Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > nmosetre | Structured version Visualization version GIF version |
Description: The set in the supremum of the operator norm definition df-nmoo 28516 is a set of reals. (Contributed by NM, 13-Nov-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmosetre.2 | ⊢ 𝑌 = (BaseSet‘𝑊) |
nmosetre.4 | ⊢ 𝑁 = (normCV‘𝑊) |
Ref | Expression |
---|---|
nmosetre | ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → {𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝑀‘𝑧) ≤ 1 ∧ 𝑥 = (𝑁‘(𝑇‘𝑧)))} ⊆ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffvelrn 6843 | . . . . . . . 8 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝑧 ∈ 𝑋) → (𝑇‘𝑧) ∈ 𝑌) | |
2 | nmosetre.2 | . . . . . . . . 9 ⊢ 𝑌 = (BaseSet‘𝑊) | |
3 | nmosetre.4 | . . . . . . . . 9 ⊢ 𝑁 = (normCV‘𝑊) | |
4 | 2, 3 | nvcl 28432 | . . . . . . . 8 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑇‘𝑧) ∈ 𝑌) → (𝑁‘(𝑇‘𝑧)) ∈ ℝ) |
5 | 1, 4 | sylan2 594 | . . . . . . 7 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑇:𝑋⟶𝑌 ∧ 𝑧 ∈ 𝑋)) → (𝑁‘(𝑇‘𝑧)) ∈ ℝ) |
6 | 5 | anassrs 470 | . . . . . 6 ⊢ (((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) ∧ 𝑧 ∈ 𝑋) → (𝑁‘(𝑇‘𝑧)) ∈ ℝ) |
7 | eleq1 2900 | . . . . . 6 ⊢ (𝑥 = (𝑁‘(𝑇‘𝑧)) → (𝑥 ∈ ℝ ↔ (𝑁‘(𝑇‘𝑧)) ∈ ℝ)) | |
8 | 6, 7 | syl5ibr 248 | . . . . 5 ⊢ (𝑥 = (𝑁‘(𝑇‘𝑧)) → (((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) ∧ 𝑧 ∈ 𝑋) → 𝑥 ∈ ℝ)) |
9 | 8 | impcom 410 | . . . 4 ⊢ ((((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) ∧ 𝑧 ∈ 𝑋) ∧ 𝑥 = (𝑁‘(𝑇‘𝑧))) → 𝑥 ∈ ℝ) |
10 | 9 | adantrl 714 | . . 3 ⊢ ((((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) ∧ 𝑧 ∈ 𝑋) ∧ ((𝑀‘𝑧) ≤ 1 ∧ 𝑥 = (𝑁‘(𝑇‘𝑧)))) → 𝑥 ∈ ℝ) |
11 | 10 | rexlimdva2 3287 | . 2 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (∃𝑧 ∈ 𝑋 ((𝑀‘𝑧) ≤ 1 ∧ 𝑥 = (𝑁‘(𝑇‘𝑧))) → 𝑥 ∈ ℝ)) |
12 | 11 | abssdv 4044 | 1 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → {𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝑀‘𝑧) ≤ 1 ∧ 𝑥 = (𝑁‘(𝑇‘𝑧)))} ⊆ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {cab 2799 ∃wrex 3139 ⊆ wss 3935 class class class wbr 5058 ⟶wf 6345 ‘cfv 6349 ℝcr 10530 1c1 10532 ≤ cle 10670 NrmCVeccnv 28355 BaseSetcba 28357 normCVcnmcv 28361 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-1st 7683 df-2nd 7684 df-vc 28330 df-nv 28363 df-va 28366 df-ba 28367 df-sm 28368 df-0v 28369 df-nmcv 28371 |
This theorem is referenced by: nmoxr 28537 nmooge0 28538 nmorepnf 28539 nmoolb 28542 nmoubi 28543 nmlno0lem 28564 nmopsetretHIL 29635 |
Copyright terms: Public domain | W3C validator |