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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 2189 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | eqid 2189 |
. . . . . 6
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5 | eqid 2189 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | eqid 2189 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 1, 2, 3, 4, 5, 6 | lmodvneg1 13607 |
. . . . 5
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8 | 7 | sneqd 3620 |
. . . 4
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9 | 8 | fveq2d 5534 |
. . 3
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10 | simpl 109 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | 3 | lmodfgrp 13573 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | eqid 2189 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
13 | 3, 12, 5 | lmod1cl 13592 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | 12, 6 | grpinvcl 12958 |
. . . . . 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 13695 |
. . . 4
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20 | 10, 16, 17, 19 | syl3anc 1249 |
. . 3
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21 | 9, 20 | eqsstrrd 3207 |
. 2
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22 | 1, 2 | lmodvnegcl 13605 |
. . . . . . 7
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23 | 1, 2, 3, 4, 5, 6 | lmodvneg1 13607 |
. . . . . . 7
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24 | 22, 23 | syldan 282 |
. . . . . 6
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25 | lmodgrp 13571 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
26 | 1, 2 | grpinvinv 12977 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 25, 26 | sylan 283 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 24, 27 | eqtrd 2222 |
. . . . 5
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29 | 28 | sneqd 3620 |
. . . 4
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30 | 29 | fveq2d 5534 |
. . 3
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31 | 3, 12, 1, 4, 18 | lspsnvsi 13695 |
. . . 4
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32 | 10, 16, 22, 31 | syl3anc 1249 |
. . 3
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33 | 30, 32 | eqsstrrd 3207 |
. 2
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34 | 21, 33 | eqssd 3187 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7920 ax-resscn 7921 ax-1cn 7922 ax-1re 7923 ax-icn 7924 ax-addcl 7925 ax-addrcl 7926 ax-mulcl 7927 ax-addcom 7929 ax-addass 7931 ax-i2m1 7934 ax-0lt1 7935 ax-0id 7937 ax-rnegex 7938 ax-pre-ltirr 7941 ax-pre-ltadd 7945 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-pnf 8012 df-mnf 8013 df-ltxr 8015 df-inn 8938 df-2 8996 df-3 8997 df-4 8998 df-5 8999 df-6 9000 df-ndx 12483 df-slot 12484 df-base 12486 df-sets 12487 df-plusg 12568 df-mulr 12569 df-sca 12571 df-vsca 12572 df-0g 12729 df-mgm 12798 df-sgrp 12831 df-mnd 12844 df-grp 12914 df-minusg 12915 df-sbg 12916 df-mgp 13236 df-ur 13275 df-ring 13313 df-lmod 13566 df-lssm 13630 df-lsp 13664 |
This theorem is referenced by: lspsnsub 13698 lmodindp1 13705 |
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