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Theorem nmounbi 29039

Description: Two ways two express that an operator is unbounded. (Contributed by NM, 11-Jan-2008.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
nmoubi.1	⊢ 𝑋 = (BaseSet‘𝑈)
nmoubi.y	⊢ 𝑌 = (BaseSet‘𝑊)
nmoubi.l	⊢ 𝐿 = (norm_CV‘𝑈)
nmoubi.m	⊢ 𝑀 = (norm_CV‘𝑊)
nmoubi.3	⊢ 𝑁 = (𝑈 normOp_OLD 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 𝑦,𝑈 𝑦,𝑊 𝑌,𝑟,𝑦 𝑀,𝑟,𝑦 𝑇,𝑟,𝑦 𝑋,𝑟,𝑦 𝑁,𝑟,𝑦 Allowed substitution hints: 𝑈(𝑟) 𝑊(𝑟)

**Proof of Theorem nmounbi**
Step	Hyp	Ref	Expression
1		nmoubi.1	. . . 4 ⊢ 𝑋 = (BaseSet‘𝑈)
2		nmoubi.y	. . . 4 ⊢ 𝑌 = (BaseSet‘𝑊)
3		nmoubi.l	. . . 4 ⊢ 𝐿 = (norm_CV‘𝑈)
4		nmoubi.m	. . . 4 ⊢ 𝑀 = (norm_CV‘𝑊)
5		nmoubi.3	. . . 4 ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊)
6		nmoubi.u	. . . 4 ⊢ 𝑈 ∈ NrmCVec
7		nmoubi.w	. . . 4 ⊢ 𝑊 ∈ NrmCVec
8	1, 2, 3, 4, 5, 6, 7	nmobndi 29038	. . 3 ⊢ (𝑇:𝑋⟶𝑌 → ((𝑁‘𝑇) ∈ ℝ ↔ ∃𝑟 ∈ ℝ ∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑟)))
9	1, 2, 5	nmorepnf 29031	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → ((𝑁‘𝑇) ∈ ℝ ↔ (𝑁‘𝑇) ≠ +∞))
10	6, 7, 9	mp3an12 1449	. . 3 ⊢ (𝑇:𝑋⟶𝑌 → ((𝑁‘𝑇) ∈ ℝ ↔ (𝑁‘𝑇) ≠ +∞))
11		ffvelrn 6941	. . . . . . . . . . . 12 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝑦 ∈ 𝑋) → (𝑇‘𝑦) ∈ 𝑌)
12	2, 4	nvcl 28924	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑇‘𝑦) ∈ 𝑌) → (𝑀‘(𝑇‘𝑦)) ∈ ℝ)
13	7, 11, 12	sylancr 586	. . . . . . . . . . 11 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝑦 ∈ 𝑋) → (𝑀‘(𝑇‘𝑦)) ∈ ℝ)
14		lenlt 10984	. . . . . . . . . . 11 ⊢ (((𝑀‘(𝑇‘𝑦)) ∈ ℝ ∧ 𝑟 ∈ ℝ) → ((𝑀‘(𝑇‘𝑦)) ≤ 𝑟 ↔ ¬ 𝑟 < (𝑀‘(𝑇‘𝑦))))
15	13, 14	sylan 579	. . . . . . . . . 10 ⊢ (((𝑇:𝑋⟶𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ) → ((𝑀‘(𝑇‘𝑦)) ≤ 𝑟 ↔ ¬ 𝑟 < (𝑀‘(𝑇‘𝑦))))
16	15	an32s 648	. . . . . . . . 9 ⊢ (((𝑇:𝑋⟶𝑌 ∧ 𝑟 ∈ ℝ) ∧ 𝑦 ∈ 𝑋) → ((𝑀‘(𝑇‘𝑦)) ≤ 𝑟 ↔ ¬ 𝑟 < (𝑀‘(𝑇‘𝑦))))
17	16	imbi2d 340	. . . . . . . 8 ⊢ (((𝑇:𝑋⟶𝑌 ∧ 𝑟 ∈ ℝ) ∧ 𝑦 ∈ 𝑋) → (((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑟) ↔ ((𝐿‘𝑦) ≤ 1 → ¬ 𝑟 < (𝑀‘(𝑇‘𝑦)))))
18		imnan 399	. . . . . . . 8 ⊢ (((𝐿‘𝑦) ≤ 1 → ¬ 𝑟 < (𝑀‘(𝑇‘𝑦))) ↔ ¬ ((𝐿‘𝑦) ≤ 1 ∧ 𝑟 < (𝑀‘(𝑇‘𝑦))))
19	17, 18	bitrdi 286	. . . . . . 7 ⊢ (((𝑇:𝑋⟶𝑌 ∧ 𝑟 ∈ ℝ) ∧ 𝑦 ∈ 𝑋) → (((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑟) ↔ ¬ ((𝐿‘𝑦) ≤ 1 ∧ 𝑟 < (𝑀‘(𝑇‘𝑦)))))
20	19	ralbidva 3119	. . . . . 6 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝑟 ∈ ℝ) → (∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑟) ↔ ∀𝑦 ∈ 𝑋 ¬ ((𝐿‘𝑦) ≤ 1 ∧ 𝑟 < (𝑀‘(𝑇‘𝑦)))))
21		ralnex 3163	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑋 ¬ ((𝐿‘𝑦) ≤ 1 ∧ 𝑟 < (𝑀‘(𝑇‘𝑦))) ↔ ¬ ∃𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 ∧ 𝑟 < (𝑀‘(𝑇‘𝑦))))
22	20, 21	bitrdi 286	. . . . 5 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝑟 ∈ ℝ) → (∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑟) ↔ ¬ ∃𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 ∧ 𝑟 < (𝑀‘(𝑇‘𝑦)))))
23	22	rexbidva 3224	. . . 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)))))
24		rexnal 3165	. . . 4 ⊢ (∃𝑟 ∈ ℝ ¬ ∃𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 ∧ 𝑟 < (𝑀‘(𝑇‘𝑦))) ↔ ¬ ∀𝑟 ∈ ℝ ∃𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 ∧ 𝑟 < (𝑀‘(𝑇‘𝑦))))
25	23, 24	bitrdi 286	. . 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)))))
26	8, 10, 25	3bitr3d 308	. 2 ⊢ (𝑇:𝑋⟶𝑌 → ((𝑁‘𝑇) ≠ +∞ ↔ ¬ ∀𝑟 ∈ ℝ ∃𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 ∧ 𝑟 < (𝑀‘(𝑇‘𝑦)))))
27	26	necon4abid 2983	1 ⊢ (𝑇:𝑋⟶𝑌 → ((𝑁‘𝑇) = +∞ ↔ ∀𝑟 ∈ ℝ ∃𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 ∧ 𝑟 < (𝑀‘(𝑇‘𝑦)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 ∃wrex 3064 class class class wbr 5070 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ℝcr 10801 1c1 10803 +∞cpnf 10937 < clt 10940 ≤ cle 10941 NrmCVeccnv 28847 BaseSetcba 28849 norm_CVcnmcv 28853 normOp_OLD cnmoo 29004

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-rep 5205 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-pre-sup 10880

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-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-sup 9131 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-3 11967 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-abs 14875 df-grpo 28756 df-gid 28757 df-ginv 28758 df-ablo 28808 df-vc 28822 df-nv 28855 df-va 28858 df-ba 28859 df-sm 28860 df-0v 28861 df-nmcv 28863 df-nmoo 29008

This theorem is referenced by: nmounbseqi 29040 nmounbseqiALT 29041

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