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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 6906 | . . . . 5 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈)) | |
| 3 | 0oval.1 | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
| 4 | 2, 3 | eqtr4di 2795 | . . . 4 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋) |
| 5 | 4 | xpeq1d 5714 | . . 3 ⊢ (𝑢 = 𝑈 → ((BaseSet‘𝑢) × {(0vec‘𝑤)}) = (𝑋 × {(0vec‘𝑤)})) |
| 6 | fveq2 6906 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (0vec‘𝑤) = (0vec‘𝑊)) | |
| 7 | 0oval.6 | . . . . . 6 ⊢ 𝑍 = (0vec‘𝑊) | |
| 8 | 6, 7 | eqtr4di 2795 | . . . . 5 ⊢ (𝑤 = 𝑊 → (0vec‘𝑤) = 𝑍) |
| 9 | 8 | sneqd 4638 | . . . 4 ⊢ (𝑤 = 𝑊 → {(0vec‘𝑤)} = {𝑍}) |
| 10 | 9 | xpeq2d 5715 | . . 3 ⊢ (𝑤 = 𝑊 → (𝑋 × {(0vec‘𝑤)}) = (𝑋 × {𝑍})) |
| 11 | df-0o 30766 | . . 3 ⊢ 0op = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ ((BaseSet‘𝑢) × {(0vec‘𝑤)})) | |
| 12 | 3 | fvexi 6920 | . . . 4 ⊢ 𝑋 ∈ V |
| 13 | snex 5436 | . . . 4 ⊢ {𝑍} ∈ V | |
| 14 | 12, 13 | xpex 7773 | . . 3 ⊢ (𝑋 × {𝑍}) ∈ V |
| 15 | 5, 10, 11, 14 | ovmpo 7593 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 0op 𝑊) = (𝑋 × {𝑍})) |
| 16 | 1, 15 | eqtrid 2789 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑂 = (𝑋 × {𝑍})) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {csn 4626 × cxp 5683 ‘cfv 6561 (class class class)co 7431 NrmCVeccnv 30603 BaseSetcba 30605 0veccn0v 30607 0op c0o 30762 |
| 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-sep 5296 ax-nul 5306 ax-pow 5365 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-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-0o 30766 |
| This theorem is referenced by: 0oval 30807 0oo 30808 lnon0 30817 blocni 30824 hh0oi 31922 |
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