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Theorem nmosetn0 29749
Description: The set in the supremum of the operator norm definition df-nmoo 29729 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 29618 . . 3 (π‘ˆ ∈ NrmCVec β†’ 𝑍 ∈ 𝑋)
4 nmosetn0.4 . . . . . 6 𝑀 = (normCVβ€˜π‘ˆ)
52, 4nvz0 29652 . . . . 5 (π‘ˆ ∈ NrmCVec β†’ (π‘€β€˜π‘) = 0)
6 0le1 11683 . . . . 5 0 ≀ 1
75, 6eqbrtrdi 5145 . . . 4 (π‘ˆ ∈ NrmCVec β†’ (π‘€β€˜π‘) ≀ 1)
8 eqid 2733 . . . 4 (π‘β€˜(π‘‡β€˜π‘)) = (π‘β€˜(π‘‡β€˜π‘))
97, 8jctir 522 . . 3 (π‘ˆ ∈ NrmCVec β†’ ((π‘€β€˜π‘) ≀ 1 ∧ (π‘β€˜(π‘‡β€˜π‘)) = (π‘β€˜(π‘‡β€˜π‘))))
10 fveq2 6843 . . . . . 6 (𝑦 = 𝑍 β†’ (π‘€β€˜π‘¦) = (π‘€β€˜π‘))
1110breq1d 5116 . . . . 5 (𝑦 = 𝑍 β†’ ((π‘€β€˜π‘¦) ≀ 1 ↔ (π‘€β€˜π‘) ≀ 1))
12 2fveq3 6848 . . . . . 6 (𝑦 = 𝑍 β†’ (π‘β€˜(π‘‡β€˜π‘¦)) = (π‘β€˜(π‘‡β€˜π‘)))
1312eqeq2d 2744 . . . . 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 632 . . . 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 3580 . . 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 585 . 2 (π‘ˆ ∈ NrmCVec β†’ βˆƒπ‘¦ ∈ 𝑋 ((π‘€β€˜π‘¦) ≀ 1 ∧ (π‘β€˜(π‘‡β€˜π‘)) = (π‘β€˜(π‘‡β€˜π‘¦))))
17 fvex 6856 . . 3 (π‘β€˜(π‘‡β€˜π‘)) ∈ V
18 eqeq1 2737 . . . . 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 630 . . . 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 3172 . . 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 3631 . 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 397   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆƒwrex 3070   class class class wbr 5106  β€˜cfv 6497  0cc0 11056  1c1 11057   ≀ cle 11195  NrmCVeccnv 29568  BaseSetcba 29570  0veccn0v 29572  normCVcnmcv 29574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133  ax-pre-sup 11134
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-sup 9383  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-div 11818  df-nn 12159  df-2 12221  df-3 12222  df-n0 12419  df-z 12505  df-uz 12769  df-rp 12921  df-seq 13913  df-exp 13974  df-cj 14990  df-re 14991  df-im 14992  df-sqrt 15126  df-abs 15127  df-grpo 29477  df-gid 29478  df-ginv 29479  df-ablo 29529  df-vc 29543  df-nv 29576  df-va 29579  df-ba 29580  df-sm 29581  df-0v 29582  df-nmcv 29584
This theorem is referenced by:  nmooge0  29751  nmorepnf  29752
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