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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 | β’ π = (BaseSetβπ) |
0oval.6 | β’ π = (0vecβπ) |
0oval.0 | β’ π = (π 0op π) |
Ref | Expression |
---|---|
0ofval | β’ ((π β NrmCVec β§ π β NrmCVec) β π = (π Γ {π})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0oval.0 | . 2 β’ π = (π 0op π) | |
2 | fveq2 6846 | . . . . 5 β’ (π’ = π β (BaseSetβπ’) = (BaseSetβπ)) | |
3 | 0oval.1 | . . . . 5 β’ π = (BaseSetβπ) | |
4 | 2, 3 | eqtr4di 2791 | . . . 4 β’ (π’ = π β (BaseSetβπ’) = π) |
5 | 4 | xpeq1d 5666 | . . 3 β’ (π’ = π β ((BaseSetβπ’) Γ {(0vecβπ€)}) = (π Γ {(0vecβπ€)})) |
6 | fveq2 6846 | . . . . . 6 β’ (π€ = π β (0vecβπ€) = (0vecβπ)) | |
7 | 0oval.6 | . . . . . 6 β’ π = (0vecβπ) | |
8 | 6, 7 | eqtr4di 2791 | . . . . 5 β’ (π€ = π β (0vecβπ€) = π) |
9 | 8 | sneqd 4602 | . . . 4 β’ (π€ = π β {(0vecβπ€)} = {π}) |
10 | 9 | xpeq2d 5667 | . . 3 β’ (π€ = π β (π Γ {(0vecβπ€)}) = (π Γ {π})) |
11 | df-0o 29738 | . . 3 β’ 0op = (π’ β NrmCVec, π€ β NrmCVec β¦ ((BaseSetβπ’) Γ {(0vecβπ€)})) | |
12 | 3 | fvexi 6860 | . . . 4 β’ π β V |
13 | snex 5392 | . . . 4 β’ {π} β V | |
14 | 12, 13 | xpex 7691 | . . 3 β’ (π Γ {π}) β V |
15 | 5, 10, 11, 14 | ovmpo 7519 | . 2 β’ ((π β NrmCVec β§ π β NrmCVec) β (π 0op π) = (π Γ {π})) |
16 | 1, 15 | eqtrid 2785 | 1 β’ ((π β NrmCVec β§ π β NrmCVec) β π = (π Γ {π})) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 {csn 4590 Γ cxp 5635 βcfv 6500 (class class class)co 7361 NrmCVeccnv 29575 BaseSetcba 29577 0veccn0v 29579 0op c0o 29734 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-iota 6452 df-fun 6502 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-0o 29738 |
This theorem is referenced by: 0oval 29779 0oo 29780 lnon0 29789 blocni 29796 hh0oi 30894 |
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