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Mirrors > Home > ILE Home > Th. List > lmodsubvs | Unicode version |
Description: Subtraction of a scalar product in terms of addition. (Contributed by NM, 9-Apr-2015.) |
Ref | Expression |
---|---|
lmodsubvs.v |
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lmodsubvs.p |
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lmodsubvs.m |
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lmodsubvs.t |
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lmodsubvs.f |
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lmodsubvs.k |
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lmodsubvs.n |
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lmodsubvs.w |
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lmodsubvs.a |
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lmodsubvs.x |
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lmodsubvs.y |
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Ref | Expression |
---|---|
lmodsubvs |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodsubvs.w |
. . 3
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2 | lmodsubvs.x |
. . 3
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3 | lmodsubvs.a |
. . . 4
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4 | lmodsubvs.y |
. . . 4
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5 | lmodsubvs.v |
. . . . 5
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6 | lmodsubvs.f |
. . . . 5
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7 | lmodsubvs.t |
. . . . 5
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8 | lmodsubvs.k |
. . . . 5
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9 | 5, 6, 7, 8 | lmodvscl 13489 |
. . . 4
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10 | 1, 3, 4, 9 | syl3anc 1248 |
. . 3
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11 | lmodsubvs.p |
. . . 4
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12 | lmodsubvs.m |
. . . 4
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13 | lmodsubvs.n |
. . . 4
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14 | eqid 2187 |
. . . 4
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15 | 5, 11, 12, 6, 7, 13, 14 | lmodvsubval2 13526 |
. . 3
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16 | 1, 2, 10, 15 | syl3anc 1248 |
. 2
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17 | 6 | lmodring 13479 |
. . . . . . . 8
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18 | 1, 17 | syl 14 |
. . . . . . 7
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19 | ringgrp 13248 |
. . . . . . 7
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20 | 18, 19 | syl 14 |
. . . . . 6
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21 | 8, 14 | ringidcl 13267 |
. . . . . . 7
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22 | 18, 21 | syl 14 |
. . . . . 6
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23 | 8, 13 | grpinvcl 12944 |
. . . . . 6
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24 | 20, 22, 23 | syl2anc 411 |
. . . . 5
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25 | eqid 2187 |
. . . . . 6
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26 | 5, 6, 7, 8, 25 | lmodvsass 13497 |
. . . . 5
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27 | 1, 24, 3, 4, 26 | syl13anc 1250 |
. . . 4
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28 | 8, 25, 14, 13, 18, 3 | ringnegl 13296 |
. . . . 5
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29 | 28 | oveq1d 5903 |
. . . 4
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30 | 27, 29 | eqtr3d 2222 |
. . 3
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31 | 30 | oveq2d 5904 |
. 2
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32 | 16, 31 | eqtrd 2220 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-addcom 7924 ax-addass 7926 ax-i2m1 7929 ax-0lt1 7930 ax-0id 7932 ax-rnegex 7933 ax-pre-ltirr 7936 ax-pre-ltadd 7940 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-pnf 8007 df-mnf 8008 df-ltxr 8010 df-inn 8933 df-2 8991 df-3 8992 df-4 8993 df-5 8994 df-6 8995 df-ndx 12478 df-slot 12479 df-base 12481 df-sets 12482 df-plusg 12563 df-mulr 12564 df-sca 12566 df-vsca 12567 df-0g 12724 df-mgm 12793 df-sgrp 12826 df-mnd 12839 df-grp 12901 df-minusg 12902 df-sbg 12903 df-mgp 13171 df-ur 13207 df-ring 13245 df-lmod 13473 |
This theorem is referenced by: (None) |
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