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Theorem strlem1 29941
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 𝐴C
strlem1.2 𝐵C
Assertion
Ref Expression
strlem1 𝐴𝐵 → ∃𝑢 ∈ (𝐴𝐵)(norm𝑢) = 1)
Distinct variable groups:   𝑢,𝐴   𝑢,𝐵

Proof of Theorem strlem1
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 neq0 4312 . . 3 (¬ (𝐴𝐵) = ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴𝐵))
2 ssdif0 4326 . . 3 (𝐴𝐵 ↔ (𝐴𝐵) = ∅)
31, 2xchnxbir 334 . 2 𝐴𝐵 ↔ ∃𝑥 𝑥 ∈ (𝐴𝐵))
4 eldifi 4106 . . . . . . . . . . 11 (𝑥 ∈ (𝐴𝐵) → 𝑥𝐴)
5 strlem1.1 . . . . . . . . . . . 12 𝐴C
65cheli 28923 . . . . . . . . . . 11 (𝑥𝐴𝑥 ∈ ℋ)
7 normcl 28816 . . . . . . . . . . 11 (𝑥 ∈ ℋ → (norm𝑥) ∈ ℝ)
84, 6, 73syl 18 . . . . . . . . . 10 (𝑥 ∈ (𝐴𝐵) → (norm𝑥) ∈ ℝ)
9 strlem1.2 . . . . . . . . . . . . . . . 16 𝐵C
10 ch0 28919 . . . . . . . . . . . . . . . 16 (𝐵C → 0𝐵)
119, 10ax-mp 5 . . . . . . . . . . . . . . 15 0𝐵
12 eldifn 4107 . . . . . . . . . . . . . . 15 (0 ∈ (𝐴𝐵) → ¬ 0𝐵)
1311, 12mt2 201 . . . . . . . . . . . . . 14 ¬ 0 ∈ (𝐴𝐵)
14 eleq1 2904 . . . . . . . . . . . . . 14 (𝑥 = 0 → (𝑥 ∈ (𝐴𝐵) ↔ 0 ∈ (𝐴𝐵)))
1513, 14mtbiri 328 . . . . . . . . . . . . 13 (𝑥 = 0 → ¬ 𝑥 ∈ (𝐴𝐵))
1615con2i 141 . . . . . . . . . . . 12 (𝑥 ∈ (𝐴𝐵) → ¬ 𝑥 = 0)
17 norm-i 28820 . . . . . . . . . . . . 13 (𝑥 ∈ ℋ → ((norm𝑥) = 0 ↔ 𝑥 = 0))
184, 6, 173syl 18 . . . . . . . . . . . 12 (𝑥 ∈ (𝐴𝐵) → ((norm𝑥) = 0 ↔ 𝑥 = 0))
1916, 18mtbird 326 . . . . . . . . . . 11 (𝑥 ∈ (𝐴𝐵) → ¬ (norm𝑥) = 0)
2019neqned 3027 . . . . . . . . . 10 (𝑥 ∈ (𝐴𝐵) → (norm𝑥) ≠ 0)
218, 20rereccld 11459 . . . . . . . . 9 (𝑥 ∈ (𝐴𝐵) → (1 / (norm𝑥)) ∈ ℝ)
2221recnd 10661 . . . . . . . 8 (𝑥 ∈ (𝐴𝐵) → (1 / (norm𝑥)) ∈ ℂ)
235chshii 28918 . . . . . . . . . 10 𝐴S
24 shmulcl 28909 . . . . . . . . . 10 ((𝐴S ∧ (1 / (norm𝑥)) ∈ ℂ ∧ 𝑥𝐴) → ((1 / (norm𝑥)) · 𝑥) ∈ 𝐴)
2523, 24mp3an1 1441 . . . . . . . . 9 (((1 / (norm𝑥)) ∈ ℂ ∧ 𝑥𝐴) → ((1 / (norm𝑥)) · 𝑥) ∈ 𝐴)
2625ex 413 . . . . . . . 8 ((1 / (norm𝑥)) ∈ ℂ → (𝑥𝐴 → ((1 / (norm𝑥)) · 𝑥) ∈ 𝐴))
2722, 26syl 17 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) → (𝑥𝐴 → ((1 / (norm𝑥)) · 𝑥) ∈ 𝐴))
288recnd 10661 . . . . . . . . . 10 (𝑥 ∈ (𝐴𝐵) → (norm𝑥) ∈ ℂ)
299chshii 28918 . . . . . . . . . . . 12 𝐵S
30 shmulcl 28909 . . . . . . . . . . . 12 ((𝐵S ∧ (norm𝑥) ∈ ℂ ∧ ((1 / (norm𝑥)) · 𝑥) ∈ 𝐵) → ((norm𝑥) · ((1 / (norm𝑥)) · 𝑥)) ∈ 𝐵)
3129, 30mp3an1 1441 . . . . . . . . . . 11 (((norm𝑥) ∈ ℂ ∧ ((1 / (norm𝑥)) · 𝑥) ∈ 𝐵) → ((norm𝑥) · ((1 / (norm𝑥)) · 𝑥)) ∈ 𝐵)
3231ex 413 . . . . . . . . . 10 ((norm𝑥) ∈ ℂ → (((1 / (norm𝑥)) · 𝑥) ∈ 𝐵 → ((norm𝑥) · ((1 / (norm𝑥)) · 𝑥)) ∈ 𝐵))
3328, 32syl 17 . . . . . . . . 9 (𝑥 ∈ (𝐴𝐵) → (((1 / (norm𝑥)) · 𝑥) ∈ 𝐵 → ((norm𝑥) · ((1 / (norm𝑥)) · 𝑥)) ∈ 𝐵))
3428, 20recidd 11403 . . . . . . . . . . . 12 (𝑥 ∈ (𝐴𝐵) → ((norm𝑥) · (1 / (norm𝑥))) = 1)
3534oveq1d 7166 . . . . . . . . . . 11 (𝑥 ∈ (𝐴𝐵) → (((norm𝑥) · (1 / (norm𝑥))) · 𝑥) = (1 · 𝑥))
364, 6syl 17 . . . . . . . . . . . 12 (𝑥 ∈ (𝐴𝐵) → 𝑥 ∈ ℋ)
37 ax-hvmulass 28698 . . . . . . . . . . . 12 (((norm𝑥) ∈ ℂ ∧ (1 / (norm𝑥)) ∈ ℂ ∧ 𝑥 ∈ ℋ) → (((norm𝑥) · (1 / (norm𝑥))) · 𝑥) = ((norm𝑥) · ((1 / (norm𝑥)) · 𝑥)))
3828, 22, 36, 37syl3anc 1365 . . . . . . . . . . 11 (𝑥 ∈ (𝐴𝐵) → (((norm𝑥) · (1 / (norm𝑥))) · 𝑥) = ((norm𝑥) · ((1 / (norm𝑥)) · 𝑥)))
39 ax-hvmulid 28697 . . . . . . . . . . . 12 (𝑥 ∈ ℋ → (1 · 𝑥) = 𝑥)
404, 6, 393syl 18 . . . . . . . . . . 11 (𝑥 ∈ (𝐴𝐵) → (1 · 𝑥) = 𝑥)
4135, 38, 403eqtr3d 2868 . . . . . . . . . 10 (𝑥 ∈ (𝐴𝐵) → ((norm𝑥) · ((1 / (norm𝑥)) · 𝑥)) = 𝑥)
4241eleq1d 2901 . . . . . . . . 9 (𝑥 ∈ (𝐴𝐵) → (((norm𝑥) · ((1 / (norm𝑥)) · 𝑥)) ∈ 𝐵𝑥𝐵))
4333, 42sylibd 240 . . . . . . . 8 (𝑥 ∈ (𝐴𝐵) → (((1 / (norm𝑥)) · 𝑥) ∈ 𝐵𝑥𝐵))
4443con3d 155 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) → (¬ 𝑥𝐵 → ¬ ((1 / (norm𝑥)) · 𝑥) ∈ 𝐵))
4527, 44anim12d 608 . . . . . 6 (𝑥 ∈ (𝐴𝐵) → ((𝑥𝐴 ∧ ¬ 𝑥𝐵) → (((1 / (norm𝑥)) · 𝑥) ∈ 𝐴 ∧ ¬ ((1 / (norm𝑥)) · 𝑥) ∈ 𝐵)))
46 eldif 3949 . . . . . 6 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
47 eldif 3949 . . . . . 6 (((1 / (norm𝑥)) · 𝑥) ∈ (𝐴𝐵) ↔ (((1 / (norm𝑥)) · 𝑥) ∈ 𝐴 ∧ ¬ ((1 / (norm𝑥)) · 𝑥) ∈ 𝐵))
4845, 46, 473imtr4g 297 . . . . 5 (𝑥 ∈ (𝐴𝐵) → (𝑥 ∈ (𝐴𝐵) → ((1 / (norm𝑥)) · 𝑥) ∈ (𝐴𝐵)))
4948pm2.43i 52 . . . 4 (𝑥 ∈ (𝐴𝐵) → ((1 / (norm𝑥)) · 𝑥) ∈ (𝐴𝐵))
50 norm-iii 28831 . . . . . 6 (((1 / (norm𝑥)) ∈ ℂ ∧ 𝑥 ∈ ℋ) → (norm‘((1 / (norm𝑥)) · 𝑥)) = ((abs‘(1 / (norm𝑥))) · (norm𝑥)))
5122, 36, 50syl2anc 584 . . . . 5 (𝑥 ∈ (𝐴𝐵) → (norm‘((1 / (norm𝑥)) · 𝑥)) = ((abs‘(1 / (norm𝑥))) · (norm𝑥)))
5215necon2ai 3049 . . . . . . . . 9 (𝑥 ∈ (𝐴𝐵) → 𝑥 ≠ 0)
53 normgt0 28818 . . . . . . . . . 10 (𝑥 ∈ ℋ → (𝑥 ≠ 0 ↔ 0 < (norm𝑥)))
544, 6, 533syl 18 . . . . . . . . 9 (𝑥 ∈ (𝐴𝐵) → (𝑥 ≠ 0 ↔ 0 < (norm𝑥)))
5552, 54mpbid 233 . . . . . . . 8 (𝑥 ∈ (𝐴𝐵) → 0 < (norm𝑥))
56 1re 10633 . . . . . . . . 9 1 ∈ ℝ
57 0le1 11155 . . . . . . . . 9 0 ≤ 1
58 divge0 11501 . . . . . . . . 9 (((1 ∈ ℝ ∧ 0 ≤ 1) ∧ ((norm𝑥) ∈ ℝ ∧ 0 < (norm𝑥))) → 0 ≤ (1 / (norm𝑥)))
5956, 57, 58mpanl12 698 . . . . . . . 8 (((norm𝑥) ∈ ℝ ∧ 0 < (norm𝑥)) → 0 ≤ (1 / (norm𝑥)))
608, 55, 59syl2anc 584 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) → 0 ≤ (1 / (norm𝑥)))
6121, 60absidd 14775 . . . . . 6 (𝑥 ∈ (𝐴𝐵) → (abs‘(1 / (norm𝑥))) = (1 / (norm𝑥)))
6261oveq1d 7166 . . . . 5 (𝑥 ∈ (𝐴𝐵) → ((abs‘(1 / (norm𝑥))) · (norm𝑥)) = ((1 / (norm𝑥)) · (norm𝑥)))
6328, 20recid2d 11404 . . . . 5 (𝑥 ∈ (𝐴𝐵) → ((1 / (norm𝑥)) · (norm𝑥)) = 1)
6451, 62, 633eqtrd 2864 . . . 4 (𝑥 ∈ (𝐴𝐵) → (norm‘((1 / (norm𝑥)) · 𝑥)) = 1)
65 fveqeq2 6675 . . . . 5 (𝑢 = ((1 / (norm𝑥)) · 𝑥) → ((norm𝑢) = 1 ↔ (norm‘((1 / (norm𝑥)) · 𝑥)) = 1))
6665rspcev 3626 . . . 4 ((((1 / (norm𝑥)) · 𝑥) ∈ (𝐴𝐵) ∧ (norm‘((1 / (norm𝑥)) · 𝑥)) = 1) → ∃𝑢 ∈ (𝐴𝐵)(norm𝑢) = 1)
6749, 64, 66syl2anc 584 . . 3 (𝑥 ∈ (𝐴𝐵) → ∃𝑢 ∈ (𝐴𝐵)(norm𝑢) = 1)
6867exlimiv 1924 . 2 (∃𝑥 𝑥 ∈ (𝐴𝐵) → ∃𝑢 ∈ (𝐴𝐵)(norm𝑢) = 1)
693, 68sylbi 218 1 𝐴𝐵 → ∃𝑢 ∈ (𝐴𝐵)(norm𝑢) = 1)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1530  wex 1773  wcel 2107  wne 3020  wrex 3143  cdif 3936  wss 3939  c0 4294   class class class wbr 5062  cfv 6351  (class class class)co 7151  cc 10527  cr 10528  0cc0 10529  1c1 10530   · cmul 10534   < clt 10667  cle 10668   / cdiv 11289  abscabs 14586  chba 28610   · csm 28612  normcno 28614  0c0v 28615   S csh 28619   C cch 28620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607  ax-hilex 28690  ax-hfvadd 28691  ax-hv0cl 28694  ax-hfvmul 28696  ax-hvmulid 28697  ax-hvmulass 28698  ax-hvmul0 28701  ax-hfi 28770  ax-his1 28773  ax-his3 28775  ax-his4 28776
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8282  df-en 8502  df-dom 8503  df-sdom 8504  df-sup 8898  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-rp 12383  df-seq 13363  df-exp 13423  df-cj 14451  df-re 14452  df-im 14453  df-sqrt 14587  df-abs 14588  df-hnorm 28659  df-sh 28898  df-ch 28912
This theorem is referenced by:  stri  29948  hstri  29956
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