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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 6663 | . . . . 5 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈)) | |
3 | 0oval.1 | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
4 | 2, 3 | syl6eqr 2871 | . . . 4 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋) |
5 | 4 | xpeq1d 5577 | . . 3 ⊢ (𝑢 = 𝑈 → ((BaseSet‘𝑢) × {(0vec‘𝑤)}) = (𝑋 × {(0vec‘𝑤)})) |
6 | fveq2 6663 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (0vec‘𝑤) = (0vec‘𝑊)) | |
7 | 0oval.6 | . . . . . 6 ⊢ 𝑍 = (0vec‘𝑊) | |
8 | 6, 7 | syl6eqr 2871 | . . . . 5 ⊢ (𝑤 = 𝑊 → (0vec‘𝑤) = 𝑍) |
9 | 8 | sneqd 4569 | . . . 4 ⊢ (𝑤 = 𝑊 → {(0vec‘𝑤)} = {𝑍}) |
10 | 9 | xpeq2d 5578 | . . 3 ⊢ (𝑤 = 𝑊 → (𝑋 × {(0vec‘𝑤)}) = (𝑋 × {𝑍})) |
11 | df-0o 28451 | . . 3 ⊢ 0op = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ ((BaseSet‘𝑢) × {(0vec‘𝑤)})) | |
12 | 3 | fvexi 6677 | . . . 4 ⊢ 𝑋 ∈ V |
13 | snex 5322 | . . . 4 ⊢ {𝑍} ∈ V | |
14 | 12, 13 | xpex 7465 | . . 3 ⊢ (𝑋 × {𝑍}) ∈ V |
15 | 5, 10, 11, 14 | ovmpo 7299 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 0op 𝑊) = (𝑋 × {𝑍})) |
16 | 1, 15 | syl5eq 2865 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑂 = (𝑋 × {𝑍})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {csn 4557 × cxp 5546 ‘cfv 6348 (class class class)co 7145 NrmCVeccnv 28288 BaseSetcba 28290 0veccn0v 28292 0op c0o 28447 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-0o 28451 |
This theorem is referenced by: 0oval 28492 0oo 28493 lnon0 28502 blocni 28509 hh0oi 29607 |
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