Theorem nmoge0 8426

Description: The norm of an operator is nonnegative.

Hypotheses

Ref	Expression
nmoxr.1	⊢ X = (Base ‘U)
nmoxr.2	⊢ Y = (Base ‘W)
nmoxr.3	⊢ N = (U normOp W)

Assertion

Ref	Expression
nmoge0	⊢ ((U ∈ NrmCVec ⋀ W ∈ NrmCVec ⋀ T:X–→Y) → 0 ≤ (N ‘T))

Proof of Theorem nmoge0

Step	Hyp	Ref	Expression
1		nmoxr.2	. . . 4 ⊢ Y = (Base ‘W)
2		eqid 1478	. . . 4 ⊢ (norm ‘W) = (norm ‘W)
3	1, 2	nvge0 8298	. . 3 ⊢ ((W ∈ NrmCVec ⋀ (T ‘(0v ‘U)) ∈ Y) → 0 ≤ ((norm ‘W) ‘(T ‘(0v ‘U))))
4		3simp2 791	. . 3 ⊢ ((U ∈ NrmCVec ⋀ W ∈ NrmCVec ⋀ T:X–→Y) → W ∈ NrmCVec)
5		ffvelrn 3820	. . . . . 6 ⊢ ((T:X–→Y ⋀ (0v ‘U) ∈ X) → (T ‘(0v ‘U)) ∈ Y)
6		nmoxr.1	. . . . . . 7 ⊢ X = (Base ‘U)
7		eqid 1478	. . . . . . 7 ⊢ (0v ‘U) = (0v ‘U)
8	6, 7	nvzcl 8251	. . . . . 6 ⊢ (U ∈ NrmCVec → (0v ‘U) ∈ X)
9	5, 8	sylan2 453	. . . . 5 ⊢ ((T:X–→Y ⋀ U ∈ NrmCVec) → (T ‘(0v ‘U)) ∈ Y)
10	9	ancoms 438	. . . 4 ⊢ ((U ∈ NrmCVec ⋀ T:X–→Y) → (T ‘(0v ‘U)) ∈ Y)
11	10	3adant2 800	. . 3 ⊢ ((U ∈ NrmCVec ⋀ W ∈ NrmCVec ⋀ T:X–→Y) → (T ‘(0v ‘U)) ∈ Y)
12	3, 4, 11	sylanc 473	. 2 ⊢ ((U ∈ NrmCVec ⋀ W ∈ NrmCVec ⋀ T:X–→Y) → 0 ≤ ((norm ‘W) ‘(T ‘(0v ‘U))))
13		supxrub 6100	. . . . . 6 ⊢ (({x∣∃z ∈ X (((norm ‘U) ‘z) ≤ 1 ⋀ x = ((norm ‘W) ‘(T ‘z)))} ⊆ ℝ* ⋀ ((norm ‘W) ‘(T ‘(0v ‘U))) ∈ {x∣∃z ∈ X (((norm ‘U) ‘z) ≤ 1 ⋀ x = ((norm ‘W) ‘(T ‘z)))}) → ((norm ‘W) ‘(T ‘(0v ‘U))) ≤ sup({x∣∃z ∈ X (((norm ‘U) ‘z) ≤ 1 ⋀ x = ((norm ‘W) ‘(T ‘z)))}, ℝ*, < ))
14	1, 2	nmosetre 8423	. . . . . . 7 ⊢ ((W ∈ NrmCVec ⋀ T:X–→Y) → {x∣∃z ∈ X (((norm ‘U) ‘z) ≤ 1 ⋀ x = ((norm ‘W) ‘(T ‘z)))} ⊆ ℝ)
15		ressxr 5510	. . . . . . . 8 ⊢ ℝ ⊆ ℝ*
16		sstr 2075	. . . . . . . 8 ⊢ (({x∣∃z ∈ X (((norm ‘U) ‘z) ≤ 1 ⋀ x = ((norm ‘W) ‘(T ‘z)))} ⊆ ℝ ⋀ ℝ ⊆ ℝ*) → {x∣∃z ∈ X (((norm ‘U) ‘z) ≤ 1 ⋀ x = ((norm ‘W) ‘(T ‘z)))} ⊆ ℝ*)
17	15, 16	mpan2 698	. . . . . . 7 ⊢ ({x∣∃z ∈ X (((norm ‘U) ‘z) ≤ 1 ⋀ x = ((norm ‘W) ‘(T ‘z)))} ⊆ ℝ → {x∣∃z ∈ X (((norm ‘U) ‘z) ≤ 1 ⋀ x = ((norm ‘W) ‘(T ‘z)))} ⊆ ℝ*)
18	14, 17	syl 10	. . . . . 6 ⊢ ((W ∈ NrmCVec ⋀ T:X–→Y) → {x∣∃z ∈ X (((norm ‘U) ‘z) ≤ 1 ⋀ x = ((norm ‘W) ‘(T ‘z)))} ⊆ ℝ*)
19		eqid 1478	. . . . . . 7 ⊢ (norm ‘U) = (norm ‘U)
20	6, 7, 19	nmosetn0 8424	. . . . . 6 ⊢ (U ∈ NrmCVec → ((norm ‘W) ‘(T ‘(0v ‘U))) ∈ {x∣∃z ∈ X (((norm ‘U) ‘z) ≤ 1 ⋀ x = ((norm ‘W) ‘(T ‘z)))})
21	13, 18, 20	syl2an 456	. . . . 5 ⊢ (((W ∈ NrmCVec ⋀ T:X–→Y) ⋀ U ∈ NrmCVec) → ((norm ‘W) ‘(T ‘(0v ‘U))) ≤ sup({x∣∃z ∈ X (((norm ‘U) ‘z) ≤ 1 ⋀ x = ((norm ‘W) ‘(T ‘z)))}, ℝ*, < ))
22	21	3impa 830	. . . 4 ⊢ ((W ∈ NrmCVec ⋀ T:X–→Y ⋀ U ∈ NrmCVec) → ((norm ‘W) ‘(T ‘(0v ‘U))) ≤ sup({x∣∃z ∈ X (((norm ‘U) ‘z) ≤ 1 ⋀ x = ((norm ‘W) ‘(T ‘z)))}, ℝ*, < ))
23	22	3comr 843	. . 3 ⊢ ((U ∈ NrmCVec ⋀ W ∈ NrmCVec ⋀ T:X–→Y) → ((norm ‘W) ‘(T ‘(0v ‘U))) ≤ sup({x∣∃z ∈ X (((norm ‘U) ‘z) ≤ 1 ⋀ x = ((norm ‘W) ‘(T ‘z)))}, ℝ*, < ))
24		nmoxr.3	. . . 4 ⊢ N = (U normOp W)
25	6, 1, 19, 2, 24	nmoval 8422	. . 3 ⊢ ((U ∈ NrmCVec ⋀ W ∈ NrmCVec ⋀ T:X–→Y) → (N ‘T) = sup({x∣∃z ∈ X (((norm ‘U) ‘z) ≤ 1 ⋀ x = ((norm ‘W) ‘(T ‘z)))}, ℝ*, < ))
26	23, 25	breqtrrd 2646	. 2 ⊢ ((U ∈ NrmCVec ⋀ W ∈ NrmCVec ⋀ T:X–→Y) → ((norm ‘W) ‘(T ‘(0v ‘U))) ≤ (N ‘T))
27		xrletrt 5576	. . 3 ⊢ ((0 ∈ ℝ* ⋀ ((norm ‘W) ‘(T ‘(0v ‘U))) ∈ ℝ* ⋀ (N ‘T) ∈ ℝ*) → ((0 ≤ ((norm ‘W) ‘(T ‘(0v ‘U))) ⋀ ((norm ‘W) ‘(T ‘(0v ‘U))) ≤ (N ‘T)) → 0 ≤ (N ‘T)))
28		0re 5452	. . . . 5 ⊢ 0 ∈ ℝ
29		rexrt 5511	. . . . 5 ⊢ (0 ∈ ℝ → 0 ∈ ℝ*)
30	28, 29	ax-mp 7	. . . 4 ⊢ 0 ∈ ℝ*
31	30	a1i 8	. . 3 ⊢ ((U ∈ NrmCVec ⋀ W ∈ NrmCVec ⋀ T:X–→Y) → 0 ∈ ℝ*)
32	1, 2	nvcl 8283	. . . . 5 ⊢ ((W ∈ NrmCVec ⋀ (T ‘(0v ‘U)) ∈ Y) → ((norm ‘W) ‘(T ‘(0v ‘U))) ∈ ℝ)
33	32, 4, 11	sylanc 473	. . . 4 ⊢ ((U ∈ NrmCVec ⋀ W ∈ NrmCVec ⋀ T:X–→Y) → ((norm ‘W) ‘(T ‘(0v ‘U))) ∈ ℝ)
34		rexrt 5511	. . . 4 ⊢ (((norm ‘W) ‘(T ‘(0v ‘U))) ∈ ℝ → ((norm ‘W) ‘(T ‘(0v ‘U))) ∈ ℝ*)
35	33, 34	syl 10	. . 3 ⊢ ((U ∈ NrmCVec ⋀ W ∈ NrmCVec ⋀ T:X–→Y) → ((norm ‘W) ‘(T ‘(0v ‘U))) ∈ ℝ*)
36	6, 1, 24	nmoxr 8425	. . 3 ⊢ ((U ∈ NrmCVec ⋀ W ∈ NrmCVec ⋀ T:X–→Y) → (N ‘T) ∈ ℝ*)
37	27, 31, 35, 36	syl3anc 860	. 2 ⊢ ((U ∈ NrmCVec ⋀ W ∈ NrmCVec ⋀ T:X–→Y) → ((0 ≤ ((norm ‘W) ‘(T ‘(0v ‘U))) ⋀ ((norm ‘W) ‘(T ‘(0v ‘U))) ≤ (N ‘T)) → 0 ≤ (N ‘T)))
38	12, 26, 37	mp2and 705	1 ⊢ ((U ∈ NrmCVec ⋀ W ∈ NrmCVec ⋀ T:X–→Y) → 0 ≤ (N ‘T))

Syntax hints:   → wi 3   ⋀ wa 223   ⋀ w3a 777   = wceq 958   ∈ wcel 960  {cab 1466  ∃wrex 1649   ⊆ wss 2050   class class class wbr 2624  –→wf 3184   ‘cfv 3188  (class class class)co 3969  supcsup 4582  ℝcr 5245  0cc0 5246  1c1 5247   ≤ cle 5307  ℝ^*cxr 5497   < clt 5498  NrmCVeccnv 8199  Basecba 8201  0_vcn0v 8203  normcnm 8205   normOp cnmo 8398

This theorem is referenced by:  nmlnogt0 8453  htthlem10 8625

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-map 4330  df-en 4374  df-dom 4375  df-sdom 4376  df-sup 4583  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-ltr 5182  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-lt 5259  df-sub 5368  df-neg 5370  df-pnf 5499  df-mnf 5500  df-xr 5501  df-ltxr 5502  df-le 5503  df-div 5715  df-2 5972  df-sqr 6671  df-re 6752  df-im 6753  df-cj 6754  df-abs 6755  df-grp 8034  df-gid 8035  df-ginv 8036  df-abl 8096  df-vc 8161  df-nv 8207  df-va 8210  df-ba 8211  df-sm 8212  df-0v 8213  df-nm 8215  df-nmo 8402

Copyright terms: Public domain