Theorem nmopsetn0 30128

Description: The set in the supremum of the operator norm definition df-nmop 30102 is nonempty. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
nmopsetn0	⊢ (normℎ‘(𝑇‘0ℎ)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}

Distinct variable group:   𝑥,𝑦,𝑇

Proof of Theorem nmopsetn0

Step	Hyp	Ref	Expression
1		ax-hv0cl 29266	. . 3 ⊢ 0ℎ ∈ ℋ
2		norm0 29391	. . . . 5 ⊢ (normℎ‘0ℎ) = 0
3		0le1 11428	. . . . 5 ⊢ 0 ≤ 1
4	2, 3	eqbrtri 5091	. . . 4 ⊢ (normℎ‘0ℎ) ≤ 1
5		eqid 2738	. . . 4 ⊢ (normℎ‘(𝑇‘0ℎ)) = (normℎ‘(𝑇‘0ℎ))
6	4, 5	pm3.2i 470	. . 3 ⊢ ((normℎ‘0ℎ) ≤ 1 ∧ (normℎ‘(𝑇‘0ℎ)) = (normℎ‘(𝑇‘0ℎ)))
7		fveq2 6756	. . . . . 6 ⊢ (𝑦 = 0ℎ → (normℎ‘𝑦) = (normℎ‘0ℎ))
8	7	breq1d 5080	. . . . 5 ⊢ (𝑦 = 0ℎ → ((normℎ‘𝑦) ≤ 1 ↔ (normℎ‘0ℎ) ≤ 1))
9		2fveq3 6761	. . . . . 6 ⊢ (𝑦 = 0ℎ → (normℎ‘(𝑇‘𝑦)) = (normℎ‘(𝑇‘0ℎ)))
10	9	eqeq2d 2749	. . . . 5 ⊢ (𝑦 = 0ℎ → ((normℎ‘(𝑇‘0ℎ)) = (normℎ‘(𝑇‘𝑦)) ↔ (normℎ‘(𝑇‘0ℎ)) = (normℎ‘(𝑇‘0ℎ))))
11	8, 10	anbi12d 630	. . . 4 ⊢ (𝑦 = 0ℎ → (((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘(𝑇‘0ℎ)) = (normℎ‘(𝑇‘𝑦))) ↔ ((normℎ‘0ℎ) ≤ 1 ∧ (normℎ‘(𝑇‘0ℎ)) = (normℎ‘(𝑇‘0ℎ)))))
12	11	rspcev 3552	. . 3 ⊢ ((0ℎ ∈ ℋ ∧ ((normℎ‘0ℎ) ≤ 1 ∧ (normℎ‘(𝑇‘0ℎ)) = (normℎ‘(𝑇‘0ℎ)))) → ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘(𝑇‘0ℎ)) = (normℎ‘(𝑇‘𝑦))))
13	1, 6, 12	mp2an 688	. 2 ⊢ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘(𝑇‘0ℎ)) = (normℎ‘(𝑇‘𝑦)))
14		fvex 6769	. . 3 ⊢ (normℎ‘(𝑇‘0ℎ)) ∈ V
15		eqeq1 2742	. . . . 5 ⊢ (𝑥 = (normℎ‘(𝑇‘0ℎ)) → (𝑥 = (normℎ‘(𝑇‘𝑦)) ↔ (normℎ‘(𝑇‘0ℎ)) = (normℎ‘(𝑇‘𝑦))))
16	15	anbi2d 628	. . . 4 ⊢ (𝑥 = (normℎ‘(𝑇‘0ℎ)) → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘(𝑇‘0ℎ)) = (normℎ‘(𝑇‘𝑦)))))
17	16	rexbidv 3225	. . 3 ⊢ (𝑥 = (normℎ‘(𝑇‘0ℎ)) → (∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘(𝑇‘0ℎ)) = (normℎ‘(𝑇‘𝑦)))))
18	14, 17	elab 3602	. 2 ⊢ ((normℎ‘(𝑇‘0ℎ)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ↔ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘(𝑇‘0ℎ)) = (normℎ‘(𝑇‘𝑦))))
19	13, 18	mpbir 230	1 ⊢ (normℎ‘(𝑇‘0ℎ)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}

Syntax hints:   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {cab 2715  ∃wrex 3064   class class class wbr 5070  ‘cfv 6418  0cc0 10802  1c1 10803   ≤ cle 10941   ℋchba 29182  norm_ℎcno 29186  0_ℎc0v 29187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-hv0cl 29266  ax-hvmul0 29273  ax-hfi 29342  ax-his3 29347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-n0 12164  df-z 12250  df-uz 12512  df-rp 12660  df-seq 13650  df-exp 13711  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-hnorm 29231

This theorem is referenced by:  nmoprepnf  30130