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Mirrors > Home > ILE Home > Th. List > lspsnneg | Unicode version |
Description: Negation does not change the span of a singleton. (Contributed by NM, 24-Apr-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lspsnneg.v |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
lspsnneg.m |
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lspsnneg.n |
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Ref | Expression |
---|---|
lspsnneg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspsnneg.v |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | lspsnneg.m |
. . . . . 6
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3 | eqid 2187 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | eqid 2187 |
. . . . . 6
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5 | eqid 2187 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | eqid 2187 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 1, 2, 3, 4, 5, 6 | lmodvneg1 13483 |
. . . . 5
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8 | 7 | sneqd 3617 |
. . . 4
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9 | 8 | fveq2d 5531 |
. . 3
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10 | simpl 109 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | 3 | lmodfgrp 13449 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | eqid 2187 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
13 | 3, 12, 5 | lmod1cl 13468 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | 12, 6 | grpinvcl 12942 |
. . . . . 6
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15 | 11, 13, 14 | syl2anc 411 |
. . . . 5
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16 | 15 | adantr 276 |
. . . 4
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17 | simpr 110 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
18 | lspsnneg.n |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | 3, 12, 1, 4, 18 | lspsnvsi 13571 |
. . . 4
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20 | 10, 16, 17, 19 | syl3anc 1248 |
. . 3
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21 | 9, 20 | eqsstrrd 3204 |
. 2
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22 | 1, 2 | lmodvnegcl 13481 |
. . . . . . 7
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23 | 1, 2, 3, 4, 5, 6 | lmodvneg1 13483 |
. . . . . . 7
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24 | 22, 23 | syldan 282 |
. . . . . 6
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25 | lmodgrp 13447 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
26 | 1, 2 | grpinvinv 12961 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 25, 26 | sylan 283 |
. . . . . 6
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28 | 24, 27 | eqtrd 2220 |
. . . . 5
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29 | 28 | sneqd 3617 |
. . . 4
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30 | 29 | fveq2d 5531 |
. . 3
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31 | 3, 12, 1, 4, 18 | lspsnvsi 13571 |
. . . 4
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32 | 10, 16, 22, 31 | syl3anc 1248 |
. . 3
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33 | 30, 32 | eqsstrrd 3204 |
. 2
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34 | 21, 33 | eqssd 3184 |
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 12837 df-grp 12899 df-minusg 12900 df-sbg 12901 df-mgp 13163 df-ur 13197 df-ring 13235 df-lmod 13442 df-lssm 13506 df-lsp 13540 |
This theorem is referenced by: lspsnsub 13574 lmodindp1 13581 |
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