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Mirrors > Home > ILE Home > Th. List > dvds2sub | Unicode version |
Description: If an integer divides each of two other integers, it divides their difference. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
dvds2sub |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3simpa 996 |
. 2
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2 | 3simpb 997 |
. 2
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3 | zsubcl 9324 |
. . . 4
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4 | 3 | anim2i 342 |
. . 3
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5 | 4 | 3impb 1201 |
. 2
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6 | zsubcl 9324 |
. . 3
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7 | 6 | adantl 277 |
. 2
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8 | zcn 9288 |
. . . . . . . 8
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9 | zcn 9288 |
. . . . . . . 8
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10 | zcn 9288 |
. . . . . . . 8
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11 | subdir 8373 |
. . . . . . . 8
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12 | 8, 9, 10, 11 | syl3an 1291 |
. . . . . . 7
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13 | 12 | 3comr 1213 |
. . . . . 6
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14 | 13 | 3expb 1206 |
. . . . 5
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15 | oveq12 5905 |
. . . . 5
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16 | 14, 15 | sylan9eq 2242 |
. . . 4
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17 | 16 | ex 115 |
. . 3
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18 | 17 | 3ad2antl1 1161 |
. 2
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19 | 1, 2, 5, 7, 18 | dvds2lem 11842 |
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-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7932 ax-resscn 7933 ax-1cn 7934 ax-1re 7935 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-addcom 7941 ax-mulcom 7942 ax-addass 7943 ax-distr 7945 ax-i2m1 7946 ax-0lt1 7947 ax-0id 7949 ax-rnegex 7950 ax-cnre 7952 ax-pre-ltirr 7953 ax-pre-ltwlin 7954 ax-pre-lttrn 7955 ax-pre-ltadd 7957 |
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-nel 2456 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 5852 df-ov 5899 df-oprab 5900 df-mpo 5901 df-pnf 8024 df-mnf 8025 df-xr 8026 df-ltxr 8027 df-le 8028 df-sub 8160 df-neg 8161 df-inn 8950 df-n0 9207 df-z 9284 df-dvds 11827 |
This theorem is referenced by: dvds2subd 11866 dvdssub2 11874 difsqpwdvds 12370 |
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