))) · (normℎ‘𝑥)) = ((1 / (normℎ‘𝑥)) · (normℎ‘𝑥)))
63	28, 20	recid2d 11930	. . . . 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 6849	. . . . 5 ⊢ (𝑢 = ((1 / (normℎ‘𝑥)) ·ℎ 𝑥) → ((normℎ‘𝑢) = 1 ↔ (normℎ‘((1 / (normℎ‘𝑥)) ·ℎ 𝑥)) = 1))
66	65	rspcev 3585	. . . 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 3908   ⊆ wss 3911  ∅c0 4292   class class class wbr 5102  ‘cfv 6499  (class class class)co 7369  ℂcc 11042  ℝcr 11043  0cc0 11044  1c1 11045   · cmul 11049   < clt 11184   ≤ cle 11185   / cdiv 11811  abscabs 15176   ℋchba 30821   ·_ℎ csm 30823  norm_ℎcno 30825  0_ℎc0v 30826   S_ℋ csh 30830   C_ℋ cch 30831

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 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121  ax-pre-sup 11122  ax-hilex 30901  ax-hfvadd 30902  ax-hv0cl 30905  ax-hfvmul 30907  ax-hvmulid 30908  ax-hvmulass 30909  ax-hvmul0 30912  ax-hfi 30981  ax-his1 30984  ax-his3 30986  ax-his4 30987

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 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-sup 9369  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-div 11812  df-nn 12163  df-2 12225  df-3 12226  df-n0 12419  df-z 12506  df-uz 12770  df-rp 12928  df-seq 13943  df-exp 14003  df-cj 15041  df-re 15042  df-im 15043  df-sqrt 15177  df-abs 15178  df-hnorm 30870  df-sh 31109  df-ch 31123

This theorem is referenced by:  stri  32159  hstri  32167