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Mirrors > Home > MPE Home > Th. List > 0oval | Structured version Visualization version GIF version |
Description: Value of the zero operator. (Contributed by NM, 28-Nov-2007.) (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 |
---|---|
0oval | β’ ((π β NrmCVec β§ π β NrmCVec β§ π΄ β π) β (πβπ΄) = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0oval.1 | . . . . 5 β’ π = (BaseSetβπ) | |
2 | 0oval.6 | . . . . 5 β’ π = (0vecβπ) | |
3 | 0oval.0 | . . . . 5 β’ π = (π 0op π) | |
4 | 1, 2, 3 | 0ofval 29778 | . . . 4 β’ ((π β NrmCVec β§ π β NrmCVec) β π = (π Γ {π})) |
5 | 4 | fveq1d 6848 | . . 3 β’ ((π β NrmCVec β§ π β NrmCVec) β (πβπ΄) = ((π Γ {π})βπ΄)) |
6 | 5 | 3adant3 1133 | . 2 β’ ((π β NrmCVec β§ π β NrmCVec β§ π΄ β π) β (πβπ΄) = ((π Γ {π})βπ΄)) |
7 | 2 | fvexi 6860 | . . . 4 β’ π β V |
8 | 7 | fvconst2 7157 | . . 3 β’ (π΄ β π β ((π Γ {π})βπ΄) = π) |
9 | 8 | 3ad2ant3 1136 | . 2 β’ ((π β NrmCVec β§ π β NrmCVec β§ π΄ β π) β ((π Γ {π})βπ΄) = π) |
10 | 6, 9 | eqtrd 2773 | 1 β’ ((π β NrmCVec β§ π β NrmCVec β§ π΄ β π) β (πβπ΄) = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = 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-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-0o 29738 |
This theorem is referenced by: 0lno 29781 nmoo0 29782 nmlno0lem 29784 |
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