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| Mirrors > Home > MPE Home > Th. List > nvo00 | Structured version Visualization version GIF version | ||
| Description: Two ways to express a zero operator. (Contributed by NM, 27-Nov-2007.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| nvo00.1 | β’ π = (BaseSetβπ) |
| Ref | Expression |
|---|---|
| nvo00 | β’ ((π β NrmCVec β§ π:πβΆπ) β (π = (π Γ {π}) β ran π = {π})) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ffn 6718 | . 2 β’ (π:πβΆπ β π Fn π) | |
| 2 | nvo00.1 | . . . 4 β’ π = (BaseSetβπ) | |
| 3 | eqid 2733 | . . . 4 β’ (0vecβπ) = (0vecβπ) | |
| 4 | 2, 3 | nvzcl 29887 | . . 3 β’ (π β NrmCVec β (0vecβπ) β π) |
| 5 | 4 | ne0d 4336 | . 2 β’ (π β NrmCVec β π β β ) |
| 6 | fconst5 7207 | . 2 β’ ((π Fn π β§ π β β ) β (π = (π Γ {π}) β ran π = {π})) | |
| 7 | 1, 5, 6 | syl2anr 598 | 1 β’ ((π β NrmCVec β§ π:πβΆπ) β (π = (π Γ {π}) β ran π = {π})) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 β wne 2941 β c0 4323 {csn 4629 Γ cxp 5675 ran crn 5678 Fn wfn 6539 βΆwf 6540 βcfv 6544 NrmCVeccnv 29837 BaseSetcba 29839 0veccn0v 29841 |
| 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-rep 5286 ax-sep 5300 ax-nul 5307 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-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 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-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-1st 7975 df-2nd 7976 df-grpo 29746 df-gid 29747 df-ablo 29798 df-vc 29812 df-nv 29845 df-va 29848 df-ba 29849 df-sm 29850 df-0v 29851 df-nmcv 29853 |
| This theorem is referenced by: (None) |
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