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Mirrors > Home > MPE Home > Th. List > bloval | Structured version Visualization version GIF version |
Description: The class of bounded linear operators between two normed complex vector spaces. (Contributed by NM, 6-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bloval.3 | ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) |
bloval.4 | ⊢ 𝐿 = (𝑈 LnOp 𝑊) |
bloval.5 | ⊢ 𝐵 = (𝑈 BLnOp 𝑊) |
Ref | Expression |
---|---|
bloval | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐵 = {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bloval.5 | . 2 ⊢ 𝐵 = (𝑈 BLnOp 𝑊) | |
2 | oveq1 7225 | . . . 4 ⊢ (𝑢 = 𝑈 → (𝑢 LnOp 𝑤) = (𝑈 LnOp 𝑤)) | |
3 | oveq1 7225 | . . . . . 6 ⊢ (𝑢 = 𝑈 → (𝑢 normOpOLD 𝑤) = (𝑈 normOpOLD 𝑤)) | |
4 | 3 | fveq1d 6724 | . . . . 5 ⊢ (𝑢 = 𝑈 → ((𝑢 normOpOLD 𝑤)‘𝑡) = ((𝑈 normOpOLD 𝑤)‘𝑡)) |
5 | 4 | breq1d 5068 | . . . 4 ⊢ (𝑢 = 𝑈 → (((𝑢 normOpOLD 𝑤)‘𝑡) < +∞ ↔ ((𝑈 normOpOLD 𝑤)‘𝑡) < +∞)) |
6 | 2, 5 | rabeqbidv 3401 | . . 3 ⊢ (𝑢 = 𝑈 → {𝑡 ∈ (𝑢 LnOp 𝑤) ∣ ((𝑢 normOpOLD 𝑤)‘𝑡) < +∞} = {𝑡 ∈ (𝑈 LnOp 𝑤) ∣ ((𝑈 normOpOLD 𝑤)‘𝑡) < +∞}) |
7 | oveq2 7226 | . . . . 5 ⊢ (𝑤 = 𝑊 → (𝑈 LnOp 𝑤) = (𝑈 LnOp 𝑊)) | |
8 | bloval.4 | . . . . 5 ⊢ 𝐿 = (𝑈 LnOp 𝑊) | |
9 | 7, 8 | eqtr4di 2796 | . . . 4 ⊢ (𝑤 = 𝑊 → (𝑈 LnOp 𝑤) = 𝐿) |
10 | oveq2 7226 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑈 normOpOLD 𝑤) = (𝑈 normOpOLD 𝑊)) | |
11 | bloval.3 | . . . . . . 7 ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) | |
12 | 10, 11 | eqtr4di 2796 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑈 normOpOLD 𝑤) = 𝑁) |
13 | 12 | fveq1d 6724 | . . . . 5 ⊢ (𝑤 = 𝑊 → ((𝑈 normOpOLD 𝑤)‘𝑡) = (𝑁‘𝑡)) |
14 | 13 | breq1d 5068 | . . . 4 ⊢ (𝑤 = 𝑊 → (((𝑈 normOpOLD 𝑤)‘𝑡) < +∞ ↔ (𝑁‘𝑡) < +∞)) |
15 | 9, 14 | rabeqbidv 3401 | . . 3 ⊢ (𝑤 = 𝑊 → {𝑡 ∈ (𝑈 LnOp 𝑤) ∣ ((𝑈 normOpOLD 𝑤)‘𝑡) < +∞} = {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞}) |
16 | df-blo 28832 | . . 3 ⊢ BLnOp = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {𝑡 ∈ (𝑢 LnOp 𝑤) ∣ ((𝑢 normOpOLD 𝑤)‘𝑡) < +∞}) | |
17 | 8 | ovexi 7252 | . . . 4 ⊢ 𝐿 ∈ V |
18 | 17 | rabex 5230 | . . 3 ⊢ {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞} ∈ V |
19 | 6, 15, 16, 18 | ovmpo 7374 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 BLnOp 𝑊) = {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞}) |
20 | 1, 19 | syl5eq 2790 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐵 = {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 {crab 3065 class class class wbr 5058 ‘cfv 6385 (class class class)co 7218 +∞cpnf 10869 < clt 10872 NrmCVeccnv 28670 LnOp clno 28826 normOpOLD cnmoo 28827 BLnOp cblo 28828 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5197 ax-nul 5204 ax-pr 5327 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3415 df-sbc 3700 df-dif 3874 df-un 3876 df-in 3878 df-ss 3888 df-nul 4243 df-if 4445 df-sn 4547 df-pr 4549 df-op 4553 df-uni 4825 df-br 5059 df-opab 5121 df-id 5460 df-xp 5562 df-rel 5563 df-cnv 5564 df-co 5565 df-dm 5566 df-iota 6343 df-fun 6387 df-fv 6393 df-ov 7221 df-oprab 7222 df-mpo 7223 df-blo 28832 |
This theorem is referenced by: isblo 28868 hhbloi 29988 |
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