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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcdsca | Structured version Visualization version Unicode version |
Description: The ring of scalars of the closed kernel dual space. (Contributed by NM, 16-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.) |
Ref | Expression |
---|---|
lcdsca.h |
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lcdsca.u |
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lcdsca.f |
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lcdsca.o |
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lcdsca.c |
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lcdsca.r |
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lcdsca.k |
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Ref | Expression |
---|---|
lcdsca |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcdsca.r |
. 2
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2 | lcdsca.h |
. . . . . 6
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3 | eqid 2461 |
. . . . . 6
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4 | lcdsca.c |
. . . . . 6
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5 | lcdsca.u |
. . . . . 6
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6 | eqid 2461 |
. . . . . 6
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7 | eqid 2461 |
. . . . . 6
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8 | eqid 2461 |
. . . . . 6
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9 | lcdsca.k |
. . . . . 6
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10 | 2, 3, 4, 5, 6, 7, 8, 9 | lcdval 35201 |
. . . . 5
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11 | 10 | fveq2d 5891 |
. . . 4
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12 | fvex 5897 |
. . . . . 6
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13 | 12 | rabex 4567 |
. . . . 5
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14 | eqid 2461 |
. . . . . 6
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15 | eqid 2461 |
. . . . . 6
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16 | 14, 15 | resssca 15323 |
. . . . 5
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17 | 13, 16 | ax-mp 5 |
. . . 4
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18 | 11, 17 | syl6eqr 2513 |
. . 3
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19 | lcdsca.f |
. . . 4
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20 | lcdsca.o |
. . . 4
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21 | 2, 5, 9 | dvhlmod 34722 |
. . . 4
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22 | 19, 20, 8, 15, 21 | ldualsca 32742 |
. . 3
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23 | 18, 22 | eqtrd 2495 |
. 2
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24 | 1, 23 | syl5eq 2507 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1679 ax-4 1692 ax-5 1768 ax-6 1815 ax-7 1861 ax-8 1899 ax-9 1906 ax-10 1925 ax-11 1930 ax-12 1943 ax-13 2101 ax-ext 2441 ax-rep 4528 ax-sep 4538 ax-nul 4547 ax-pow 4594 ax-pr 4652 ax-un 6609 ax-cnex 9620 ax-resscn 9621 ax-1cn 9622 ax-icn 9623 ax-addcl 9624 ax-addrcl 9625 ax-mulcl 9626 ax-mulrcl 9627 ax-mulcom 9628 ax-addass 9629 ax-mulass 9630 ax-distr 9631 ax-i2m1 9632 ax-1ne0 9633 ax-1rid 9634 ax-rnegex 9635 ax-rrecex 9636 ax-cnre 9637 ax-pre-lttri 9638 ax-pre-lttrn 9639 ax-pre-ltadd 9640 ax-pre-mulgt0 9641 ax-riotaBAD 32569 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3or 992 df-3an 993 df-tru 1457 df-fal 1460 df-ex 1674 df-nf 1678 df-sb 1808 df-eu 2313 df-mo 2314 df-clab 2448 df-cleq 2454 df-clel 2457 df-nfc 2591 df-ne 2634 df-nel 2635 df-ral 2753 df-rex 2754 df-reu 2755 df-rmo 2756 df-rab 2757 df-v 3058 df-sbc 3279 df-csb 3375 df-dif 3418 df-un 3420 df-in 3422 df-ss 3429 df-pss 3431 df-nul 3743 df-if 3893 df-pw 3964 df-sn 3980 df-pr 3982 df-tp 3984 df-op 3986 df-uni 4212 df-int 4248 df-iun 4293 df-iin 4294 df-br 4416 df-opab 4475 df-mpt 4476 df-tr 4511 df-eprel 4763 df-id 4767 df-po 4773 df-so 4774 df-fr 4811 df-we 4813 df-xp 4858 df-rel 4859 df-cnv 4860 df-co 4861 df-dm 4862 df-rn 4863 df-res 4864 df-ima 4865 df-pred 5398 df-ord 5444 df-on 5445 df-lim 5446 df-suc 5447 df-iota 5564 df-fun 5602 df-fn 5603 df-f 5604 df-f1 5605 df-fo 5606 df-f1o 5607 df-fv 5608 df-riota 6276 df-ov 6317 df-oprab 6318 df-mpt2 6319 df-of 6557 df-om 6719 df-1st 6819 df-2nd 6820 df-tpos 6998 df-undef 7045 df-wrecs 7053 df-recs 7115 df-rdg 7153 df-1o 7207 df-oadd 7211 df-er 7388 df-map 7499 df-en 7595 df-dom 7596 df-sdom 7597 df-fin 7598 df-pnf 9702 df-mnf 9703 df-xr 9704 df-ltxr 9705 df-le 9706 df-sub 9887 df-neg 9888 df-nn 10637 df-2 10695 df-3 10696 df-4 10697 df-5 10698 df-6 10699 df-n0 10898 df-z 10966 df-uz 11188 df-fz 11813 df-struct 15171 df-ndx 15172 df-slot 15173 df-base 15174 df-sets 15175 df-ress 15176 df-plusg 15251 df-mulr 15252 df-sca 15254 df-vsca 15255 df-0g 15388 df-preset 16221 df-poset 16239 df-plt 16252 df-lub 16268 df-glb 16269 df-join 16270 df-meet 16271 df-p0 16333 df-p1 16334 df-lat 16340 df-clat 16402 df-mgm 16536 df-sgrp 16575 df-mnd 16585 df-grp 16721 df-minusg 16722 df-mgp 17772 df-ur 17784 df-ring 17830 df-oppr 17899 df-dvdsr 17917 df-unit 17918 df-invr 17948 df-dvr 17959 df-drng 18025 df-lmod 18141 df-lvec 18374 df-ldual 32734 df-oposet 32786 df-ol 32788 df-oml 32789 df-covers 32876 df-ats 32877 df-atl 32908 df-cvlat 32932 df-hlat 32961 df-llines 33107 df-lplanes 33108 df-lvols 33109 df-lines 33110 df-psubsp 33112 df-pmap 33113 df-padd 33405 df-lhyp 33597 df-laut 33598 df-ldil 33713 df-ltrn 33714 df-trl 33769 df-tendo 34366 df-edring 34368 df-dvech 34691 df-lcdual 35199 |
This theorem is referenced by: lcdsbase 35212 lcdsadd 35213 lcdsmul 35214 lcd0 35220 lcd1 35221 lcdneg 35222 lcdvsub 35229 |
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