Theorem nmounbi 8435

Description: Two ways two express that an operator is unbounded.

Hypotheses

Ref	Expression
nmoubi.1	⊢ X = (Base ‘U)
nmoubi.y	⊢ Y = (Base ‘W)
nmoubi.l	⊢ L = (norm ‘U)
nmoubi.m	⊢ M = (norm ‘W)
nmoubi.3	⊢ N = (U normOp W)
nmoubi.u	⊢ U ∈ NrmCVec
nmoubi.w	⊢ W ∈ NrmCVec

Assertion

Ref	Expression
nmounbi	⊢ (T:X–→Y → ((N ‘T) = +∞ ↔ ∀r ∈ ℝ ∃y ∈ X ((L ‘y) ≤ 1 ⋀ r < (M ‘(T ‘y)))))

Distinct variable groups:   y,r,L   M,r,y   N,r,y   T,r,y   y,U   y,W   X,r,y   Y,r,y

Proof of Theorem nmounbi

Step	Hyp	Ref	Expression
1		nmoubi.1	. . . 4 ⊢ X = (Base ‘U)
2		nmoubi.y	. . . 4 ⊢ Y = (Base ‘W)
3		nmoubi.l	. . . 4 ⊢ L = (norm ‘U)
4		nmoubi.m	. . . 4 ⊢ M = (norm ‘W)
5		nmoubi.3	. . . 4 ⊢ N = (U normOp W)
6		nmoubi.u	. . . 4 ⊢ U ∈ NrmCVec
7		nmoubi.w	. . . 4 ⊢ W ∈ NrmCVec
8	1, 2, 3, 4, 5, 6, 7	nmobndi 8434	. . 3 ⊢ (T:X–→Y → ((N ‘T) ∈ ℝ ↔ ∃r ∈ ℝ ∀y ∈ X ((L ‘y) ≤ 1 → (M ‘(T ‘y)) ≤ r)))
9	1, 2, 5	nmorepnf 8427	. . . 4 ⊢ ((U ∈ NrmCVec ⋀ W ∈ NrmCVec ⋀ T:X–→Y) → ((N ‘T) ∈ ℝ ↔ (N ‘T) ≠ +∞))
10	6, 7, 9	mp3an12 908	. . 3 ⊢ (T:X–→Y → ((N ‘T) ∈ ℝ ↔ (N ‘T) ≠ +∞))
11		lenltt 5522	. . . . . . . . . . 11 ⊢ (((M ‘(T ‘y)) ∈ ℝ ⋀ r ∈ ℝ) → ((M ‘(T ‘y)) ≤ r ↔ ¬ r < (M ‘(T ‘y))))
12		ffvelrn 3820	. . . . . . . . . . . 12 ⊢ ((T:X–→Y ⋀ y ∈ X) → (T ‘y) ∈ Y)
13	2, 4	nvcl 8283	. . . . . . . . . . . . 13 ⊢ ((W ∈ NrmCVec ⋀ (T ‘y) ∈ Y) → (M ‘(T ‘y)) ∈ ℝ)
14	7, 13	mpan 697	. . . . . . . . . . . 12 ⊢ ((T ‘y) ∈ Y → (M ‘(T ‘y)) ∈ ℝ)
15	12, 14	syl 10	. . . . . . . . . . 11 ⊢ ((T:X–→Y ⋀ y ∈ X) → (M ‘(T ‘y)) ∈ ℝ)
16	11, 15	sylan 450	. . . . . . . . . 10 ⊢ (((T:X–→Y ⋀ y ∈ X) ⋀ r ∈ ℝ) → ((M ‘(T ‘y)) ≤ r ↔ ¬ r < (M ‘(T ‘y))))
17	16	an1rs 491	. . . . . . . . 9 ⊢ (((T:X–→Y ⋀ r ∈ ℝ) ⋀ y ∈ X) → ((M ‘(T ‘y)) ≤ r ↔ ¬ r < (M ‘(T ‘y))))
18	17	imbi2d 614	. . . . . . . 8 ⊢ (((T:X–→Y ⋀ r ∈ ℝ) ⋀ y ∈ X) → (((L ‘y) ≤ 1 → (M ‘(T ‘y)) ≤ r) ↔ ((L ‘y) ≤ 1 → ¬ r < (M ‘(T ‘y)))))
19		imnan 242	. . . . . . . 8 ⊢ (((L ‘y) ≤ 1 → ¬ r < (M ‘(T ‘y))) ↔ ¬ ((L ‘y) ≤ 1 ⋀ r < (M ‘(T ‘y))))
20	18, 19	syl6bb 538	. . . . . . 7 ⊢ (((T:X–→Y ⋀ r ∈ ℝ) ⋀ y ∈ X) → (((L ‘y) ≤ 1 → (M ‘(T ‘y)) ≤ r) ↔ ¬ ((L ‘y) ≤ 1 ⋀ r < (M ‘(T ‘y)))))
21	20	ralbidva 1662	. . . . . 6 ⊢ ((T:X–→Y ⋀ r ∈ ℝ) → (∀y ∈ X ((L ‘y) ≤ 1 → (M ‘(T ‘y)) ≤ r) ↔ ∀y ∈ X ¬ ((L ‘y) ≤ 1 ⋀ r < (M ‘(T ‘y)))))
22		ralnex 1656	. . . . . 6 ⊢ (∀y ∈ X ¬ ((L ‘y) ≤ 1 ⋀ r < (M ‘(T ‘y))) ↔ ¬ ∃y ∈ X ((L ‘y) ≤ 1 ⋀ r < (M ‘(T ‘y))))
23	21, 22	syl6bb 538	. . . . 5 ⊢ ((T:X–→Y ⋀ r ∈ ℝ) → (∀y ∈ X ((L ‘y) ≤ 1 → (M ‘(T ‘y)) ≤ r) ↔ ¬ ∃y ∈ X ((L ‘y) ≤ 1 ⋀ r < (M ‘(T ‘y)))))
24	23	rexbidva 1663	. . . 4 ⊢ (T:X–→Y → (∃r ∈ ℝ ∀y ∈ X ((L ‘y) ≤ 1 → (M ‘(T ‘y)) ≤ r) ↔ ∃r ∈ ℝ ¬ ∃y ∈ X ((L ‘y) ≤ 1 ⋀ r < (M ‘(T ‘y)))))
25		rexnal 1657	. . . 4 ⊢ (∃r ∈ ℝ ¬ ∃y ∈ X ((L ‘y) ≤ 1 ⋀ r < (M ‘(T ‘y))) ↔ ¬ ∀r ∈ ℝ ∃y ∈ X ((L ‘y) ≤ 1 ⋀ r < (M ‘(T ‘y))))
26	24, 25	syl6bb 538	. . 3 ⊢ (T:X–→Y → (∃r ∈ ℝ ∀y ∈ X ((L ‘y) ≤ 1 → (M ‘(T ‘y)) ≤ r) ↔ ¬ ∀r ∈ ℝ ∃y ∈ X ((L ‘y) ≤ 1 ⋀ r < (M ‘(T ‘y)))))
27	8, 10, 26	3bitr3d 550	. 2 ⊢ (T:X–→Y → ((N ‘T) ≠ +∞ ↔ ¬ ∀r ∈ ℝ ∃y ∈ X ((L ‘y) ≤ 1 ⋀ r < (M ‘(T ‘y)))))
28	27	necon4abid 1632	1 ⊢ (T:X–→Y → ((N ‘T) = +∞ ↔ ∀r ∈ ℝ ∃y ∈ X ((L ‘y) ≤ 1 ⋀ r < (M ‘(T ‘y)))))

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146   ⋀ wa 223   = wceq 958   ∈ wcel 960   ≠ wne 1588  ∀wral 1648  ∃wrex 1649   class class class wbr 2624  –→wf 3184   ‘cfv 3188  (class class class)co 3969  ℝcr 5245  1c1 5247   ≤ cle 5307   +∞cpnf 5495   < clt 5498  NrmCVeccnv 8199  Basecba 8201  normcnm 8205   normOp cnmo 8398

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-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

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