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Mirrors > Home > ILE Home > Th. List > modqmuladd | Unicode version |
Description: Decomposition of an integer into a multiple of a modulus and a remainder. (Contributed by Jim Kingdon, 23-Oct-2021.) |
Ref | Expression |
---|---|
modqmuladd.a |
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modqmuladd.bq |
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modqmuladd.b |
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modqmuladd.m |
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modqmuladd.mgt0 |
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Ref | Expression |
---|---|
modqmuladd |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | modqmuladd.a |
. . . . . . 7
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2 | zq 9109 |
. . . . . . 7
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3 | 1, 2 | syl 14 |
. . . . . 6
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4 | modqmuladd.m |
. . . . . 6
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5 | modqmuladd.mgt0 |
. . . . . . 7
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6 | 5 | gt0ne0d 7988 |
. . . . . 6
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7 | qdivcl 9126 |
. . . . . 6
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8 | 3, 4, 6, 7 | syl3anc 1174 |
. . . . 5
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9 | 8 | flqcld 9680 |
. . . 4
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10 | oveq1 5659 |
. . . . . . 7
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11 | 10 | oveq1d 5667 |
. . . . . 6
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12 | 11 | eqeq2d 2099 |
. . . . 5
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13 | 12 | adantl 271 |
. . . 4
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14 | flqpmodeq 9730 |
. . . . . 6
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15 | 3, 4, 5, 14 | syl3anc 1174 |
. . . . 5
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16 | 15 | eqcomd 2093 |
. . . 4
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17 | 9, 13, 16 | rspcedvd 2728 |
. . 3
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18 | oveq2 5660 |
. . . . . 6
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19 | 18 | eqeq2d 2099 |
. . . . 5
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20 | 19 | eqcoms 2091 |
. . . 4
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21 | 20 | rexbidv 2381 |
. . 3
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22 | 17, 21 | syl5ibrcom 155 |
. 2
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23 | oveq1 5659 |
. . . . . 6
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24 | 23 | adantl 271 |
. . . . 5
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25 | simplr 497 |
. . . . . 6
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26 | 4 | ad2antrr 472 |
. . . . . 6
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27 | modqmuladd.bq |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
28 | 27 | ad2antrr 472 |
. . . . . 6
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29 | modqmuladd.b |
. . . . . . 7
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30 | 29 | ad2antrr 472 |
. . . . . 6
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31 | mulqaddmodid 9767 |
. . . . . 6
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32 | 25, 26, 28, 30, 31 | syl22anc 1175 |
. . . . 5
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33 | 24, 32 | eqtrd 2120 |
. . . 4
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34 | 33 | ex 113 |
. . 3
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35 | 34 | rexlimdva 2489 |
. 2
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36 | 22, 35 | impbid 127 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-setind 4353 ax-cnex 7434 ax-resscn 7435 ax-1cn 7436 ax-1re 7437 ax-icn 7438 ax-addcl 7439 ax-addrcl 7440 ax-mulcl 7441 ax-mulrcl 7442 ax-addcom 7443 ax-mulcom 7444 ax-addass 7445 ax-mulass 7446 ax-distr 7447 ax-i2m1 7448 ax-0lt1 7449 ax-1rid 7450 ax-0id 7451 ax-rnegex 7452 ax-precex 7453 ax-cnre 7454 ax-pre-ltirr 7455 ax-pre-ltwlin 7456 ax-pre-lttrn 7457 ax-pre-apti 7458 ax-pre-ltadd 7459 ax-pre-mulgt0 7460 ax-pre-mulext 7461 ax-arch 7462 |
This theorem depends on definitions: df-bi 115 df-3or 925 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-nel 2351 df-ral 2364 df-rex 2365 df-reu 2366 df-rmo 2367 df-rab 2368 df-v 2621 df-sbc 2841 df-csb 2934 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-int 3689 df-iun 3732 df-br 3846 df-opab 3900 df-mpt 3901 df-id 4120 df-po 4123 df-iso 4124 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-res 4450 df-ima 4451 df-iota 4980 df-fun 5017 df-fn 5018 df-f 5019 df-fv 5023 df-riota 5608 df-ov 5655 df-oprab 5656 df-mpt2 5657 df-1st 5911 df-2nd 5912 df-pnf 7522 df-mnf 7523 df-xr 7524 df-ltxr 7525 df-le 7526 df-sub 7653 df-neg 7654 df-reap 8050 df-ap 8057 df-div 8138 df-inn 8421 df-n0 8672 df-z 8749 df-q 9103 df-rp 9133 df-ico 9310 df-fl 9673 df-mod 9726 |
This theorem is referenced by: modqmuladdim 9770 |
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