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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 9335 |
. . . 4
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2 | 1 | anim1i 340 |
. . 3
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3 | 2 | 3impa 1196 |
. 2
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4 | 3simpc 998 |
. 2
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5 | zmulcl 9335 |
. . . 4
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6 | 5 | ancoms 268 |
. . 3
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7 | 6 | 3ad2antl1 1161 |
. 2
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8 | zcn 9287 |
. . . . . . . 8
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9 | zcn 9287 |
. . . . . . . 8
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10 | zcn 9287 |
. . . . . . . 8
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11 | mulass 7971 |
. . . . . . . 8
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12 | 8, 9, 10, 11 | syl3an 1291 |
. . . . . . 7
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13 | 12 | 3coml 1212 |
. . . . . 6
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14 | 13 | 3expa 1205 |
. . . . 5
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15 | 14 | 3adantl3 1157 |
. . . 4
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16 | 15 | eqeq1d 2198 |
. . 3
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17 | 16 | biimprd 158 |
. 2
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18 | 3, 4, 7, 17 | dvds1lem 11840 |
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-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-setind 4554 ax-cnex 7931 ax-resscn 7932 ax-1cn 7933 ax-1re 7934 ax-icn 7935 ax-addcl 7936 ax-addrcl 7937 ax-mulcl 7938 ax-mulrcl 7939 ax-addcom 7940 ax-mulcom 7941 ax-addass 7942 ax-mulass 7943 ax-distr 7944 ax-i2m1 7945 ax-1rid 7947 ax-0id 7948 ax-rnegex 7949 ax-cnre 7951 |
This theorem depends on definitions: df-bi 117 df-3or 981 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-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5851 df-ov 5898 df-oprab 5899 df-mpo 5900 df-sub 8159 df-neg 8160 df-inn 8949 df-n0 9206 df-z 9283 df-dvds 11826 |
This theorem is referenced by: (None) |
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