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Mirrors > Home > ILE Home > Th. List > muldvds2 | Unicode version |
Description: If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
muldvds2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zmulcl 8793 |
. . . 4
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2 | 1 | anim1i 333 |
. . 3
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3 | 2 | 3impa 1138 |
. 2
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4 | 3simpc 942 |
. 2
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5 | zmulcl 8793 |
. . . 4
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6 | 5 | ancoms 264 |
. . 3
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7 | 6 | 3ad2antl1 1105 |
. 2
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8 | zcn 8745 |
. . . . . . . 8
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9 | zcn 8745 |
. . . . . . . 8
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10 | zcn 8745 |
. . . . . . . 8
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11 | mulass 7463 |
. . . . . . . 8
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12 | 8, 9, 10, 11 | syl3an 1216 |
. . . . . . 7
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13 | 12 | 3coml 1150 |
. . . . . 6
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14 | 13 | 3expa 1143 |
. . . . 5
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15 | 14 | 3adantl3 1101 |
. . . 4
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16 | 15 | eqeq1d 2096 |
. . 3
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17 | 16 | biimprd 156 |
. 2
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18 | 3, 4, 7, 17 | dvds1lem 11072 |
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-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3955 ax-pow 4007 ax-pr 4034 ax-setind 4351 ax-cnex 7426 ax-resscn 7427 ax-1cn 7428 ax-1re 7429 ax-icn 7430 ax-addcl 7431 ax-addrcl 7432 ax-mulcl 7433 ax-mulrcl 7434 ax-addcom 7435 ax-mulcom 7436 ax-addass 7437 ax-mulass 7438 ax-distr 7439 ax-i2m1 7440 ax-1rid 7442 ax-0id 7443 ax-rnegex 7444 ax-cnre 7446 |
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-ral 2364 df-rex 2365 df-reu 2366 df-rab 2368 df-v 2621 df-sbc 2841 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-pw 3429 df-sn 3450 df-pr 3451 df-op 3453 df-uni 3652 df-int 3687 df-br 3844 df-opab 3898 df-id 4118 df-xp 4442 df-rel 4443 df-cnv 4444 df-co 4445 df-dm 4446 df-iota 4975 df-fun 5012 df-fv 5018 df-riota 5600 df-ov 5647 df-oprab 5648 df-mpt2 5649 df-sub 7645 df-neg 7646 df-inn 8413 df-n0 8664 df-z 8741 df-dvds 11062 |
This theorem is referenced by: (None) |
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