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Theorem nmoo0 27030
 Description: The operator norm of the zero operator. (Contributed by NM, 27-Nov-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nmoo0.3 𝑁 = (𝑈 normOpOLD 𝑊)
nmoo0.0 𝑍 = (𝑈 0op 𝑊)
Assertion
Ref Expression
nmoo0 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑁𝑍) = 0)

Proof of Theorem nmoo0
Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . . . . 5 (BaseSet‘𝑈) = (BaseSet‘𝑈)
2 eqid 2610 . . . . 5 (BaseSet‘𝑊) = (BaseSet‘𝑊)
3 nmoo0.0 . . . . 5 𝑍 = (𝑈 0op 𝑊)
41, 2, 30oo 27028 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍:(BaseSet‘𝑈)⟶(BaseSet‘𝑊))
5 eqid 2610 . . . . 5 (normCV𝑈) = (normCV𝑈)
6 eqid 2610 . . . . 5 (normCV𝑊) = (normCV𝑊)
7 nmoo0.3 . . . . 5 𝑁 = (𝑈 normOpOLD 𝑊)
81, 2, 5, 6, 7nmooval 27002 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑍:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)) → (𝑁𝑍) = sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑈)(((normCV𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑊)‘(𝑍𝑧)))}, ℝ*, < ))
94, 8mpd3an3 1417 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑁𝑍) = sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑈)(((normCV𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑊)‘(𝑍𝑧)))}, ℝ*, < ))
10 df-sn 4126 . . . . 5 {0} = {𝑥𝑥 = 0}
11 eqid 2610 . . . . . . . . . . 11 (0vec𝑈) = (0vec𝑈)
121, 11nvzcl 26873 . . . . . . . . . 10 (𝑈 ∈ NrmCVec → (0vec𝑈) ∈ (BaseSet‘𝑈))
1311, 5nvz0 26907 . . . . . . . . . . 11 (𝑈 ∈ NrmCVec → ((normCV𝑈)‘(0vec𝑈)) = 0)
14 0le1 10430 . . . . . . . . . . 11 0 ≤ 1
1513, 14syl6eqbr 4622 . . . . . . . . . 10 (𝑈 ∈ NrmCVec → ((normCV𝑈)‘(0vec𝑈)) ≤ 1)
16 fveq2 6103 . . . . . . . . . . . 12 (𝑧 = (0vec𝑈) → ((normCV𝑈)‘𝑧) = ((normCV𝑈)‘(0vec𝑈)))
1716breq1d 4593 . . . . . . . . . . 11 (𝑧 = (0vec𝑈) → (((normCV𝑈)‘𝑧) ≤ 1 ↔ ((normCV𝑈)‘(0vec𝑈)) ≤ 1))
1817rspcev 3282 . . . . . . . . . 10 (((0vec𝑈) ∈ (BaseSet‘𝑈) ∧ ((normCV𝑈)‘(0vec𝑈)) ≤ 1) → ∃𝑧 ∈ (BaseSet‘𝑈)((normCV𝑈)‘𝑧) ≤ 1)
1912, 15, 18syl2anc 691 . . . . . . . . 9 (𝑈 ∈ NrmCVec → ∃𝑧 ∈ (BaseSet‘𝑈)((normCV𝑈)‘𝑧) ≤ 1)
2019biantrurd 528 . . . . . . . 8 (𝑈 ∈ NrmCVec → (𝑥 = 0 ↔ (∃𝑧 ∈ (BaseSet‘𝑈)((normCV𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = 0)))
2120adantr 480 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑥 = 0 ↔ (∃𝑧 ∈ (BaseSet‘𝑈)((normCV𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = 0)))
22 eqid 2610 . . . . . . . . . . . . . . 15 (0vec𝑊) = (0vec𝑊)
231, 22, 30oval 27027 . . . . . . . . . . . . . 14 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑧 ∈ (BaseSet‘𝑈)) → (𝑍𝑧) = (0vec𝑊))
24233expa 1257 . . . . . . . . . . . . 13 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → (𝑍𝑧) = (0vec𝑊))
2524fveq2d 6107 . . . . . . . . . . . 12 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((normCV𝑊)‘(𝑍𝑧)) = ((normCV𝑊)‘(0vec𝑊)))
2622, 6nvz0 26907 . . . . . . . . . . . . 13 (𝑊 ∈ NrmCVec → ((normCV𝑊)‘(0vec𝑊)) = 0)
2726ad2antlr 759 . . . . . . . . . . . 12 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((normCV𝑊)‘(0vec𝑊)) = 0)
2825, 27eqtrd 2644 . . . . . . . . . . 11 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((normCV𝑊)‘(𝑍𝑧)) = 0)
2928eqeq2d 2620 . . . . . . . . . 10 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → (𝑥 = ((normCV𝑊)‘(𝑍𝑧)) ↔ 𝑥 = 0))
3029anbi2d 736 . . . . . . . . 9 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((((normCV𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑊)‘(𝑍𝑧))) ↔ (((normCV𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = 0)))
3130rexbidva 3031 . . . . . . . 8 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (∃𝑧 ∈ (BaseSet‘𝑈)(((normCV𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑊)‘(𝑍𝑧))) ↔ ∃𝑧 ∈ (BaseSet‘𝑈)(((normCV𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = 0)))
32 r19.41v 3070 . . . . . . . 8 (∃𝑧 ∈ (BaseSet‘𝑈)(((normCV𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = 0) ↔ (∃𝑧 ∈ (BaseSet‘𝑈)((normCV𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = 0))
3331, 32syl6rbb 276 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → ((∃𝑧 ∈ (BaseSet‘𝑈)((normCV𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = 0) ↔ ∃𝑧 ∈ (BaseSet‘𝑈)(((normCV𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑊)‘(𝑍𝑧)))))
3421, 33bitrd 267 . . . . . 6 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑥 = 0 ↔ ∃𝑧 ∈ (BaseSet‘𝑈)(((normCV𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑊)‘(𝑍𝑧)))))
3534abbidv 2728 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → {𝑥𝑥 = 0} = {𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑈)(((normCV𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑊)‘(𝑍𝑧)))})
3610, 35syl5req 2657 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → {𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑈)(((normCV𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑊)‘(𝑍𝑧)))} = {0})
3736supeq1d 8235 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑈)(((normCV𝑈)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑊)‘(𝑍𝑧)))}, ℝ*, < ) = sup({0}, ℝ*, < ))
389, 37eqtrd 2644 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑁𝑍) = sup({0}, ℝ*, < ))
39 xrltso 11850 . . 3 < Or ℝ*
40 0xr 9965 . . 3 0 ∈ ℝ*
41 supsn 8261 . . 3 (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0)
4239, 40, 41mp2an 704 . 2 sup({0}, ℝ*, < ) = 0
4338, 42syl6eq 2660 1 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑁𝑍) = 0)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∃wrex 2897  {csn 4125   class class class wbr 4583   Or wor 4958  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  supcsup 8229  0cc0 9815  1c1 9816  ℝ*cxr 9952   < clt 9953   ≤ cle 9954  NrmCVeccnv 26823  BaseSetcba 26825  0veccn0v 26827  normCVcnmcv 26829   normOpOLD cnmoo 26980   0op c0o 26982 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-grpo 26731  df-gid 26732  df-ginv 26733  df-ablo 26783  df-vc 26798  df-nv 26831  df-va 26834  df-ba 26835  df-sm 26836  df-0v 26837  df-nmcv 26839  df-nmoo 26984  df-0o 26986 This theorem is referenced by:  0blo  27031  nmlno0lem  27032
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