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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 6103 | . . . . 5 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈)) | |
3 | 0oval.1 | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
4 | 2, 3 | syl6eqr 2662 | . . . 4 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋) |
5 | 4 | xpeq1d 5062 | . . 3 ⊢ (𝑢 = 𝑈 → ((BaseSet‘𝑢) × {(0vec‘𝑤)}) = (𝑋 × {(0vec‘𝑤)})) |
6 | fveq2 6103 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (0vec‘𝑤) = (0vec‘𝑊)) | |
7 | 0oval.6 | . . . . . 6 ⊢ 𝑍 = (0vec‘𝑊) | |
8 | 6, 7 | syl6eqr 2662 | . . . . 5 ⊢ (𝑤 = 𝑊 → (0vec‘𝑤) = 𝑍) |
9 | 8 | sneqd 4137 | . . . 4 ⊢ (𝑤 = 𝑊 → {(0vec‘𝑤)} = {𝑍}) |
10 | 9 | xpeq2d 5063 | . . 3 ⊢ (𝑤 = 𝑊 → (𝑋 × {(0vec‘𝑤)}) = (𝑋 × {𝑍})) |
11 | df-0o 26986 | . . 3 ⊢ 0op = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ ((BaseSet‘𝑢) × {(0vec‘𝑤)})) | |
12 | fvex 6113 | . . . . 5 ⊢ (BaseSet‘𝑈) ∈ V | |
13 | 3, 12 | eqeltri 2684 | . . . 4 ⊢ 𝑋 ∈ V |
14 | snex 4835 | . . . 4 ⊢ {𝑍} ∈ V | |
15 | 13, 14 | xpex 6860 | . . 3 ⊢ (𝑋 × {𝑍}) ∈ V |
16 | 5, 10, 11, 15 | ovmpt2 6694 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 0op 𝑊) = (𝑋 × {𝑍})) |
17 | 1, 16 | syl5eq 2656 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑂 = (𝑋 × {𝑍})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 {csn 4125 × cxp 5036 ‘cfv 5804 (class class class)co 6549 NrmCVeccnv 26823 BaseSetcba 26825 0veccn0v 26827 0op c0o 26982 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-0o 26986 |
This theorem is referenced by: 0oval 27027 0oo 27028 lnon0 27037 blocni 27044 hh0oi 28146 |
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