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Theorem nmosetn0 30603
Description: The set in the supremum of the operator norm definition df-nmoo 30583 is nonempty. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nmosetn0.1 𝑋 = (BaseSetβ€˜π‘ˆ)
nmosetn0.5 𝑍 = (0vecβ€˜π‘ˆ)
nmosetn0.4 𝑀 = (normCVβ€˜π‘ˆ)
Assertion
Ref Expression
nmosetn0 (π‘ˆ ∈ NrmCVec β†’ (π‘β€˜(π‘‡β€˜π‘)) ∈ {π‘₯ ∣ βˆƒπ‘¦ ∈ 𝑋 ((π‘€β€˜π‘¦) ≀ 1 ∧ π‘₯ = (π‘β€˜(π‘‡β€˜π‘¦)))})
Distinct variable groups:   π‘₯,𝑦,𝑀   π‘₯,𝑁,𝑦   π‘₯,𝑇,𝑦   π‘₯,𝑋,𝑦   π‘₯,𝑍,𝑦
Allowed substitution hints:   π‘ˆ(π‘₯,𝑦)

Proof of Theorem nmosetn0
StepHypRef Expression
1 nmosetn0.1 . . . 4 𝑋 = (BaseSetβ€˜π‘ˆ)
2 nmosetn0.5 . . . 4 𝑍 = (0vecβ€˜π‘ˆ)
31, 2nvzcl 30472 . . 3 (π‘ˆ ∈ NrmCVec β†’ 𝑍 ∈ 𝑋)
4 nmosetn0.4 . . . . . 6 𝑀 = (normCVβ€˜π‘ˆ)
52, 4nvz0 30506 . . . . 5 (π‘ˆ ∈ NrmCVec β†’ (π‘€β€˜π‘) = 0)
6 0le1 11777 . . . . 5 0 ≀ 1
75, 6eqbrtrdi 5191 . . . 4 (π‘ˆ ∈ NrmCVec β†’ (π‘€β€˜π‘) ≀ 1)
8 eqid 2728 . . . 4 (π‘β€˜(π‘‡β€˜π‘)) = (π‘β€˜(π‘‡β€˜π‘))
97, 8jctir 519 . . 3 (π‘ˆ ∈ NrmCVec β†’ ((π‘€β€˜π‘) ≀ 1 ∧ (π‘β€˜(π‘‡β€˜π‘)) = (π‘β€˜(π‘‡β€˜π‘))))
10 fveq2 6902 . . . . . 6 (𝑦 = 𝑍 β†’ (π‘€β€˜π‘¦) = (π‘€β€˜π‘))
1110breq1d 5162 . . . . 5 (𝑦 = 𝑍 β†’ ((π‘€β€˜π‘¦) ≀ 1 ↔ (π‘€β€˜π‘) ≀ 1))
12 2fveq3 6907 . . . . . 6 (𝑦 = 𝑍 β†’ (π‘β€˜(π‘‡β€˜π‘¦)) = (π‘β€˜(π‘‡β€˜π‘)))
1312eqeq2d 2739 . . . . 5 (𝑦 = 𝑍 β†’ ((π‘β€˜(π‘‡β€˜π‘)) = (π‘β€˜(π‘‡β€˜π‘¦)) ↔ (π‘β€˜(π‘‡β€˜π‘)) = (π‘β€˜(π‘‡β€˜π‘))))
1411, 13anbi12d 630 . . . 4 (𝑦 = 𝑍 β†’ (((π‘€β€˜π‘¦) ≀ 1 ∧ (π‘β€˜(π‘‡β€˜π‘)) = (π‘β€˜(π‘‡β€˜π‘¦))) ↔ ((π‘€β€˜π‘) ≀ 1 ∧ (π‘β€˜(π‘‡β€˜π‘)) = (π‘β€˜(π‘‡β€˜π‘)))))
1514rspcev 3611 . . 3 ((𝑍 ∈ 𝑋 ∧ ((π‘€β€˜π‘) ≀ 1 ∧ (π‘β€˜(π‘‡β€˜π‘)) = (π‘β€˜(π‘‡β€˜π‘)))) β†’ βˆƒπ‘¦ ∈ 𝑋 ((π‘€β€˜π‘¦) ≀ 1 ∧ (π‘β€˜(π‘‡β€˜π‘)) = (π‘β€˜(π‘‡β€˜π‘¦))))
163, 9, 15syl2anc 582 . 2 (π‘ˆ ∈ NrmCVec β†’ βˆƒπ‘¦ ∈ 𝑋 ((π‘€β€˜π‘¦) ≀ 1 ∧ (π‘β€˜(π‘‡β€˜π‘)) = (π‘β€˜(π‘‡β€˜π‘¦))))
17 fvex 6915 . . 3 (π‘β€˜(π‘‡β€˜π‘)) ∈ V
18 eqeq1 2732 . . . . 5 (π‘₯ = (π‘β€˜(π‘‡β€˜π‘)) β†’ (π‘₯ = (π‘β€˜(π‘‡β€˜π‘¦)) ↔ (π‘β€˜(π‘‡β€˜π‘)) = (π‘β€˜(π‘‡β€˜π‘¦))))
1918anbi2d 628 . . . 4 (π‘₯ = (π‘β€˜(π‘‡β€˜π‘)) β†’ (((π‘€β€˜π‘¦) ≀ 1 ∧ π‘₯ = (π‘β€˜(π‘‡β€˜π‘¦))) ↔ ((π‘€β€˜π‘¦) ≀ 1 ∧ (π‘β€˜(π‘‡β€˜π‘)) = (π‘β€˜(π‘‡β€˜π‘¦)))))
2019rexbidv 3176 . . 3 (π‘₯ = (π‘β€˜(π‘‡β€˜π‘)) β†’ (βˆƒπ‘¦ ∈ 𝑋 ((π‘€β€˜π‘¦) ≀ 1 ∧ π‘₯ = (π‘β€˜(π‘‡β€˜π‘¦))) ↔ βˆƒπ‘¦ ∈ 𝑋 ((π‘€β€˜π‘¦) ≀ 1 ∧ (π‘β€˜(π‘‡β€˜π‘)) = (π‘β€˜(π‘‡β€˜π‘¦)))))
2117, 20elab 3669 . 2 ((π‘β€˜(π‘‡β€˜π‘)) ∈ {π‘₯ ∣ βˆƒπ‘¦ ∈ 𝑋 ((π‘€β€˜π‘¦) ≀ 1 ∧ π‘₯ = (π‘β€˜(π‘‡β€˜π‘¦)))} ↔ βˆƒπ‘¦ ∈ 𝑋 ((π‘€β€˜π‘¦) ≀ 1 ∧ (π‘β€˜(π‘‡β€˜π‘)) = (π‘β€˜(π‘‡β€˜π‘¦))))
2216, 21sylibr 233 1 (π‘ˆ ∈ NrmCVec β†’ (π‘β€˜(π‘‡β€˜π‘)) ∈ {π‘₯ ∣ βˆƒπ‘¦ ∈ 𝑋 ((π‘€β€˜π‘¦) ≀ 1 ∧ π‘₯ = (π‘β€˜(π‘‡β€˜π‘¦)))})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  {cab 2705  βˆƒwrex 3067   class class class wbr 5152  β€˜cfv 6553  0cc0 11148  1c1 11149   ≀ cle 11289  NrmCVeccnv 30422  BaseSetcba 30424  0veccn0v 30426  normCVcnmcv 30428
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748  ax-cnex 11204  ax-resscn 11205  ax-1cn 11206  ax-icn 11207  ax-addcl 11208  ax-addrcl 11209  ax-mulcl 11210  ax-mulrcl 11211  ax-mulcom 11212  ax-addass 11213  ax-mulass 11214  ax-distr 11215  ax-i2m1 11216  ax-1ne0 11217  ax-1rid 11218  ax-rnegex 11219  ax-rrecex 11220  ax-cnre 11221  ax-pre-lttri 11222  ax-pre-lttrn 11223  ax-pre-ltadd 11224  ax-pre-mulgt0 11225  ax-pre-sup 11226
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7879  df-1st 8001  df-2nd 8002  df-frecs 8295  df-wrecs 8326  df-recs 8400  df-rdg 8439  df-er 8733  df-en 8973  df-dom 8974  df-sdom 8975  df-sup 9475  df-pnf 11290  df-mnf 11291  df-xr 11292  df-ltxr 11293  df-le 11294  df-sub 11486  df-neg 11487  df-div 11912  df-nn 12253  df-2 12315  df-3 12316  df-n0 12513  df-z 12599  df-uz 12863  df-rp 13017  df-seq 14009  df-exp 14069  df-cj 15088  df-re 15089  df-im 15090  df-sqrt 15224  df-abs 15225  df-grpo 30331  df-gid 30332  df-ginv 30333  df-ablo 30383  df-vc 30397  df-nv 30430  df-va 30433  df-ba 30434  df-sm 30435  df-0v 30436  df-nmcv 30438
This theorem is referenced by:  nmooge0  30605  nmorepnf  30606
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