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Theorem nmounbi 29718

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 29717	. . 3 ⊢ (𝑇:𝑋⟶𝑌 → ((𝑁‘𝑇) ∈ ℝ ↔ ∃𝑟 ∈ ℝ ∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑟)))
9	1, 2, 5	nmorepnf 29710	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → ((𝑁‘𝑇) ∈ ℝ ↔ (𝑁‘𝑇) ≠ +∞))
10	6, 7, 9	mp3an12 1451	. . 3 ⊢ (𝑇:𝑋⟶𝑌 → ((𝑁‘𝑇) ∈ ℝ ↔ (𝑁‘𝑇) ≠ +∞))
11		ffvelcdm 7032	. . . . . . . . . . . 12 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝑦 ∈ 𝑋) → (𝑇‘𝑦) ∈ 𝑌)
12	2, 4	nvcl 29603	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑇‘𝑦) ∈ 𝑌) → (𝑀‘(𝑇‘𝑦)) ∈ ℝ)
13	7, 11, 12	sylancr 587	. . . . . . . . . . 11 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝑦 ∈ 𝑋) → (𝑀‘(𝑇‘𝑦)) ∈ ℝ)
14		lenlt 11233	. . . . . . . . . . 11 ⊢ (((𝑀‘(𝑇‘𝑦)) ∈ ℝ ∧ 𝑟 ∈ ℝ) → ((𝑀‘(𝑇‘𝑦)) ≤ 𝑟 ↔ ¬ 𝑟 < (𝑀‘(𝑇‘𝑦))))
15	13, 14	sylan 580	. . . . . . . . . 10 ⊢ (((𝑇:𝑋⟶𝑌 ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ) → ((𝑀‘(𝑇‘𝑦)) ≤ 𝑟 ↔ ¬ 𝑟 < (𝑀‘(𝑇‘𝑦))))
16	15	an32s 650	. . . . . . . . 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 400	. . . . . . . 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 3172	. . . . . 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 3075	. . . . . 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 3173	. . . 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 3103	. . . 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 2984	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 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3073 class class class wbr 5105 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ℝcr 11050 1c1 11052 +∞cpnf 11186 < clt 11189 ≤ cle 11190 NrmCVeccnv 29526 BaseSetcba 29528 norm_CVcnmcv 29532 normOp_OLD cnmoo 29683

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-sup 9378 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-n0 12414 df-z 12500 df-uz 12764 df-rp 12916 df-seq 13907 df-exp 13968 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-grpo 29435 df-gid 29436 df-ginv 29437 df-ablo 29487 df-vc 29501 df-nv 29534 df-va 29537 df-ba 29538 df-sm 29539 df-0v 29540 df-nmcv 29542 df-nmoo 29687

This theorem is referenced by: nmounbseqi 29719 nmounbseqiALT 29720

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