Theorem nmopsetn0 31897

Description: The set in the supremum of the operator norm definition df-nmop 31871 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 31035	. . 3 ⊢ 0ℎ ∈ ℋ
2		norm0 31160	. . . . 5 ⊢ (normℎ‘0ℎ) = 0
3		0le1 11813	. . . . 5 ⊢ 0 ≤ 1
4	2, 3	eqbrtri 5187	. . . 4 ⊢ (normℎ‘0ℎ) ≤ 1
5		eqid 2740	. . . 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 6920	. . . . . 6 ⊢ (𝑦 = 0ℎ → (normℎ‘𝑦) = (normℎ‘0ℎ))
8	7	breq1d 5176	. . . . 5 ⊢ (𝑦 = 0ℎ → ((normℎ‘𝑦) ≤ 1 ↔ (normℎ‘0ℎ) ≤ 1))
9		2fveq3 6925	. . . . . 6 ⊢ (𝑦 = 0ℎ → (normℎ‘(𝑇‘𝑦)) = (normℎ‘(𝑇‘0ℎ)))
10	9	eqeq2d 2751	. . . . 5 ⊢ (𝑦 = 0ℎ → ((normℎ‘(𝑇‘0ℎ)) = (normℎ‘(𝑇‘𝑦)) ↔ (normℎ‘(𝑇‘0ℎ)) = (normℎ‘(𝑇‘0ℎ))))
11	8, 10	anbi12d 631	. . . 4 ⊢ (𝑦 = 0ℎ → (((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘(𝑇‘0ℎ)) = (normℎ‘(𝑇‘𝑦))) ↔ ((normℎ‘0ℎ) ≤ 1 ∧ (normℎ‘(𝑇‘0ℎ)) = (normℎ‘(𝑇‘0ℎ)))))
12	11	rspcev 3635	. . 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 691	. 2 ⊢ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘(𝑇‘0ℎ)) = (normℎ‘(𝑇‘𝑦)))
14		fvex 6933	. . 3 ⊢ (normℎ‘(𝑇‘0ℎ)) ∈ V
15		eqeq1 2744	. . . . 5 ⊢ (𝑥 = (normℎ‘(𝑇‘0ℎ)) → (𝑥 = (normℎ‘(𝑇‘𝑦)) ↔ (normℎ‘(𝑇‘0ℎ)) = (normℎ‘(𝑇‘𝑦))))
16	15	anbi2d 629	. . . 4 ⊢ (𝑥 = (normℎ‘(𝑇‘0ℎ)) → (((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦))) ↔ ((normℎ‘𝑦) ≤ 1 ∧ (normℎ‘(𝑇‘0ℎ)) = (normℎ‘(𝑇‘𝑦)))))
17	16	rexbidv 3185	. . 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 3694	. 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 231	1 ⊢ (normℎ‘(𝑇‘0ℎ)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}

Syntax hints:   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {cab 2717  ∃wrex 3076   class class class wbr 5166  ‘cfv 6573  0cc0 11184  1c1 11185   ≤ cle 11325   ℋchba 30951  norm_ℎcno 30955  0_ℎc0v 30956

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-hv0cl 31035  ax-hvmul0 31042  ax-hfi 31111  ax-his3 31116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-n0 12554  df-z 12640  df-uz 12904  df-rp 13058  df-seq 14053  df-exp 14113  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-hnorm 31000

This theorem is referenced by:  nmoprepnf  31899