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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 30040 | . . . 4 β’ ((π β NrmCVec β§ π β NrmCVec) β π = (π Γ {π})) |
| 5 | 4 | fveq1d 6894 | . . 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 6906 | . . . 4 β’ π β V |
| 8 | 7 | fvconst2 7205 | . . 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 4629 Γ cxp 5675 βcfv 6544 (class class class)co 7409 NrmCVeccnv 29837 BaseSetcba 29839 0veccn0v 29841 0op c0o 29996 |
| 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 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
| 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 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-0o 30000 |
| This theorem is referenced by: 0lno 30043 nmoo0 30044 nmlno0lem 30046 |
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