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Mirrors > Home > MPE Home > Th. List > nmounbi | Structured version Visualization version GIF version |
Description: Two ways two express that an operator is unbounded. (Contributed by NM, 11-Jan-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmoubi.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
nmoubi.y | ⊢ 𝑌 = (BaseSet‘𝑊) |
nmoubi.l | ⊢ 𝐿 = (normCV‘𝑈) |
nmoubi.m | ⊢ 𝑀 = (normCV‘𝑊) |
nmoubi.3 | ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) |
nmoubi.u | ⊢ 𝑈 ∈ NrmCVec |
nmoubi.w | ⊢ 𝑊 ∈ NrmCVec |
Ref | Expression |
---|---|
nmounbi | ⊢ (𝑇:𝑋⟶𝑌 → ((𝑁‘𝑇) = +∞ ↔ ∀𝑟 ∈ ℝ ∃𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 ∧ 𝑟 < (𝑀‘(𝑇‘𝑦))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmoubi.1 | . . . 4 ⊢ 𝑋 = (BaseSet‘𝑈) | |
2 | nmoubi.y | . . . 4 ⊢ 𝑌 = (BaseSet‘𝑊) | |
3 | nmoubi.l | . . . 4 ⊢ 𝐿 = (normCV‘𝑈) | |
4 | nmoubi.m | . . . 4 ⊢ 𝑀 = (normCV‘𝑊) | |
5 | nmoubi.3 | . . . 4 ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) | |
6 | nmoubi.u | . . . 4 ⊢ 𝑈 ∈ NrmCVec | |
7 | nmoubi.w | . . . 4 ⊢ 𝑊 ∈ NrmCVec | |
8 | 1, 2, 3, 4, 5, 6, 7 | nmobndi 30452 | . . 3 ⊢ (𝑇:𝑋⟶𝑌 → ((𝑁‘𝑇) ∈ ℝ ↔ ∃𝑟 ∈ ℝ ∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑟))) |
9 | 1, 2, 5 | nmorepnf 30445 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → ((𝑁‘𝑇) ∈ ℝ ↔ (𝑁‘𝑇) ≠ +∞)) |
10 | 6, 7, 9 | mp3an12 1447 | . . 3 ⊢ (𝑇:𝑋⟶𝑌 → ((𝑁‘𝑇) ∈ ℝ ↔ (𝑁‘𝑇) ≠ +∞)) |
11 | ffvelcdm 7073 | . . . . . . . . . . . 12 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝑦 ∈ 𝑋) → (𝑇‘𝑦) ∈ 𝑌) | |
12 | 2, 4 | nvcl 30338 | . . . . . . . . . . . 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 11288 | . . . . . . . . . . 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 649 | . . . . . . . . 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 287 | . . . . . . 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 3167 | . . . . . 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 3064 | . . . . . 6 ⊢ (∀𝑦 ∈ 𝑋 ¬ ((𝐿‘𝑦) ≤ 1 ∧ 𝑟 < (𝑀‘(𝑇‘𝑦))) ↔ ¬ ∃𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 ∧ 𝑟 < (𝑀‘(𝑇‘𝑦)))) | |
22 | 20, 21 | bitrdi 287 | . . . . 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 3168 | . . . 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 3092 | . . . 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 287 | . . 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 309 | . 2 ⊢ (𝑇:𝑋⟶𝑌 → ((𝑁‘𝑇) ≠ +∞ ↔ ¬ ∀𝑟 ∈ ℝ ∃𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 ∧ 𝑟 < (𝑀‘(𝑇‘𝑦))))) |
27 | 26 | necon4abid 2973 | 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 1533 ∈ wcel 2098 ≠ wne 2932 ∀wral 3053 ∃wrex 3062 class class class wbr 5138 ⟶wf 6529 ‘cfv 6533 (class class class)co 7401 ℝcr 11104 1c1 11106 +∞cpnf 11241 < clt 11244 ≤ cle 11245 NrmCVeccnv 30261 BaseSetcba 30263 normCVcnmcv 30267 normOpOLD cnmoo 30418 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8698 df-map 8817 df-en 8935 df-dom 8936 df-sdom 8937 df-sup 9432 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-grpo 30170 df-gid 30171 df-ginv 30172 df-ablo 30222 df-vc 30236 df-nv 30269 df-va 30272 df-ba 30273 df-sm 30274 df-0v 30275 df-nmcv 30277 df-nmoo 30422 |
This theorem is referenced by: nmounbseqi 30454 nmounbseqiALT 30455 |
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