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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lindsind | Unicode version |
Description: A linearly independent set is independent: no nonzero element multiple can be expressed as a linear combination of the others. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
Ref | Expression |
---|---|
lindfind.s |
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lindfind.n |
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lindfind.l |
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lindfind.z |
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lindfind.k |
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Ref | Expression |
---|---|
lindsind |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplr 732 |
. 2
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2 | eldifsn 3891 |
. . . 4
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3 | 2 | biimpri 198 |
. . 3
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4 | 3 | adantl 453 |
. 2
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5 | elfvdm 5720 |
. . . . . 6
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6 | eqid 2408 |
. . . . . . 7
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7 | lindfind.s |
. . . . . . 7
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8 | lindfind.n |
. . . . . . 7
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9 | lindfind.l |
. . . . . . 7
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10 | lindfind.k |
. . . . . . 7
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11 | lindfind.z |
. . . . . . 7
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12 | 6, 7, 8, 9, 10, 11 | islinds2 27155 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
13 | 5, 12 | syl 16 |
. . . . 5
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14 | 13 | ibi 233 |
. . . 4
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15 | 14 | simprd 450 |
. . 3
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16 | 15 | ad2antrr 707 |
. 2
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17 | oveq2 6052 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
18 | sneq 3789 |
. . . . . . 7
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19 | 18 | difeq2d 3429 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | 19 | fveq2d 5695 |
. . . . 5
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21 | 17, 20 | eleq12d 2476 |
. . . 4
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22 | 21 | notbid 286 |
. . 3
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23 | oveq1 6051 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | 23 | eleq1d 2474 |
. . . 4
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25 | 24 | notbid 286 |
. . 3
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26 | 22, 25 | rspc2va 3023 |
. 2
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27 | 1, 4, 16, 26 | syl21anc 1183 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1662 ax-8 1683 ax-13 1723 ax-14 1725 ax-6 1740 ax-7 1745 ax-11 1757 ax-12 1946 ax-ext 2389 ax-sep 4294 ax-nul 4302 ax-pow 4341 ax-pr 4367 ax-un 4664 |
This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-eu 2262 df-mo 2263 df-clab 2395 df-cleq 2401 df-clel 2404 df-nfc 2533 df-ne 2573 df-ral 2675 df-rex 2676 df-rab 2679 df-v 2922 df-sbc 3126 df-dif 3287 df-un 3289 df-in 3291 df-ss 3298 df-nul 3593 df-if 3704 df-pw 3765 df-sn 3784 df-pr 3785 df-op 3787 df-uni 3980 df-br 4177 df-opab 4231 df-mpt 4232 df-id 4462 df-xp 4847 df-rel 4848 df-cnv 4849 df-co 4850 df-dm 4851 df-rn 4852 df-res 4853 df-ima 4854 df-iota 5381 df-fun 5419 df-fn 5420 df-f 5421 df-f1 5422 df-fo 5423 df-f1o 5424 df-fv 5425 df-ov 6047 df-lindf 27148 df-linds 27149 |
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