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Mirrors > Home > MPE Home > Th. List > 0ofval | Structured version Visualization version GIF version |
Description: The zero operator between two normed complex vector spaces. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
0oval.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
0oval.6 | ⊢ 𝑍 = (0vec‘𝑊) |
0oval.0 | ⊢ 𝑂 = (𝑈 0op 𝑊) |
Ref | Expression |
---|---|
0ofval | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑂 = (𝑋 × {𝑍})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0oval.0 | . 2 ⊢ 𝑂 = (𝑈 0op 𝑊) | |
2 | fveq2 6191 | . . . . 5 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈)) | |
3 | 0oval.1 | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
4 | 2, 3 | syl6eqr 2674 | . . . 4 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋) |
5 | 4 | xpeq1d 5138 | . . 3 ⊢ (𝑢 = 𝑈 → ((BaseSet‘𝑢) × {(0vec‘𝑤)}) = (𝑋 × {(0vec‘𝑤)})) |
6 | fveq2 6191 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (0vec‘𝑤) = (0vec‘𝑊)) | |
7 | 0oval.6 | . . . . . 6 ⊢ 𝑍 = (0vec‘𝑊) | |
8 | 6, 7 | syl6eqr 2674 | . . . . 5 ⊢ (𝑤 = 𝑊 → (0vec‘𝑤) = 𝑍) |
9 | 8 | sneqd 4189 | . . . 4 ⊢ (𝑤 = 𝑊 → {(0vec‘𝑤)} = {𝑍}) |
10 | 9 | xpeq2d 5139 | . . 3 ⊢ (𝑤 = 𝑊 → (𝑋 × {(0vec‘𝑤)}) = (𝑋 × {𝑍})) |
11 | df-0o 27602 | . . 3 ⊢ 0op = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ ((BaseSet‘𝑢) × {(0vec‘𝑤)})) | |
12 | fvex 6201 | . . . . 5 ⊢ (BaseSet‘𝑈) ∈ V | |
13 | 3, 12 | eqeltri 2697 | . . . 4 ⊢ 𝑋 ∈ V |
14 | snex 4908 | . . . 4 ⊢ {𝑍} ∈ V | |
15 | 13, 14 | xpex 6962 | . . 3 ⊢ (𝑋 × {𝑍}) ∈ V |
16 | 5, 10, 11, 15 | ovmpt2 6796 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 0op 𝑊) = (𝑋 × {𝑍})) |
17 | 1, 16 | syl5eq 2668 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑂 = (𝑋 × {𝑍})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 {csn 4177 × cxp 5112 ‘cfv 5888 (class class class)co 6650 NrmCVeccnv 27439 BaseSetcba 27441 0veccn0v 27443 0op c0o 27598 |
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-8 1992 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-pow 4843 ax-pr 4906 ax-un 6949 |
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-pw 4160 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-0o 27602 |
This theorem is referenced by: 0oval 27643 0oo 27644 lnon0 27653 blocni 27660 hh0oi 28762 |
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