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Mirrors > Home > ILE Home > Th. List > absmulgcd | Unicode version |
Description: Distribute absolute value of multiplication over gcd. Theorem 1.4(c) in [ApostolNT] p. 16. (Contributed by Paul Chapman, 22-Jun-2011.) |
Ref | Expression |
---|---|
absmulgcd |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gcdcl 11385 |
. . . . 5
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2 | nn0re 8780 |
. . . . . 6
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3 | nn0ge0 8796 |
. . . . . 6
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4 | 2, 3 | absidd 10715 |
. . . . 5
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5 | 1, 4 | syl 14 |
. . . 4
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6 | 5 | oveq2d 5706 |
. . 3
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7 | 6 | 3adant1 964 |
. 2
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8 | zcn 8853 |
. . . 4
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9 | 1 | nn0cnd 8826 |
. . . 4
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10 | absmul 10617 |
. . . 4
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11 | 8, 9, 10 | syl2an 284 |
. . 3
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12 | 11 | 3impb 1142 |
. 2
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13 | zcn 8853 |
. . . . 5
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14 | zcn 8853 |
. . . . 5
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15 | absmul 10617 |
. . . . . . 7
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16 | absmul 10617 |
. . . . . . 7
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17 | 15, 16 | oveqan12d 5709 |
. . . . . 6
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18 | 17 | 3impdi 1236 |
. . . . 5
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19 | 8, 13, 14, 18 | syl3an 1223 |
. . . 4
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20 | zmulcl 8901 |
. . . . . 6
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21 | zmulcl 8901 |
. . . . . 6
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22 | gcdabs 11406 |
. . . . . 6
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23 | 20, 21, 22 | syl2an 284 |
. . . . 5
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24 | 23 | 3impdi 1236 |
. . . 4
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25 | nn0abscl 10633 |
. . . . 5
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26 | zabscl 10634 |
. . . . 5
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27 | zabscl 10634 |
. . . . 5
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28 | mulgcd 11432 |
. . . . 5
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29 | 25, 26, 27, 28 | syl3an 1223 |
. . . 4
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30 | 19, 24, 29 | 3eqtr3d 2135 |
. . 3
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31 | gcdabs 11406 |
. . . . 5
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32 | 31 | 3adant1 964 |
. . . 4
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33 | 32 | oveq2d 5706 |
. . 3
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34 | 30, 33 | eqtrd 2127 |
. 2
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35 | 7, 12, 34 | 3eqtr4rd 2138 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-coll 3975 ax-sep 3978 ax-nul 3986 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-iinf 4431 ax-cnex 7533 ax-resscn 7534 ax-1cn 7535 ax-1re 7536 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-mulrcl 7541 ax-addcom 7542 ax-mulcom 7543 ax-addass 7544 ax-mulass 7545 ax-distr 7546 ax-i2m1 7547 ax-0lt1 7548 ax-1rid 7549 ax-0id 7550 ax-rnegex 7551 ax-precex 7552 ax-cnre 7553 ax-pre-ltirr 7554 ax-pre-ltwlin 7555 ax-pre-lttrn 7556 ax-pre-apti 7557 ax-pre-ltadd 7558 ax-pre-mulgt0 7559 ax-pre-mulext 7560 ax-arch 7561 ax-caucvg 7562 |
This theorem depends on definitions: df-bi 116 df-dc 784 df-3or 928 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-nel 2358 df-ral 2375 df-rex 2376 df-reu 2377 df-rmo 2378 df-rab 2379 df-v 2635 df-sbc 2855 df-csb 2948 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-if 3414 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-iun 3754 df-br 3868 df-opab 3922 df-mpt 3923 df-tr 3959 df-id 4144 df-po 4147 df-iso 4148 df-iord 4217 df-on 4219 df-ilim 4220 df-suc 4222 df-iom 4434 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-f1 5054 df-fo 5055 df-f1o 5056 df-fv 5057 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-1st 5949 df-2nd 5950 df-recs 6108 df-frec 6194 df-sup 6759 df-pnf 7621 df-mnf 7622 df-xr 7623 df-ltxr 7624 df-le 7625 df-sub 7752 df-neg 7753 df-reap 8149 df-ap 8156 df-div 8237 df-inn 8521 df-2 8579 df-3 8580 df-4 8581 df-n0 8772 df-z 8849 df-uz 9119 df-q 9204 df-rp 9234 df-fz 9574 df-fzo 9703 df-fl 9826 df-mod 9879 df-seqfrec 10001 df-exp 10070 df-cj 10391 df-re 10392 df-im 10393 df-rsqrt 10546 df-abs 10547 df-dvds 11224 df-gcd 11366 |
This theorem is referenced by: (None) |
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