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Mirrors > Home > ILE Home > Th. List > ghmsub | Unicode version |
Description: Linearity of subtraction through a group homomorphism. (Contributed by Stefan O'Rear, 31-Dec-2014.) |
Ref | Expression |
---|---|
ghmsub.b |
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ghmsub.m |
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ghmsub.n |
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Ref | Expression |
---|---|
ghmsub |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ghmgrp1 13209 |
. . . . . 6
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2 | 1 | 3ad2ant1 1020 |
. . . . 5
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3 | simp3 1001 |
. . . . 5
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4 | ghmsub.b |
. . . . . 6
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5 | eqid 2189 |
. . . . . 6
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6 | 4, 5 | grpinvcl 13015 |
. . . . 5
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7 | 2, 3, 6 | syl2anc 411 |
. . . 4
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8 | eqid 2189 |
. . . . 5
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9 | eqid 2189 |
. . . . 5
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10 | 4, 8, 9 | ghmlin 13212 |
. . . 4
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11 | 7, 10 | syld3an3 1294 |
. . 3
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12 | eqid 2189 |
. . . . . 6
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13 | 4, 5, 12 | ghminv 13214 |
. . . . 5
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14 | 13 | 3adant2 1018 |
. . . 4
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15 | 14 | oveq2d 5916 |
. . 3
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16 | 11, 15 | eqtrd 2222 |
. 2
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17 | ghmsub.m |
. . . . 5
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18 | 4, 8, 5, 17 | grpsubval 13013 |
. . . 4
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19 | 18 | fveq2d 5541 |
. . 3
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20 | 19 | 3adant1 1017 |
. 2
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21 | eqid 2189 |
. . . . . 6
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22 | 4, 21 | ghmf 13211 |
. . . . 5
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23 | ffvelcdm 5673 |
. . . . . 6
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24 | ffvelcdm 5673 |
. . . . . 6
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25 | 23, 24 | anim12dan 600 |
. . . . 5
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26 | 22, 25 | sylan 283 |
. . . 4
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27 | 26 | 3impb 1201 |
. . 3
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28 | ghmsub.n |
. . . 4
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29 | 21, 9, 12, 28 | grpsubval 13013 |
. . 3
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30 | 27, 29 | syl 14 |
. 2
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31 | 16, 20, 30 | 3eqtr4d 2232 |
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 4136 ax-sep 4139 ax-pow 4195 ax-pr 4230 ax-un 4454 ax-setind 4557 ax-cnex 7937 ax-resscn 7938 ax-1re 7940 ax-addrcl 7943 |
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-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-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-int 3863 df-iun 3906 df-br 4022 df-opab 4083 df-mpt 4084 df-id 4314 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-rn 4658 df-res 4659 df-ima 4660 df-iota 5199 df-fun 5240 df-fn 5241 df-f 5242 df-f1 5243 df-fo 5244 df-f1o 5245 df-fv 5246 df-riota 5855 df-ov 5903 df-oprab 5904 df-mpo 5905 df-1st 6169 df-2nd 6170 df-inn 8955 df-2 9013 df-ndx 12526 df-slot 12527 df-base 12529 df-plusg 12613 df-0g 12774 df-mgm 12843 df-sgrp 12888 df-mnd 12901 df-grp 12971 df-minusg 12972 df-sbg 12973 df-ghm 13205 |
This theorem is referenced by: ghmnsgima 13232 ghmnsgpreima 13233 ghmeqker 13235 ghmf1 13237 |
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