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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 6412 | . . . . 5 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈)) | |
3 | 0oval.1 | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
4 | 2, 3 | syl6eqr 2852 | . . . 4 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋) |
5 | 4 | xpeq1d 5342 | . . 3 ⊢ (𝑢 = 𝑈 → ((BaseSet‘𝑢) × {(0vec‘𝑤)}) = (𝑋 × {(0vec‘𝑤)})) |
6 | fveq2 6412 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (0vec‘𝑤) = (0vec‘𝑊)) | |
7 | 0oval.6 | . . . . . 6 ⊢ 𝑍 = (0vec‘𝑊) | |
8 | 6, 7 | syl6eqr 2852 | . . . . 5 ⊢ (𝑤 = 𝑊 → (0vec‘𝑤) = 𝑍) |
9 | 8 | sneqd 4381 | . . . 4 ⊢ (𝑤 = 𝑊 → {(0vec‘𝑤)} = {𝑍}) |
10 | 9 | xpeq2d 5343 | . . 3 ⊢ (𝑤 = 𝑊 → (𝑋 × {(0vec‘𝑤)}) = (𝑋 × {𝑍})) |
11 | df-0o 28126 | . . 3 ⊢ 0op = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ ((BaseSet‘𝑢) × {(0vec‘𝑤)})) | |
12 | 3 | fvexi 6426 | . . . 4 ⊢ 𝑋 ∈ V |
13 | snex 5100 | . . . 4 ⊢ {𝑍} ∈ V | |
14 | 12, 13 | xpex 7197 | . . 3 ⊢ (𝑋 × {𝑍}) ∈ V |
15 | 5, 10, 11, 14 | ovmpt2 7031 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 0op 𝑊) = (𝑋 × {𝑍})) |
16 | 1, 15 | syl5eq 2846 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑂 = (𝑋 × {𝑍})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 {csn 4369 × cxp 5311 ‘cfv 6102 (class class class)co 6879 NrmCVeccnv 27963 BaseSetcba 27965 0veccn0v 27967 0op c0o 28122 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3388 df-sbc 3635 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-nul 4117 df-if 4279 df-pw 4352 df-sn 4370 df-pr 4372 df-op 4376 df-uni 4630 df-br 4845 df-opab 4907 df-id 5221 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-iota 6065 df-fun 6104 df-fv 6110 df-ov 6882 df-oprab 6883 df-mpt2 6884 df-0o 28126 |
This theorem is referenced by: 0oval 28167 0oo 28168 lnon0 28177 blocni 28184 hh0oi 29286 |
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