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Mirrors > Home > ILE Home > Th. List > subdi | Unicode version |
Description: Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 18-Nov-2004.) |
Ref | Expression |
---|---|
subdi |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 962 |
. . . . 5
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2 | simp3 964 |
. . . . 5
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3 | subcl 7878 |
. . . . . 6
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4 | 3 | 3adant1 980 |
. . . . 5
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5 | 1, 2, 4 | adddid 7708 |
. . . 4
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6 | pncan3 7887 |
. . . . . . 7
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7 | 6 | ancoms 266 |
. . . . . 6
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8 | 7 | 3adant1 980 |
. . . . 5
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9 | 8 | oveq2d 5742 |
. . . 4
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10 | 5, 9 | eqtr3d 2147 |
. . 3
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11 | mulcl 7665 |
. . . . 5
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12 | 11 | 3adant3 982 |
. . . 4
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13 | mulcl 7665 |
. . . . 5
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14 | 13 | 3adant2 981 |
. . . 4
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15 | mulcl 7665 |
. . . . . 6
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16 | 3, 15 | sylan2 282 |
. . . . 5
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17 | 16 | 3impb 1158 |
. . . 4
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18 | 12, 14, 17 | subaddd 8008 |
. . 3
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19 | 10, 18 | mpbird 166 |
. 2
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20 | 19 | eqcomd 2118 |
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 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-setind 4410 ax-resscn 7631 ax-1cn 7632 ax-icn 7634 ax-addcl 7635 ax-addrcl 7636 ax-mulcl 7637 ax-addcom 7639 ax-addass 7641 ax-distr 7643 ax-i2m1 7644 ax-0id 7647 ax-rnegex 7648 ax-cnre 7650 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-ral 2393 df-rex 2394 df-reu 2395 df-rab 2397 df-v 2657 df-sbc 2877 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-br 3894 df-opab 3948 df-id 4173 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-iota 5044 df-fun 5081 df-fv 5087 df-riota 5682 df-ov 5729 df-oprab 5730 df-mpo 5731 df-sub 7852 |
This theorem is referenced by: subdir 8061 subdii 8082 subdid 8089 expubnd 10237 subsq 10286 cos01bnd 11310 modmulconst 11367 odd2np1 11412 omoe 11435 omeo 11437 phiprmpw 11737 |
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