Theorem nmopsetn0 29423

Description: The set in the supremum of the operator norm definition df-nmop 29397 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 28559	. . 3 ⊢ 0ℎ ∈ ℋ
2		norm0 28684	. . . . 5 ⊢ (normℎ‘0ℎ) = 0
3		0le1 10964	. . . . 5 ⊢ 0 ≤ 1
4	2, 3	eqbrtri 4950	. . . 4 ⊢ (normℎ‘0ℎ) ≤ 1
5		eqid 2778	. . . 4 ⊢ (normℎ‘(𝑇‘0ℎ)) = (normℎ‘(𝑇‘0ℎ))
6	4, 5	pm3.2i 463	. . 3 ⊢ ((normℎ‘0ℎ) ≤ 1 ∧ (normℎ‘(𝑇‘0ℎ)) = (normℎ‘(𝑇‘0ℎ)))
7		fveq2 6499	. . . . . 6 ⊢ (𝑦 = 0ℎ → (normℎ‘𝑦) = (normℎ‘0ℎ))
8	7	breq1d 4939	. . . . 5 ⊢ (𝑦 = 0ℎ → ((normℎ‘𝑦) ≤ 1 ↔ (normℎ‘0ℎ) ≤ 1))
9		2fveq3 6504	. . . . . 6 ⊢ (𝑦 = 0ℎ → (normℎ‘(𝑇‘𝑦)) = (normℎ‘(𝑇‘0ℎ)))
10	9	eqeq2d 2788	. . . . 5 ⊢ (𝑦 = 0ℎ → ((normℎ‘(𝑇‘0ℎ)) = (normℎ‘(𝑇‘𝑦)) ↔ (normℎ‘(𝑇‘0ℎ)) = (normℎ‘(𝑇‘0ℎ))))
11	8, 10	anbi12d 621	. . . 4 ⊢ (𝑦 = 0ℎ → (((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘(𝑇‘0ℎ)) = (normℎ‘(𝑇‘𝑦))) ↔ ((normℎ‘0ℎ) ≤ 1 ∧ (normℎ‘(𝑇‘0ℎ)) = (normℎ‘(𝑇‘0ℎ)))))
12	11	rspcev 3535	. . 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 679	. 2 ⊢ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘(𝑇‘0ℎ)) = (normℎ‘(𝑇‘𝑦)))
14		fvex 6512	. . 3 ⊢ (normℎ‘(𝑇‘0ℎ)) ∈ V
15		eqeq1 2782	. . . . 5 ⊢ (𝑥 = (normℎ‘(𝑇‘0ℎ)) → (𝑥 = (normℎ‘(𝑇‘𝑦)) ↔ (normℎ‘(𝑇‘0ℎ)) = (normℎ‘(𝑇‘𝑦))))
16	15	anbi2d 619	. . . 4 ⊢ (𝑥 = (normℎ‘(𝑇‘0ℎ)) → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘(𝑇‘0ℎ)) = (normℎ‘(𝑇‘𝑦)))))
17	16	rexbidv 3242	. . 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 3582	. 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 223	1 ⊢ (normℎ‘(𝑇‘0ℎ)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}

Syntax hints:   ∧ wa 387   = wceq 1507   ∈ wcel 2050  {cab 2758  ∃wrex 3089   class class class wbr 4929  ‘cfv 6188  0cc0 10335  1c1 10336   ≤ cle 10475   ℋchba 28475  norm_ℎcno 28479  0_ℎc0v 28480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-cnex 10391  ax-resscn 10392  ax-1cn 10393  ax-icn 10394  ax-addcl 10395  ax-addrcl 10396  ax-mulcl 10397  ax-mulrcl 10398  ax-mulcom 10399  ax-addass 10400  ax-mulass 10401  ax-distr 10402  ax-i2m1 10403  ax-1ne0 10404  ax-1rid 10405  ax-rnegex 10406  ax-rrecex 10407  ax-cnre 10408  ax-pre-lttri 10409  ax-pre-lttrn 10410  ax-pre-ltadd 10411  ax-pre-mulgt0 10412  ax-hv0cl 28559  ax-hvmul0 28566  ax-hfi 28635  ax-his3 28640

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-nel 3074  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-riota 6937  df-ov 6979  df-oprab 6980  df-mpo 6981  df-om 7397  df-2nd 7502  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-er 8089  df-en 8307  df-dom 8308  df-sdom 8309  df-pnf 10476  df-mnf 10477  df-xr 10478  df-ltxr 10479  df-le 10480  df-sub 10672  df-neg 10673  df-div 11099  df-nn 11440  df-2 11503  df-n0 11708  df-z 11794  df-uz 12059  df-rp 12205  df-seq 13185  df-exp 13245  df-cj 14319  df-re 14320  df-im 14321  df-sqrt 14455  df-hnorm 28524

This theorem is referenced by:  nmoprepnf  29425