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| 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 30764 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 | ffvelcdm 7101 | . . . . . . . 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 30680 | . . . . . . . 8 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑇‘𝑧) ∈ 𝑌) → (𝑁‘(𝑇‘𝑧)) ∈ ℝ) |
| 5 | 1, 4 | sylan2 593 | . . . . . . 7 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑇:𝑋⟶𝑌 ∧ 𝑧 ∈ 𝑋)) → (𝑁‘(𝑇‘𝑧)) ∈ ℝ) |
| 6 | 5 | anassrs 467 | . . . . . 6 ⊢ (((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) ∧ 𝑧 ∈ 𝑋) → (𝑁‘(𝑇‘𝑧)) ∈ ℝ) |
| 7 | eleq1 2829 | . . . . . 6 ⊢ (𝑥 = (𝑁‘(𝑇‘𝑧)) → (𝑥 ∈ ℝ ↔ (𝑁‘(𝑇‘𝑧)) ∈ ℝ)) | |
| 8 | 6, 7 | imbitrrid 246 | . . . . 5 ⊢ (𝑥 = (𝑁‘(𝑇‘𝑧)) → (((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) ∧ 𝑧 ∈ 𝑋) → 𝑥 ∈ ℝ)) |
| 9 | 8 | impcom 407 | . . . 4 ⊢ ((((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) ∧ 𝑧 ∈ 𝑋) ∧ 𝑥 = (𝑁‘(𝑇‘𝑧))) → 𝑥 ∈ ℝ) |
| 10 | 9 | adantrl 716 | . . 3 ⊢ ((((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) ∧ 𝑧 ∈ 𝑋) ∧ ((𝑀‘𝑧) ≤ 1 ∧ 𝑥 = (𝑁‘(𝑇‘𝑧)))) → 𝑥 ∈ ℝ) |
| 11 | 10 | rexlimdva2 3157 | . 2 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (∃𝑧 ∈ 𝑋 ((𝑀‘𝑧) ≤ 1 ∧ 𝑥 = (𝑁‘(𝑇‘𝑧))) → 𝑥 ∈ ℝ)) |
| 12 | 11 | abssdv 4068 | 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 395 = wceq 1540 ∈ wcel 2108 {cab 2714 ∃wrex 3070 ⊆ wss 3951 class class class wbr 5143 ⟶wf 6557 ‘cfv 6561 ℝcr 11154 1c1 11156 ≤ cle 11296 NrmCVeccnv 30603 BaseSetcba 30605 normCVcnmcv 30609 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-1st 8014 df-2nd 8015 df-vc 30578 df-nv 30611 df-va 30614 df-ba 30615 df-sm 30616 df-0v 30617 df-nmcv 30619 |
| This theorem is referenced by: nmoxr 30785 nmooge0 30786 nmorepnf 30787 nmoolb 30790 nmoubi 30791 nmlno0lem 30812 nmopsetretHIL 31883 |
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