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Mirrors > Home > MPE Home > Th. List > lspval | Unicode version |
Description: The span of a set of vectors (in a left module). (spanval 22792 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lspval.v |
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lspval.s |
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lspval.n |
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Ref | Expression |
---|---|
lspval |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspval.v |
. . . . 5
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2 | lspval.s |
. . . . 5
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3 | lspval.n |
. . . . 5
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4 | 1, 2, 3 | lspfval 16008 |
. . . 4
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5 | 4 | fveq1d 5693 |
. . 3
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6 | 5 | adantr 452 |
. 2
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7 | simpr 448 |
. . . 4
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8 | fvex 5705 |
. . . . . 6
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9 | 1, 8 | eqeltri 2478 |
. . . . 5
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10 | 9 | elpw2 4328 |
. . . 4
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11 | 7, 10 | sylibr 204 |
. . 3
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12 | 1, 2 | lss1 15974 |
. . . . 5
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13 | sseq2 3334 |
. . . . . 6
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14 | 13 | rspcev 3016 |
. . . . 5
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15 | 12, 14 | sylan 458 |
. . . 4
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16 | intexrab 4323 |
. . . 4
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17 | 15, 16 | sylib 189 |
. . 3
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18 | sseq1 3333 |
. . . . . 6
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19 | 18 | rabbidv 2912 |
. . . . 5
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20 | 19 | inteqd 4019 |
. . . 4
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21 | eqid 2408 |
. . . 4
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22 | 20, 21 | fvmptg 5767 |
. . 3
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23 | 11, 17, 22 | syl2anc 643 |
. 2
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24 | 6, 23 | eqtrd 2440 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem is referenced by: lspid 16017 lspss 16019 lspssid 16020 dochspss 31865 |
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-rep 4284 ax-sep 4294 ax-nul 4302 ax-pow 4341 ax-pr 4367 |
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-reu 2677 df-rmo 2678 df-rab 2679 df-v 2922 df-sbc 3126 df-csb 3216 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-int 4015 df-iun 4059 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-riota 6512 df-0g 13686 df-mnd 14649 df-grp 14771 df-lmod 15911 df-lss 15968 df-lsp 16007 |
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