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Mirrors > Home > ILE Home > Th. List > dvrfvald | Unicode version |
Description: Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Proof shortened by AV, 2-Mar-2024.) |
Ref | Expression |
---|---|
dvrfvald.b |
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dvrfvald.t |
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dvrfvald.u |
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dvrfvald.i |
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dvrfvald.d |
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dvrfvald.r |
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Ref | Expression |
---|---|
dvrfvald |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dvr 13339 |
. . 3
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2 | fveq2 5517 |
. . . 4
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3 | fveq2 5517 |
. . . 4
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4 | fveq2 5517 |
. . . . 5
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5 | eqidd 2178 |
. . . . 5
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6 | fveq2 5517 |
. . . . . 6
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7 | 6 | fveq1d 5519 |
. . . . 5
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8 | 4, 5, 7 | oveq123d 5899 |
. . . 4
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9 | 2, 3, 8 | mpoeq123dv 5940 |
. . 3
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10 | dvrfvald.r |
. . . 4
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11 | 10 | elexd 2752 |
. . 3
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12 | basfn 12523 |
. . . . 5
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13 | funfvex 5534 |
. . . . . 6
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14 | 13 | funfni 5318 |
. . . . 5
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15 | 12, 11, 14 | sylancr 414 |
. . . 4
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16 | eqidd 2178 |
. . . . . 6
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17 | eqidd 2178 |
. . . . . 6
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18 | 16, 17, 10 | unitssd 13316 |
. . . . 5
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19 | 15, 18 | ssexd 4145 |
. . . 4
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20 | mpoexga 6216 |
. . . 4
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21 | 15, 19, 20 | syl2anc 411 |
. . 3
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22 | 1, 9, 11, 21 | fvmptd3 5612 |
. 2
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23 | dvrfvald.d |
. 2
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24 | dvrfvald.b |
. . 3
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25 | dvrfvald.u |
. . 3
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26 | dvrfvald.t |
. . . 4
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27 | eqidd 2178 |
. . . 4
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28 | dvrfvald.i |
. . . . 5
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29 | 28 | fveq1d 5519 |
. . . 4
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30 | 26, 27, 29 | oveq123d 5899 |
. . 3
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31 | 24, 25, 30 | mpoeq123dv 5940 |
. 2
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32 | 22, 23, 31 | 3eqtr4d 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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7905 ax-resscn 7906 ax-1cn 7907 ax-1re 7908 ax-icn 7909 ax-addcl 7910 ax-addrcl 7911 ax-mulcl 7912 ax-addcom 7914 ax-addass 7916 ax-i2m1 7919 ax-0lt1 7920 ax-0id 7922 ax-rnegex 7923 ax-pre-ltirr 7926 ax-pre-ltadd 7930 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-riota 5834 df-ov 5881 df-oprab 5882 df-mpo 5883 df-1st 6144 df-2nd 6145 df-pnf 7997 df-mnf 7998 df-ltxr 8000 df-inn 8923 df-2 8981 df-3 8982 df-ndx 12468 df-slot 12469 df-base 12471 df-sets 12472 df-plusg 12552 df-mulr 12553 df-0g 12713 df-mgm 12782 df-sgrp 12815 df-mnd 12826 df-mgp 13147 df-srg 13185 df-dvdsr 13296 df-unit 13297 df-dvr 13339 |
This theorem is referenced by: dvrvald 13341 |
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