Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6657 | . . . 4 ⊢ (𝑢 = 𝑈 → (𝑢 LnOp 𝑤) = (𝑈 LnOp 𝑤)) | |
3 | oveq1 6657 | . . . . . 6 ⊢ (𝑢 = 𝑈 → (𝑢 normOpOLD 𝑤) = (𝑈 normOpOLD 𝑤)) | |
4 | 3 | fveq1d 6193 | . . . . 5 ⊢ (𝑢 = 𝑈 → ((𝑢 normOpOLD 𝑤)‘𝑡) = ((𝑈 normOpOLD 𝑤)‘𝑡)) |
5 | 4 | breq1d 4663 | . . . 4 ⊢ (𝑢 = 𝑈 → (((𝑢 normOpOLD 𝑤)‘𝑡) < +∞ ↔ ((𝑈 normOpOLD 𝑤)‘𝑡) < +∞)) |
6 | 2, 5 | rabeqbidv 3195 | . . 3 ⊢ (𝑢 = 𝑈 → {𝑡 ∈ (𝑢 LnOp 𝑤) ∣ ((𝑢 normOpOLD 𝑤)‘𝑡) < +∞} = {𝑡 ∈ (𝑈 LnOp 𝑤) ∣ ((𝑈 normOpOLD 𝑤)‘𝑡) < +∞}) |
7 | oveq2 6658 | . . . . 5 ⊢ (𝑤 = 𝑊 → (𝑈 LnOp 𝑤) = (𝑈 LnOp 𝑊)) | |
8 | bloval.4 | . . . . 5 ⊢ 𝐿 = (𝑈 LnOp 𝑊) | |
9 | 7, 8 | syl6eqr 2674 | . . . 4 ⊢ (𝑤 = 𝑊 → (𝑈 LnOp 𝑤) = 𝐿) |
10 | oveq2 6658 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑈 normOpOLD 𝑤) = (𝑈 normOpOLD 𝑊)) | |
11 | bloval.3 | . . . . . . 7 ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) | |
12 | 10, 11 | syl6eqr 2674 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑈 normOpOLD 𝑤) = 𝑁) |
13 | 12 | fveq1d 6193 | . . . . 5 ⊢ (𝑤 = 𝑊 → ((𝑈 normOpOLD 𝑤)‘𝑡) = (𝑁‘𝑡)) |
14 | 13 | breq1d 4663 | . . . 4 ⊢ (𝑤 = 𝑊 → (((𝑈 normOpOLD 𝑤)‘𝑡) < +∞ ↔ (𝑁‘𝑡) < +∞)) |
15 | 9, 14 | rabeqbidv 3195 | . . 3 ⊢ (𝑤 = 𝑊 → {𝑡 ∈ (𝑈 LnOp 𝑤) ∣ ((𝑈 normOpOLD 𝑤)‘𝑡) < +∞} = {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞}) |
16 | df-blo 27601 | . . 3 ⊢ BLnOp = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {𝑡 ∈ (𝑢 LnOp 𝑤) ∣ ((𝑢 normOpOLD 𝑤)‘𝑡) < +∞}) | |
17 | ovex 6678 | . . . . 5 ⊢ (𝑈 LnOp 𝑊) ∈ V | |
18 | 8, 17 | eqeltri 2697 | . . . 4 ⊢ 𝐿 ∈ V |
19 | 18 | rabex 4813 | . . 3 ⊢ {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞} ∈ V |
20 | 6, 15, 16, 19 | ovmpt2 6796 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 BLnOp 𝑊) = {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞}) |
21 | 1, 20 | syl5eq 2668 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐵 = {𝑡 ∈ 𝐿 ∣ (𝑁‘𝑡) < +∞}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {crab 2916 Vcvv 3200 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 +∞cpnf 10071 < clt 10074 NrmCVeccnv 27439 LnOp clno 27595 normOpOLD cnmoo 27596 BLnOp cblo 27597 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-blo 27601 |
This theorem is referenced by: isblo 27637 hhbloi 28761 |
Copyright terms: Public domain | W3C validator |